3685:
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58:
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314:
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1276:
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19256:
9190:{\displaystyle {\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}}{(2n+1)!}}x^{2n+1}\\&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}
9203:
8587:{\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}}
21795:{\displaystyle {\begin{aligned}{\frac {d\cos x}{dx}}&={\frac {d}{dx}}\sin(\pi /2-x)=-\cos(\pi /2-x)=-\sin x\,,\\{\frac {d\csc x}{dx}}&={\frac {d}{dx}}\sec(\pi /2-x)=-\sec(\pi /2-x)\tan(\pi /2-x)=-\csc x\cot x\,,\\{\frac {d\cot x}{dx}}&={\frac {d}{dx}}\tan(\pi /2-x)=-\sec ^{2}(\pi /2-x)=-\csc ^{2}x\,.\end{aligned}}}
23805:
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17473:
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9844:
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10638:
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475:
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for
11509:
7371:
that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
9831:{\displaystyle {\begin{aligned}\sec x&=\sum _{n=0}^{\infty }{\frac {U_{2n}}{(2n)!}}x^{2n}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\\&=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}
15535:
23766:
4889:
19265:
18996:
15198:
15678:
26230:. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications.
2957:
2709:
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13125:
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It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in
16942:
9501:{\displaystyle {\begin{aligned}\csc x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\&=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}
12221:
17339:
24112:
18978:
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to
12023:
13237:
16097:, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of
19883:{\displaystyle {\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}
8620:
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
23305:
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
18110:
17334:
11766:
13329:
of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group
14275:
19997:
12964:
7557:
5808:
22995:
10111:{\displaystyle {\begin{aligned}\cot x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\&=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}
2489:
18694:{\displaystyle {\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}}
3069:
2821:
17821:
3366:
3176:
2298:
11885:
8101:
24206:. With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant.
11312:
5725:
18059:
11043:{\displaystyle \tan x={\cfrac {x}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-\ddots }}}}}}}}={\cfrac {1}{{\cfrac {1}{x}}-{\cfrac {1}{{\cfrac {3}{x}}-{\cfrac {1}{{\cfrac {5}{x}}-{\cfrac {1}{{\cfrac {7}{x}}-\ddots }}}}}}}}}
11617:
25253:
Whittaker, E. T., & Watson, G. N. (1920). A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. University
23423:
10391:
10136:
23831:, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of
15380:
11174:
14038:
18076:. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval , see
15375:
15890:
15042:
1796:
1697:
1596:
1497:
15540:
11323:
5950:
23088:
13940:
1984:
1891:
5876:
24218:(853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. The trigonometric functions were later studied by mathematicians including
2573:
8186:
4882:
By applying the
Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
3449:
3259:
2381:
7757:
7657:
7357:
22799:, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "
22776:
20002:
19545:
19270:
19001:
17063:
8793:
5639:
5047:{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.}
19520:{\displaystyle {\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}}
19251:{\displaystyle {\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}}
16745:
5311:
5250:
23627:
16641:
16490:
16121:
is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding
Hyperbolic functions highlights the relationships between the two.
4031:
14962:
23969:
16740:
2851:
2603:
9849:
9519:
9208:
23983:
22008:
15953:
12984:
12851:
23555:
22299:
23286:
5327:. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that
22601:
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5480:
15809:
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7034:
6990:
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18115:
17239:
14606:
4741:
4446:
20249:
14528:
4370:
4218:
4140:
21174:
14854:
14795:
13407:
4318:
21292:
20952:
20826:
12086:
25528:
The
Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, pp. 529–530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.
13574:
882:
19945:
14133:
14736:
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24116:
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a
22894:
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18967:
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24868:
21251:
18364:{\displaystyle {\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}}
18088:
of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
12609:
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6603:
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1143:
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17468:{\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .}
14427:
14339:
14128:
12767:
7095:
6772:
6633:
6487:
20177:{\displaystyle {\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}}
14393:
12534:
1028:
13533:
4257:
4058:
1105:
23808:
Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (
22112:
21958:
21104:
6858:
6543:
3897:
3864:
22145:
20765:
12846:
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7065:
6457:
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6125:
6092:
4588:
3944:
2054:
995:
792:
20891:
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17234:
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698:
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represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
953:
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20361:
18842:
18809:
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6217:
2963:
2715:
659:
612:
571:
455:
1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as
21840:, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
20493:
15281:
5116:
3289:
3099:
2221:
22669:
22504:
22367:
22213:
22076:
21922:
20585:
16344:
16309:
16272:
16237:
16200:
16165:
11772:
7974:
530:
25890:
Lehrbuch der ebenen und sphärischen
Trigonometrie. Zum Gebrauch bei Selbstunterricht und in Schulen, besonders als Vorbereitung auf Geodäsie und sphärische Astronomie
20552:
19965:
17197:
17095:
13718:
13660:
7244:
3555:
for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
23004:
21200:
21130:
20981:
20855:
20726:
20614:
20522:
20457:
20430:
17941:
17703:
15936:
15706:
13856:
13611:
12078:
12052:
26338:
Gal, Shmuel and
Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991).
17496:
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14877:
13823:
13686:
11207:
8698:
7887:
5371:
5348:
5186:
5159:
25312:
Lambert, Johann
Heinrich (2004) , "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", in Berggren, Lennart;
20326:
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13960:
7907:
5650:
5423:
5395:
1176:
909:
24134:
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The
17595:
17121:
13800:
4166:
4088:
26458:
17655:
10633:{\displaystyle \cos x={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}}
10378:{\displaystyle \sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}}
19989:
16115:
13851:
13631:
13486:
13327:
11515:
7165:
7144:
6742:
6280:
6259:
6238:
6189:
23326:
26429:
24625:
in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.
14882:
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First,
22806:
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of
24215:
3540:, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions
26585:
16947:
202:
11083:
11504:{\displaystyle \pi \csc \pi x=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{x+n}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}},}
22853:
denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
16538:
16387:
15820:
1710:
1611:
1510:
1411:
14885:
13246:
can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity
5882:
25632:
23889:
16646:
1904:
1811:
5814:
25042:
25166:
2495:
26578:
25046:
3372:
3182:
2304:
25718:
23206:
15530:{\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )}
7662:
25794:
7565:
7249:
26074:
24810:
24667:
24357:
23761:{\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.}
5577:
198:
22719:
15193:{\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}}
25212:
15673:{\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).}
5975:
of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by
5255:
5194:
25415:
25376:
25592:
25558:
24795:
24785:
13365:
3975:
2952:{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}
2704:{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}
208:
25487:"Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt"
25131:
4536:, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius
23129:
7379:
Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically.
3872:
the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
26467:
26447:
26434:
25825:
23827:
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe
21964:
16083:{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}}
13120:{\displaystyle {\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}}
5405:
already return to their original position, so that the tangent function and the cotangent function have a fundamental period of
23478:
7375:
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
300:
25489:[Representation of an analytical function of a complex variable, whose absolute value lies between two given limits].
22255:
21832:. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus
6027:
For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are
26702:
26235:
26007:
25931:
25684:
25586:
25552:
25401:
25370:
25198:
25141:
16937:{\displaystyle \cos ^{2}(z+\pi )=\cos ^{2}(z),\quad \sin ^{2}(z+\pi )=\sin(z),\quad \cos(z+\pi )\sin(z+\pi )=\cos(z)\sin(z).}
14967:
5976:
26661:
25204:
24935:
24159:
23842:. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light
22554:
5485:
5431:
25973:
25916:
Trigonometrie für
Maschinenbauer und Elektrotechniker - Ein Lehr- und Aufgabenbuch für den Unterricht und zum Selbststudium
25407:
24929:
24355:
of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his
15733:
13723:
5563:, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.
4792:
26388:
26372:
26191:
23567:
15203:
6996:
6952:
6388:
6344:
4837:
4746:
4640:
4451:
488:
in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with
14533:
12216:{\displaystyle \cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}
4704:
4409:
21321:
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
20193:
14458:
13305:
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of
5982:
For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the
5086:
4323:
4171:
4093:
26114:
25513:
21135:
18077:
14800:
14741:
7388:
By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.
4274:
26455:
21256:
20897:
20771:
26402:
26381:
26354:
26316:
26295:
25856:
25615:
25333:
25018:
24107:{\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
13538:
6045:
797:
19903:
12018:{\displaystyle \sin z=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}
26687:
25676:
25325:
24425:
18073:
14663:
13243:
13232:{\displaystyle \cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).}
2176:
122:
26533:
14050:
5057:
3570:
subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete
26625:
26143:
24805:
21811:
20255:
12625:
500:. Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression
485:
425:
100:
22675:
22609:
22510:
22444:
22307:
26717:
21024:
20654:
13333:
7760:
5967:
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
2059:
998:
22373:
22219:
17126:
12418:
4499:
467:, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
26526:
26516:
21297:
18914:
18850:
18719:
18072:
interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see
17828:
17504:
625:
330:
24844:
24658:", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word
21205:
17329:{\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .}
12539:
12345:
11761:{\displaystyle \pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},}
11054:
6806:
6697:
6667:
6579:
6549:
6050:
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
5964:
of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.
1110:
26635:
24634:
16117:
becomes larger (since the color white represents infinity), and the fact that the functions contain simple
7367:
703:
26442:
25642:
23108:, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
12776:
7919:
435:
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for
26707:
26521:
26131:
8635:
6891:
3812:
3684:
1033:
25948:
21836:
from this interval to its image by the function. The common choice for this interval, called the set of
13412:
13249:
8628:, that is functions that are holomorphic in the whole complex plane, except some isolated points called
6288:
4593:
25958:
25820:
25158:
24815:
24211:
14398:
14312:
14270:{\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}}
14101:
12712:
11066:
7073:
6750:
6611:
6465:
5972:
476:
cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular
293:
244:
193:
127:
25581:(Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77ff.
14347:
13833:
Another way to define the trigonometric functions in analysis is using integration. For a real number
12959:{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x.\end{aligned}}}
12484:
1004:
25894:
25888:
24790:
24188:
19894:
13495:
11317:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
8624:
Being defined as fractions of entire functions, the other trigonometric functions may be extended to
7552:{\displaystyle {\frac {d}{dx}}\sin x=\cos x,\ {\frac {d}{dx}}\cos x=-\sin x,\ \sin(0)=0,\ \cos(0)=1.}
5803:{\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}
4235:
4036:
3777:
1993:
1075:
17:
26538:
26086:
26003:
25952:
25722:
22990:{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},}
22082:
21928:
21077:
6836:
6521:
3878:
2484:{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}
26276:
25830:
24917:
24800:
20732:
12269:
12237:
8112:
7040:
6432:
6131:
6098:
6065:
4547:
3920:
3842:
3461:
In geometric applications, the argument of a trigonometric function is generally the measure of an
3064:{\displaystyle \cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}}
2816:{\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}
2027:
958:
756:
106:
25790:
25486:
24231:
22118:
20861:
17816:{\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
17600:
17202:
15255:
14637:
14611:
14432:
12301:
3949:
3361:{\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}
3171:{\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}
2293:{\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}
664:
24352:
24269:
24253:
24129:
23832:
22153:
22016:
21862:
21315:
20987:
20291:
14280:
11880:{\displaystyle \pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.}
11074:
8720:
8606:
8096:{\displaystyle {\frac {d}{dx}}\tan x={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}=1+\tan ^{2}x\,,}
7843:
7101:
6924:
6864:
6778:
6639:
6493:
6316:
922:
72:
67:
25703:
25061:
24706:(the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from
22409:
20620:
18814:
18781:
7768:
6195:
631:
576:
535:
25668:
24707:
24243:
24151:
23828:
20463:
20367:
18069:
11070:
5994:
allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.
5095:
4376:
of these points give the values of all trigonometric functions for any arbitrary real value of
3474:
916:
409:
25412:
25356:
25209:
24279:
in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.
22642:
22477:
22340:
22186:
22049:
21895:
20558:
16322:
16287:
16250:
16215:
16178:
16143:
11307:{\displaystyle \pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}.}
503:
424:
functions, which are less used. Each of these six trigonometric functions has a corresponding
26697:
26273:
Complex
Analysis: an introduction to the theory of analytic functions of one complex variable
26225:
25954:
Erläuterungen und
Beispiele für den Gebrauch der vierstelligen Tafeln zum praktischen Rechnen
25574:
25542:
24921:
24907:
24885:
24257:
20528:
19950:
19531:
17170:
17068:
13694:
13636:
8712:
8598:
7401:
7217:
6028:
6017:
5720:{\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}}
4373:
3695:
3509:
3485:
1367:
460:
362:
326:
286:
25949:"§ 14. Erläuterungen u. Beispiele zu T. 13: lg sin X; lg cos X und T. 14: lg tg x; lg ctg X"
23122:
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the
21179:
21109:
20960:
20834:
20705:
20593:
20501:
20436:
20409:
18054:{\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
17918:
17680:
15906:
15683:
13579:
12057:
12031:
3646:. In this way, the degree symbol can be regarded as a mathematical constant such that 1° =
26630:
26253:
25263:
Bartle, R. G., & Sherbert, D. R. (2000). Introduction to real analysis (3rd ed). Wiley.
24748:
24366:
24261:
24227:
23095:
22819:
21829:
20332:
18084:, one may use directly the differential equations, in a way that is similar to that of the
17478:
16495:
16364:
14859:
13805:
13665:
12336:
11198:
8680:
8625:
7869:
5568:
5353:
5330:
5168:
5141:
4631:
3505:
456:
440:
429:
213:
132:
77:
25745:"The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates"
25610:
Boyer, Carl B. (1991). A History of
Mathematics (Second ed.). John Wiley & Sons, Inc.
21828:, one can define an inverse function, and this defines inverse trigonometric functions as
20302:
17898:
17660:
16518:
13945:
11612:{\displaystyle \pi ^{2}\csc ^{2}\pi x=\sum _{n=-\infty }^{\infty }{\frac {1}{(x+n)^{2}}},}
7892:
5408:
5380:
1156:
889:
8:
26570:
26304:
26106:
25884:
25816:
25721:. School of Mathematics and Statistics University of St Andrews, Scotland. Archived from
25637:
25313:
24768:
24195:
23418:{\displaystyle {\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}}
23299:
23123:
19530:
When the two angles are equal, the sum formulas reduce to simpler equations known as the
18710:
17574:
17100:
15939:
15724:
15717:
13779:
7196:
7184:
4145:
4067:
489:
378:
259:
175:
38:
26360:
Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions,"
23298:. This theorem can be proved by dividing the triangle into two right ones and using the
17637:
3600:
rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g.,
365:
to ratios of two side lengths. They are widely used in all sciences that are related to
26327:
25845:
25772:
25764:
25089:
25036:
24370:
24362:
24336:
24276:
23793:
23469:
21817:
19974:
18984:
16100:
13836:
13616:
13471:
13312:
12332:
12231:
10127:
7150:
7129:
6727:
6265:
6244:
6223:
6174:
6032:
3772:
3516:), without reference to any geometric notions. The other four trigonometric functions (
1999:
In a right-angled triangle, the sum of the two acute angles is a right angle, that is,
57:
25493:(in German). Vol. 1. Berlin: Mayer & Müller (published 1894). pp. 51–66.
24968:
24654:, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a
7211:
26484:
26418:
26414:
26398:
26377:
26350:
26312:
26291:
26257:
26241:
26231:
26217:
26110:
26079:
26066:
25969:
25927:
25852:
25776:
25680:
25611:
25582:
25548:
25509:
25397:
25366:
25329:
25194:
25137:
25136:. Spectrum series. Washington, DC: Mathematical Association of America. p. 119.
25112:
25093:
25081:
25024:
25014:
24925:
24348:
24325:
24207:
24203:
23839:
23435:
22807:
19968:
18380:
16358:
13306:
5374:
5162:
3488:, the trigonometric functions are generally regarded more abstractly as functions of
1150:
386:
264:
168:
31:
25352:
18409:
18390:. This is the smallest period, except for the tangent and the cotangent, which have
11169:{\displaystyle \pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.}
26283:
26053:
25961:
25919:
25756:
25482:
25449:
25317:
25073:
24916:. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.).
24712:
24401:
24172:
24135:
23820:
21821:
14041:
14033:{\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}}
8758:
8601:
of these series is infinite. Therefore, the sine and the cosine can be extended to
6002:
3914:
1279:
Plot of the six trigonometric functions, the unit circle, and a line for the angle
912:
617:
390:
25997:
25133:
Twenty years before the blackboard: the lessons and humor of a mathematics teacher
15680:
So the sine and cosine functions are related by translation over a quarter period
15370:{\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).}
14395:. The definition can be extended to all real numbers by first observing that, as
3596:, etc. refer to the value of the trigonometric functions evaluated at an angle of
26712:
26618:
26462:
26249:
26227:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
25419:
25391:
25362:
25216:
25188:
24913:
Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis
24911:
24687:
24681:
24247:
24183:
23865:
23781:
23315:
21837:
16118:
16094:
15885:{\displaystyle 0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.}
11180:
8629:
8602:
5998:
4061:
3816:
1791:{\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}}
1692:{\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}
1591:{\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}}
1492:{\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}
1370:
to each other. This means that the ratio of any two side lengths depends only on
374:
163:
92:
88:
24219:
20286:
in the following table can be verified by differentiating them. The number
20278:
of trigonometric functions result from those of sine and cosine by applying the
8175:
The basic trigonometric functions can be defined by the power series expansions
5945:{\displaystyle \sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1}
794:
This differs from the (historically later) general functional notation in which
26692:
26480:
26323:
26187:
25538:
25062:"Defining Exponential and Trigonometric Functions Using Differential Equations"
24671:
24400:
A few functions were common historically, but are now seldom used, such as the
24344:
24265:
24187:), via translation from Sanskrit to Arabic and then from Arabic to Latin. (See
23978:
23861:
23117:
23083:{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}
20283:
20263:
18085:
17657:. Again, owing to the periodicity, the zeros are all the integer multiples of
15901:
13935:{\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t}
12973:
in sine and cosine, one can express them in terms of the exponential function:
11896:
8741:
5987:
3900:
3668:(theta) can be constructed geometrically in terms of a unit circle centered at
3563:
3552:
3493:
3470:
2190:
1979:{\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}
1886:{\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}
621:
318:
274:
269:
158:
25923:
25760:
24223:
13662:, we get the measure in radians, and the usual trigonometric functions. When
5871:{\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}}
2160:
Graph of sine function versus angle. Angles from the top panel are identified.
313:
26681:
26557:
26346:
26168:
25664:
25348:
25085:
25077:
24890:
Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis
24752:
24662:, meaning "pocket" or "fold" in the twelfth-century translations of works by
24302:
24291:
24239:
24117:
23804:
23104:
20279:
18097:
12970:
12619:
8716:
8708:
8614:
7913:
5991:
5642:
4229:
3473:, in which a right angle is 90° and a complete turn is 360° (particularly in
464:
463:. This allows extending the domain of sine and cosine functions to the whole
358:
254:
26495:
17705:, all having the same residue. The poles correspond to vertical asymptotes
3688:
In this illustration, the six trigonometric functions of an arbitrary angle
2568:{\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}
484:
of an arbitrary circle, and later to indicate ratios of lengths, but as the
26544:
26268:
26221:
25028:
24199:
24169:
22862:
18101:
8775:
6021:
6013:
5961:
5555:
4221:
3835:
allow for the definition of the trigonometric functions for angles between
3571:
3466:
1378:, which are the trigonometric functions. In the following definitions, the
481:
477:
396:
The trigonometric functions most widely used in modern mathematics are the
389:, and as such are also widely used for studying periodic phenomena through
153:
49:
37:"Logarithmic cosine" redirects here. For the Clausen-related function, see
25965:
25547:(Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82.
24700:
24692:
24214:
discovered the cotangent, and produced tables of tangents and cotangents.
7382:
By a power series, which is particularly well-suited to complex variables.
4630:
on the unit circle, this definition of cosine and sine also satisfies the
3444:{\displaystyle \csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}}
3254:{\displaystyle \sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}}
2376:{\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}
26028:
25999:
A reconstruction of Peters's table of 7-place logarithms (volume 2, 1940)
24893:
24177:
23974:
23797:
13773:
8610:
7752:{\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x}
7362:
5953:
5560:
4698:
The other trigonometric functions can be found along the unit circle as
3820:
3699:
3567:
3489:
448:
436:
338:
322:
239:
137:
30:"Logarithmic sine" redirects here. For the Clausen-related function, see
26541:
Visualization of the unit circle, trigonometric and hyperbolic functions
25768:
25744:
25008:
24210:(c. 780–850) produced tables of sines, cosines and tangents. Circa 830,
23838:
Trigonometric functions also prove to be useful in the study of general
17236:
in the same domain. Because of the periodicity, the zeros of sine are
7652:{\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x}
1306:
represent the length of the line segment from the origin to that point.
26061:
24820:
24663:
24421:
24139:
23776:
20275:
8707:
Recurrences relations may also be computed for the coefficients of the
7352:{\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}
5138:
are the same for two angles whose difference is an integer multiple of
4541:
3548:
1379:
1186:
370:
249:
229:
26139:
26097:
See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.
24650:
24361:(1748). His method was to show that the sine and cosine functions are
23815:) have additional nodes. The oscillation seen about the sawtooth when
16515:
as their period. The functions sine and cosine also have semiperiods
7180:
1275:
26565:
26552:
26500:
24622:
24413:
24409:
24235:
22800:
22771:{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}
21833:
21825:
17058:{\displaystyle \cos(x+\pi /2)=-\sin(x),\quad \sin(x+\pi /2)=\cos(x).}
8106:
so the tangent function satisfies the ordinary differential equation
5983:
5971:
For an angle which, measured in degrees, is a multiple of three, the
5634:{\displaystyle \sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0}
444:
26261:
17475:
All of the zeros are simple zeros, and each function has derivative
14098:, the trigonometric functions are defined by inverting the relation
620:
appearing as a superscript after the symbol of the function denotes
26656:
26561:
26548:
25673:
Mathematics Across Cultures: The History of Non-western Mathematics
24417:
23788:
23461:
20259:
18081:
6006:
5306:{\displaystyle \quad \cos \theta =\cos \left(\theta +2k\pi \right)}
5245:{\displaystyle \sin \theta =\sin \left(\theta +2k\pi \right)\quad }
3547:. Moreover, these definitions result in simple expressions for the
3481:
366:
184:
24674:. The choice was based on a misreading of the Arabic written form
24143:
22867:
The law of sines states that for an arbitrary triangle with sides
16636:{\displaystyle \cos(z+\pi )=-\cos(z),\quad \sin(z+\pi )=-\sin(z).}
16485:{\displaystyle \cos(z+2\pi )=\cos(z),\quad \sin(z+2\pi )=\sin(z).}
13465:, and this isomorphism is unique up to taking complex conjugates.
26651:
24405:
24155:
24147:
21824:. However, on each interval on which a trigonometric function is
18980:
8711:
of the other trigonometric functions. These series have a finite
7207:
Animation for the approximation of cosine via Taylor polynomials.
5321:
4026:{\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).}
382:
234:
26488:
26245:
25116:
14957:{\displaystyle \arctan s+\arctan t=\arctan({\frac {s+t}{1-st}})}
12028:
This may be obtained from the partial fraction decomposition of
3676:
2104:
1190:
In this right triangle, denoting the measure of angle BAC as A:
26422:
24621:
Historically, trigonometric functions were often combined with
23964:{\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
16735:{\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z)}
12226:
3866:
3824:
3658:
3559:
2184:
2020:
452:
401:
15716:
One can also define the trigonometric functions using various
11077:
of the cotangent function and the reciprocal functions match:
7866:
are periodic, having the same period. Writing this period as
26666:
26093:
International Symposium on History of Machines and Mechanisms
25918:(in German) (6 ed.). Winterthur, Switzerland: Springer.
25266:
24710:
24646:
24194:
All six trigonometric functions in current use were known in
23860:
can be expressed as a sum of sine waves or cosine waves in a
15723:
For example, the sine and the cosine form the unique pair of
14856:
define the sine and cosine as periodic functions with period
13633:
is used as the base for measuring angles. For example, when
8605:(also called "sine" and "cosine"), which are (by definition)
7400:
Sine and cosine can be defined as the unique solution to the
7203:
7191:
3462:
439:. To extend the sine and cosine functions to functions whose
26393:
13688:, we get the sine and cosine of angles measured in degrees.
5118:
does not change the position or size of a shape, the points
3562:(rad) are employed, the angle is given as the length of the
26343:
The Crest of the Peacock: Non-European Roots of Mathematics
25544:
Partial differential equations for scientists and engineers
24655:
23843:
22003:{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}
17895:
has a simple pole of residue 1 at the integer multiples of
16313:
16278:
16241:
16206:
16169:
16134:
7199:
of degree 7 (pink) for a full cycle centered on the origin.
5377:
of these functions). However, after a rotation by an angle
3618:
for the trigonometric functions satisfies the relationship
397:
26600:
20266:
of trigonometric functions to that of rational fractions.
13802:. This fact can, in turn, be used to define the constant
26376:, Princeton Univ. Press. (1998). Reprint edition (2002):
23550:{\displaystyle r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}}
22294:{\textstyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}
15711:
13942:
where this defines this inverse tangent function. Also,
5350:
is the smallest value for which they are periodic (i.e.,
3680:
Sine function on unit circle (top) and its graph (bottom)
2165:
Summary of relationships between trigonometric functions
1362:
is given, then any right triangles that have an angle of
26188:"A reconstruction of Gunter's Canon triangulorum (1620)"
23281:{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
21816:
The trigonometric functions are periodic, and hence not
7195:
The sine function (blue) is closely approximated by its
26334:. Vol. 2. Open Court. pp. 142–179 (¶511–537).
25506:
Functional Equations and Inequalities with Applications
24365:
formed from the even and odd terms respectively of the
22596:{\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}
18394:
as smallest period. This means that, for every integer
15032:{\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)}
12405:
This formula is commonly considered for real values of
11001:
10989:
10981:
10969:
10950:
10938:
10930:
10918:
10899:
10887:
10879:
10867:
10848:
10836:
10828:
10816:
10776:
10757:
10743:
10724:
10710:
10691:
10677:
10665:
10577:
10549:
10516:
10488:
10455:
10436:
10422:
10410:
10322:
10294:
10261:
10233:
10200:
10181:
10167:
10155:
7395:
3504:
can be defined for all complex numbers in terms of the
22841:
denote the three (interior) angles of a triangle, and
22722:
22557:
22412:
22258:
22121:
21967:
20370:
19920:
15938:
can be expressed in terms of real sines, cosines, and
11895:
The following infinite product for the sine is due to
11837:
11718:
11004:
10992:
10984:
10972:
10953:
10941:
10933:
10921:
10902:
10890:
10882:
10870:
10851:
10839:
10831:
10819:
10779:
10760:
10746:
10727:
10713:
10694:
10680:
10668:
10580:
10552:
10519:
10491:
10458:
10439:
10425:
10413:
10325:
10297:
10264:
10236:
10203:
10184:
10170:
10158:
8638:
7665:
7568:
5529:{\displaystyle \quad \cot \theta =\cot(\theta +k\pi )}
5475:{\displaystyle \tan \theta =\tan(\theta +k\pi )\quad }
3845:
3832:
24847:
24751:" (in "cosine", "cotangent", "cosecant") is found in
24351:, Leibnitz result established that they are actually
23986:
23892:
23849:
Under rather general conditions, a periodic function
23784:, a figure formed with a trigonometry-based function.
23717:
23679:
23641:
23630:
23570:
23481:
23329:
23209:
23132:
23007:
22897:
22678:
22645:
22612:
22513:
22480:
22447:
22376:
22343:
22310:
22222:
22189:
22156:
22085:
22052:
22019:
21931:
21898:
21865:
21330:
21300:
21259:
21208:
21182:
21138:
21112:
21080:
21027:
20990:
20963:
20900:
20864:
20837:
20774:
20735:
20708:
20657:
20623:
20596:
20561:
20531:
20504:
20466:
20439:
20412:
20335:
20305:
20196:
20000:
19977:
19953:
19906:
19543:
19268:
18999:
18917:
18853:
18817:
18784:
18722:
18407:
18113:
17949:
17921:
17901:
17831:
17711:
17683:
17663:
17640:
17603:
17577:
17507:
17481:
17342:
17242:
17205:
17173:
17129:
17103:
17071:
16950:
16748:
16649:
16541:
16521:
16498:
16390:
16367:
16325:
16290:
16253:
16218:
16181:
16146:
16103:
15951:
15909:
15823:
15804:{\displaystyle \cos(x-y)=\cos x\cos y+\sin x\sin y\,}
15736:
15686:
15543:
15383:
15284:
15258:
15206:
15045:
14970:
14888:
14862:
14803:
14744:
14666:
14640:
14614:
14536:
14461:
14435:
14401:
14350:
14315:
14283:
14136:
14104:
14053:
13968:
13948:
13859:
13839:
13808:
13782:
13726:
13697:
13668:
13639:
13619:
13582:
13541:
13498:
13474:
13415:
13368:
13336:
13315:
13252:
13143:
12982:
12849:
12779:
12715:
12628:
12542:
12487:
12421:
12348:
12304:
12272:
12240:
12089:
12060:
12034:
11908:
11775:
11626:
11518:
11326:
11210:
11086:
10649:
10394:
10139:
9847:
9517:
9206:
8791:
8683:
8184:
8115:
7977:
7922:
7895:
7872:
7846:
7771:
7413:
7252:
7220:
7153:
7132:
7104:
7076:
7043:
6999:
6955:
6927:
6894:
6867:
6839:
6809:
6781:
6753:
6730:
6700:
6670:
6642:
6614:
6582:
6552:
6524:
6496:
6468:
6435:
6391:
6347:
6319:
6291:
6268:
6247:
6226:
6198:
6177:
6134:
6101:
6068:
5885:
5817:
5734:
5653:
5580:
5488:
5434:
5411:
5383:
5356:
5333:
5258:
5197:
5171:
5144:
5098:
4892:
4840:
4827:{\displaystyle \csc \theta \ =y_{\mathrm {D} }\quad }
4795:
4749:
4707:
4643:
4596:
4550:
4502:
4454:
4412:
4326:
4277:
4238:
4174:
4148:
4096:
4070:
4039:
3978:
3952:
3923:
3881:
3614:, etc.). Using this standard notation, the argument
3545:
the argument is regarded as an angle given in radians
3375:
3292:
3185:
3102:
2966:
2854:
2718:
2606:
2498:
2414:
2307:
2224:
2062:
2030:
1907:
1814:
1713:
1614:
1513:
1414:
1159:
1113:
1078:
1036:
1007:
961:
925:
892:
800:
759:
706:
667:
634:
579:
538:
506:
26311:, John Wiley & Sons, Inc., 2nd edition. (1991).
26149:
See Maor (1998), chapter 3, regarding the etymology.
26088:
A Note on the History of the Trigonometric Functions
26045:
25006:
24640:
23611:{\displaystyle \cot {\frac {A}{2}}={\frac {s-a}{r}}}
15245:{\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}}
12054:
given above, which is the logarithmic derivative of
7759:, so both sine and cosine are solutions of the same
7029:{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
6985:{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
6421:{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
6377:{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
4872:{\displaystyle \quad \sec \theta =x_{\mathrm {E} }.}
4781:{\displaystyle \quad \cot \theta =x_{\mathrm {C} },}
4688:{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}
4486:{\displaystyle \quad \sin \theta =y_{\mathrm {A} }.}
3972:). This ray intersects the unit circle at the point
1382:
is the length of the side opposite the right angle,
1181:
26475:Protter, Murray H.; Morrey, Charles B. Jr. (1970),
26411:
Logarithmic and Trigonometric Tables to Five Places
25663:
25662:Jacques Sesiano, "Islamic mathematics", p. 157, in
18709:The Pythagorean identity, is the expression of the
14601:{\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1}
4736:{\displaystyle \tan \theta =y_{\mathrm {B} }\quad }
4441:{\displaystyle \cos \theta =x_{\mathrm {A} }\quad }
3776:Signs of trigonometric functions in each quadrant.
25844:
25293:Stanley, Enumerative Combinatorics, Vol I., p. 149
24862:
24305:made the first published use of the abbreviations
24290:were first introduced by the Danish mathematician
24106:
23963:
23760:
23610:
23549:
23417:
23280:
23195:
23082:
22989:
22770:
22707:
22663:
22630:
22595:
22542:
22498:
22465:
22430:
22397:
22361:
22328:
22293:
22243:
22207:
22174:
22139:
22106:
22070:
22037:
22002:
21952:
21916:
21883:
21794:
21306:
21286:
21245:
21194:
21168:
21124:
21098:
21061:
21012:
20975:
20946:
20885:
20849:
20820:
20759:
20720:
20691:
20642:
20608:
20579:
20546:
20516:
20487:
20451:
20424:
20395:
20355:
20320:
20269:
20244:{\displaystyle d\theta ={\frac {2}{1+t^{2}}}\,dt,}
20243:
20176:
19983:
19959:
19939:
19882:
19519:
19250:
18961:
18897:
18836:
18803:
18766:
18693:
18363:
18080:). For non-geometrical proofs using only tools of
18053:
17935:
17907:
17887:
17815:
17697:
17669:
17649:
17626:
17589:
17563:
17490:
17467:
17328:
17228:
17191:
17159:
17115:
17089:
17057:
16936:
16734:
16635:
16527:
16507:
16484:
16376:
16338:
16303:
16266:
16231:
16194:
16159:
16109:
16082:
15930:
15884:
15803:
15700:
15672:
15529:
15369:
15270:
15244:
15192:
15031:
14956:
14871:
14848:
14789:
14730:
14652:
14626:
14600:
14523:{\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0}
14522:
14447:
14421:
14387:
14333:
14301:
14269:
14122:
14090:
14032:
13954:
13934:
13845:
13817:
13794:
13764:
13712:
13680:
13654:
13625:
13605:
13568:
13527:
13480:
13457:
13401:
13354:
13321:
13294:
13231:
13119:
12958:
12829:
12761:
12701:
12603:
12528:
12473:
12394:
12320:
12290:
12258:
12215:
12072:
12046:
12017:
11899:, and is of great importance in complex analysis:
11879:
11760:
11611:
11503:
11306:
11168:
11042:
10632:
10377:
10110:
9830:
9500:
9189:
8692:
8669:
8586:
8149:
8095:
7960:
7901:
7881:
7858:
7798:
7751:
7651:
7551:
7351:
7238:
7159:
7138:
7117:
7089:
7059:
7028:
6984:
6940:
6912:
6877:
6852:
6824:
6794:
6766:
6736:
6715:
6685:
6655:
6627:
6597:
6567:
6537:
6509:
6481:
6451:
6420:
6376:
6332:
6304:
6274:
6253:
6232:
6211:
6183:
6152:
6119:
6086:
5944:
5870:
5802:
5719:
5633:
5528:
5474:
5417:
5389:
5365:
5342:
5305:
5244:
5180:
5153:
5110:
5046:
4871:
4826:
4780:
4735:
4687:
4622:
4582:
4528:
4485:
4440:
4365:{\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).}
4364:
4312:
4251:
4213:{\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).}
4212:
4160:
4135:{\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),}
4134:
4082:
4052:
4025:
3964:
3938:
3891:
3858:
3811:The six trigonometric functions can be defined as
3443:
3360:
3253:
3170:
3063:
2951:
2815:
2703:
2567:
2483:
2375:
2292:
2093:
2048:
1978:
1885:
1790:
1691:
1590:
1491:
1170:
1137:
1099:
1064:
1022:
989:
947:
903:
876:
786:
745:
692:
653:
606:
565:
524:
26082:'s 1145 translation of the tables of al-Khwārizmī
26027:The anglicized form is first recorded in 1593 in
25814:
25716:
25626:
25624:
24892:(in German). Vol. 1 (3rd ed.). Berlin:
22824:
21169:{\displaystyle -\operatorname {arsinh} (\cot x),}
14849:{\displaystyle \sin(\theta +\pi )=-\sin(\theta )}
14790:{\displaystyle \cos(\theta +\pi )=-\cos(\theta )}
14130:. Thus we define the trigonometric functions by
13402:{\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.}
7840:One can then prove, as a theorem, that solutions
4313:{\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}
3532:) can be defined as quotients and reciprocals of
26679:
26216:
21287:{\displaystyle \operatorname {arsinh} (\tan x),}
20947:{\displaystyle \ln \left|\sec x+\tan x\right|+C}
20821:{\displaystyle \ln \left|\csc x-\cot x\right|+C}
18983:. One can also produce them algebraically using
18002:
17951:
17764:
17713:
16492:Consequently, the secant and cosecant also have
11106:
11053:The last one was used in the historically first
3662:All of the trigonometric functions of the angle
1374:. Thus these six ratios define six functions of
25193:(9th ed.). Cengage Learning. p. 153.
25129:
23001:is the area of the triangle, or, equivalently,
17943:. The poles correspond to vertical asymptotes
13569:{\displaystyle \mathbb {R} /a\mathbb {Z} \to U}
10121:
6046:Exact trigonometric values § Common angles
5997:For an angle which, expressed in degrees, is a
877:{\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
385:, and many others. They are among the simplest
26282:
25940:
25893:(in German) (2 ed.). Stuttgart, Germany:
25621:
25497:
25272:
24428:shows more relations between these functions.
24275:The tangent function was brought to Europe by
19940:{\displaystyle t=\tan {\tfrac {1}{2}}\theta ,}
18973:
12409:, but it remains true for all complex values.
8632:. Here, the poles are the numbers of the form
5571:for the most important angles are as follows:
4064:if necessary, intersects the line of equation
3496:, rather than angles. In fact, the functions
1326:are the heights to the line starting from the
26586:
26474:
25836:
25742:
25698:
25696:
25606:
25604:
25602:
25347:
24991:
24955:
24408:(which appeared in the earliest tables), the
24079:
24051:
16352:
14731:{\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1}
13828:
13720:is the unique value at which the derivative
11890:
11060:
1386:represents the side opposite the given angle
447:, geometrical definitions using the standard
294:
26123:
25565:
25383:
25041:: CS1 maint: multiple names: authors list (
24811:List of integrals of trigonometric functions
24679:
24358:Introduction to the Analysis of the Infinite
24343:. Though introduced as ratios of sides of a
24154:(90–165 CE). The functions of sine and
23196:{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}
21820:, so strictly speaking, they do not have an
18713:in terms of trigonometric functions. It is
16127:Trigonometric functions in the complex plane
14344:This defines the trigonometric functions on
14091:{\displaystyle -\pi /2<\theta <\pi /2}
12227:Euler's formula and the exponential function
26179:
26161:
25989:
25957:(in German) (1 ed.). Berlin, Germany:
25877:
25641:. Vol. 254. p. 74. Archived from
25531:
25481:
25341:
25180:
19893:These identities can be used to derive the
12702:{\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0}
7246:together with the first Taylor polynomials
3807:angent are positive from quadrants I to IV.
1996:can be used to remember these definitions.
26593:
26579:
26135:, Princeton University Press, 2009, p. 257
25883:
25693:
25599:
25522:
25156:
25130:Stueben, Michael; Sandford, Diane (1998).
25045:) CS1 maint: numeric names: authors list (
24796:Differentiation of trigonometric functions
22758:
22708:{\displaystyle x<-1{\text{ or }}x>1}
22631:{\displaystyle y=\operatorname {arccsc} x}
22576:
22543:{\displaystyle x<-1{\text{ or }}x>1}
22466:{\displaystyle y=\operatorname {arcsec} x}
22329:{\displaystyle y=\operatorname {arccot} x}
8782:one has the following series expansions:
8700:for the cotangent and the cosecant, where
7175:
6039:
3653:
329:, so their corresponding side lengths are
301:
287:
26534:Visionlearning Module on Wave Mathematics
25946:
25710:
25630:
25503:
24900:
24878:
24373:", as well as near-modern abbreviations (
23875:, the expansion of the periodic function
21784:
21635:
21462:
21062:{\displaystyle \ln \left|\sin x\right|+C}
20692:{\displaystyle \ln \left|\sec x\right|+C}
20386:
20231:
17458:
17360:
17319:
17247:
16384:, which is the smallest positive period:
16335:
16300:
16263:
16228:
16191:
16156:
15800:
13556:
13543:
13386:
13376:
13355:{\displaystyle \mathbb {R} /\mathbb {Z} }
13348:
13338:
12643:
12206:
12080:. From this, it can be deduced also that
12008:
8170:
8143:
8089:
7792:
3456:
2564:
2480:
2094:{\displaystyle \cos(90^{\circ }-\theta )}
26190:(Research report). HAL. inria-00543938.
26044:Various sources credit the first use of
25907:
25842:
25808:
25448:
25234:
25232:
25230:
25228:
25226:
25224:
24635:History of trigonometry § Etymology
24216:Muhammad ibn Jābir al-Harrānī al-Battānī
23803:
23787:
23775:
22398:{\displaystyle -\infty <x<\infty }
22244:{\displaystyle -\infty <x<\infty }
18100:; the other trigonometric functions are
17634:, where it has a simple pole of residue
17160:{\displaystyle -\pi <\Re (z)<\pi }
16312:
16277:
16240:
16205:
16168:
16133:
12474:{\displaystyle f_{1}(x)=\cos x+i\sin x,}
12230:
7889:is then a definition of the real number
7391:As solutions of a differential equation.
7210:
7202:
7190:
7179:
5554:
5056:
4529:{\displaystyle 0\leq \theta \leq \pi /2}
3771:
3683:
3675:
3657:
2103:
1274:
1185:
1149:be considered as denoting a composed or
312:
26477:College Calculus with Analytic Geometry
26468:MacTutor History of Mathematics archive
26448:MacTutor History of Mathematics archive
26435:MacTutor History of Mathematics archive
26408:
25871:
25826:MacTutor History of Mathematics Archive
25656:
25572:
25389:
25311:
25059:
24786:Bhaskara I's sine approximation formula
24260:of trigonometric functions in terms of
24158:(1 – cosine) can be traced back to the
21307:{\displaystyle \operatorname {arsinh} }
18962:{\displaystyle 1+\cot ^{2}x=\csc ^{2}x}
18898:{\displaystyle \tan ^{2}x+1=\sec ^{2}x}
18767:{\displaystyle \sin ^{2}x+\cos ^{2}x=1}
18704:
17888:{\displaystyle \cot(z)=\cos(z)/\sin(z)}
17564:{\displaystyle \tan(z)=\sin(z)/\cos(z)}
15895:
15252:. In particular, the limiting case as
14341:and the positive square root is taken.
7385:By using an infinite product expansion.
3735:, respectively, while the abscissas of
3508:, via power series, or as solutions to
1153:, but negative superscripts other than
573:so parentheses are required to express
532:would typically be interpreted to mean
14:
26680:
26601:Trigonometric and hyperbolic functions
26441:O'Connor, J. J., and E. F. Robertson,
26428:O'Connor, J. J., and E. F. Robertson,
26322:
26185:
26167:
25995:
25847:Elements of the History of Mathematics
25783:
25706:. Encyclopedia Britannica. 2023-11-17.
25537:
25466:
25186:
25123:
25106:
24863:{\displaystyle {\frac {1}{\sqrt {2}}}}
24301:The 17th century French mathematician
23800:with an increasing number of harmonics
21246:{\displaystyle -\pi /2<x<\pi /2}
15712:Definitions using functional equations
12604:{\displaystyle df_{j}(x)/dx=if_{j}(x)}
12395:{\displaystyle e^{ix}=\cos x+i\sin x.}
10130:are valid in the whole complex plane:
6825:{\displaystyle {\frac {\sqrt {3}}{2}}}
6716:{\displaystyle {\frac {\sqrt {2}}{2}}}
6686:{\displaystyle {\frac {\sqrt {2}}{2}}}
6598:{\displaystyle {\frac {\sqrt {3}}{3}}}
6568:{\displaystyle {\frac {\sqrt {3}}{2}}}
6012:. This results from the fact that the
1394:represents the side between the angle
1138:{\displaystyle \theta \cdot \sin x=1.}
26574:
26033:Horologiographia, the Art of Dialling
25749:Archive for History of Exact Sciences
25579:Fourier Analysis and its Applications
25238:
25221:
24906:
24884:
23771:
17915:and simple zeros at odd multiples of
13765:{\displaystyle {\frac {d}{dt}}e(t/a)}
12769:is a constant function, which equals
11179:This identity can be proved with the
7823:; cosine is the unique solution with
6005:, which may be expressed in terms of
3827:of radius one centered at the origin
746:{\displaystyle \sin(x)\cdot \sin(x),}
26100:
25111:. Addison-Wesley. pp. 256–257.
25002:
25000:
24256:(c. 1400) made early strides in the
23429:
21805:
16742:as well as other identities such as
15727:that satisfy the difference formula
13535:defines an isomorphism of the group
12830:{\displaystyle f_{1}(0)=f_{2}(0)=1.}
11065:There is a series representation as
8670:{\textstyle (2k+1){\frac {\pi }{2}}}
7961:{\displaystyle \tan x=\sin x/\cos x}
7396:Definition by differential equations
26397:. Oxford University Press, (1999).
26332:A History of Mathematical Notations
25913:
25010:Principles of mathematical analysis
24823:– a generalization to vertex angles
24732:the circle of unit radius, whereas
24728:meaning "touching", since the line
20258:, which reduces the computation of
18063:
17677:and the poles are odd multiples of
13576:. The real and imaginary parts of
13134:is real, this may be rewritten as
12298:are the real and imaginary part of
8677:for the tangent and the secant, or
6913:{\displaystyle {\frac {5\pi }{12}}}
5550:
5161:. Thus trigonometric functions are
3469:is convenient. One common unit is
1065:{\displaystyle \theta =\sin ^{-1}x}
24:
25885:von Hammer, Ernst Hermann Heinrich
25717:O'Connor, J. J.; Robertson, E. F.
25257:
25247:
24035:
23924:
23468:is the triangle's area. Therefore
23444:is the triangle's semiperimeter, (
23309:
22967:
22392:
22380:
22238:
22226:
18078:Proofs of trigonometric identities
18045:
17994:
17807:
17756:
17139:
16357:The cosine and sine functions are
15352:
15265:
14660:are extended continuously so that
14442:
13995:
13458:{\displaystyle e(t)=\exp(2\pi it)}
13295:{\displaystyle e^{a+b}=e^{a}e^{b}}
12118:
11940:
11816:
11661:
11570:
11565:
11443:
11364:
11359:
11264:
11116:
9884:
9625:
9554:
9243:
8928:
8830:
8517:
8315:
7909:which is independent of geometry.
6305:{\displaystyle {\frac {\pi }{12}}}
4860:
4817:
4769:
4726:
4623:{\displaystyle \mathrm {P} =(x,y)}
4598:
4474:
4431:
4391:are defined, respectively, as the
4344:
4328:
4301:
4279:
4241:
4192:
4176:
4120:
4098:
4042:
4011:
3996:
3980:
3884:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
25:
26729:
26509:
26328:"§2.2.1. Trigonometric Notations"
26186:Roegel, Denis, ed. (2010-12-06).
25996:Roegel, Denis, ed. (2016-08-30).
25100:
24997:
23561:The law of cotangents says that:
23111:
14422:{\displaystyle \theta \to \pi /2}
14334:{\displaystyle \theta =\arctan t}
14123:{\displaystyle \theta =\arctan t}
12762:{\displaystyle f_{1}(x)/f_{2}(x)}
7809:Sine is the unique solution with
7090:{\displaystyle {\frac {\pi }{2}}}
6767:{\displaystyle {\frac {\pi }{3}}}
6628:{\displaystyle {\frac {\pi }{4}}}
6482:{\displaystyle {\frac {\pi }{6}}}
3903:obtained by rotating by an angle
3833:right-angled triangle definitions
3831:of this coordinate system. While
3512:given particular initial values (
1182:Right-angled triangle definitions
26146:from the original on 2008-06-15.
26002:. Vandoeuvre-lès-Nancy, France:
25595:from the original on 2015-03-19.
25561:from the original on 2015-03-20.
25410:from the original on 2015-03-20.
25379:from the original on 2014-03-08.
25243:(8th ed.), pp. 432–438
25207:from the original on 2018-02-15.
24686:), which itself originated as a
24426:list of trigonometric identities
23460:is the radius of the triangle's
22879:and angles opposite those sides
18379:All trigonometric functions are
18074:List of trigonometric identities
14388:{\displaystyle (-\pi /2,\pi /2)}
13776:with positive imaginary part at
12529:{\displaystyle f_{2}(x)=e^{ix}.}
7168:
5092:Since a rotation of an angle of
4224:to the unit circle at the point
1023:{\displaystyle \arcsin x\colon }
56:
26194:from the original on 2017-07-28
26152:
26038:
26021:
26010:from the original on 2024-02-06
25865:
25797:from the original on 2017-01-07
25736:
25677:Springer Science+Business Media
25504:Kannappan, Palaniappan (2009).
25475:
25460:
25442:
25433:
25424:
25305:
25296:
25287:
25278:
25169:from the original on 2017-12-29
25157:Bityutskov, V.I. (2011-02-07).
25150:
24938:from the original on 2018-02-15
24835:
24806:Generating trigonometric tables
24698:, which along with its synonym
24198:by the 9th century, as was the
23320:The law of tangents says that:
22856:
22813:
21812:Inverse trigonometric functions
20270:Derivatives and antiderivatives
20256:tangent half-angle substitution
19947:all trigonometric functions of
18000:
17762:
17004:
16855:
16805:
16692:
16587:
16436:
15866:
15860:
15339:
14211:
14155:
14040:a definition that goes back to
13613:are the cosine and sine, where
13528:{\displaystyle t\mapsto e(t/a)}
13186:
12335:relates sine and cosine to the
12198:
12000:
10070:
9789:
9460:
9148:
8719:interpretation: they enumerate
8160:It is the unique solution with
5609:
5489:
5471:
5259:
5241:
5010:
4976:
4934:
4841:
4823:
4750:
4732:
4455:
4437:
4252:{\displaystyle {\mathcal {L}},}
4053:{\displaystyle {\mathcal {L}},}
3578:(≈ 6.28) rad. For real number
1354:-axis starting from the origin.
1100:{\displaystyle \sin \theta =x,}
911:is commonly used to denote the
480:or their lengths related to an
26286:; Sherbert, Donald R. (1999).
25575:"Convergence and completeness"
25107:Spivak, Michael (1967). "15".
25053:
24985:
24961:
24949:
24324:In a paper published in 1682,
24071:
24056:
24003:
23997:
23955:
23949:
23902:
23896:
23864:. Denoting the sine or cosine
23542:
23530:
23527:
23515:
23512:
23500:
22825:Angles and sides of a triangle
22107:{\displaystyle -1\leq x\leq 1}
21953:{\displaystyle -1\leq x\leq 1}
21759:
21739:
21717:
21697:
21608:
21588:
21579:
21559:
21544:
21524:
21444:
21424:
21409:
21389:
21278:
21266:
21160:
21148:
21099:{\displaystyle 0<x<\pi }
20383:
20377:
20350:
20344:
20315:
20309:
19447:
19435:
19178:
19166:
18667:
18647:
18620:
18600:
18573:
18556:
18529:
18512:
18485:
18465:
18438:
18418:
18335:
18326:
18294:
18285:
18253:
18244:
18212:
18203:
18174:
18165:
18133:
18124:
18096:The cosine and the secant are
18036:
18030:
18009:
17985:
17979:
17958:
17882:
17876:
17862:
17856:
17844:
17838:
17798:
17792:
17771:
17747:
17741:
17720:
17558:
17552:
17538:
17532:
17520:
17514:
17186:
17180:
17148:
17142:
17084:
17078:
17049:
17043:
17031:
17011:
16998:
16992:
16977:
16957:
16928:
16922:
16913:
16907:
16895:
16883:
16874:
16862:
16849:
16843:
16831:
16819:
16799:
16793:
16774:
16762:
16729:
16723:
16711:
16699:
16686:
16680:
16668:
16656:
16627:
16621:
16606:
16594:
16581:
16575:
16560:
16548:
16476:
16470:
16458:
16443:
16430:
16424:
16412:
16397:
15755:
15743:
15664:
15658:
15609:
15591:
15524:
15518:
15459:
15441:
15361:
15346:
15333:
15316:
15262:
15210:
15026:
14995:
14951:
14919:
14843:
14837:
14822:
14810:
14784:
14778:
14763:
14751:
14719:
14705:
14687:
14673:
14592:
14572:
14552:
14514:
14494:
14474:
14439:
14405:
14382:
14351:
14296:
14284:
14247:
14227:
14188:
14168:
13869:
13863:
13759:
13745:
13600:
13586:
13560:
13522:
13508:
13502:
13452:
13437:
13425:
13419:
13390:
12818:
12812:
12796:
12790:
12756:
12750:
12732:
12726:
12690:
12687:
12681:
12663:
12657:
12644:
12598:
12592:
12562:
12556:
12504:
12498:
12438:
12432:
12285:
12279:
12253:
12247:
12168:
12147:
11849:
11827:
11730:
11708:
11703:
11688:
11676:
11666:
11594:
11581:
11461:
11451:
11382:
11372:
11113:
10091:
10083:
9948:
9939:
9902:
9892:
9804:
9796:
9676:
9667:
9643:
9633:
9584:
9575:
9481:
9473:
9338:
9329:
9261:
9251:
9163:
9155:
9027:
9018:
8946:
8936:
8872:
8857:
8654:
8639:
8555:
8546:
8535:
8525:
8359:
8344:
8333:
8323:
7761:ordinary differential equation
7540:
7534:
7513:
7507:
7340:
7331:
7306:
7296:
7269:
7263:
7233:
7227:
6853:{\displaystyle {\frac {1}{2}}}
6538:{\displaystyle {\frac {1}{2}}}
6147:
6141:
6114:
6108:
6081:
6075:
6001:, the sine and the cosine are
5523:
5508:
5468:
5453:
4617:
4605:
4356:
4335:
4307:
4286:
4204:
4183:
4126:
4105:
4017:
3987:
3892:{\displaystyle {\mathcal {L}}}
3859:{\textstyle {\frac {\pi }{2}}}
2088:
2069:
2043:
2037:
1145:In this case, the superscript
999:inverse trigonometric function
984:
978:
868:
865:
859:
853:
844:
838:
835:
823:
817:
811:
778:
766:
737:
731:
719:
713:
687:
681:
598:
586:
551:
545:
317:Basis of trigonometry: if two
13:
1:
26662:Jyā, koti-jyā and utkrama-jyā
26288:Introduction to Real Analysis
26209:
24771:) and proceeds to define the
23294:is opposite to the side
23290:In this formula the angle at
22140:{\textstyle 0\leq y\leq \pi }
20760:{\displaystyle -\csc x\cot x}
12291:{\displaystyle \sin(\theta )}
12259:{\displaystyle \cos(\theta )}
11073:are summed up, such that the
8150:{\displaystyle y'=1+y^{2}\,.}
7060:{\displaystyle 2+{\sqrt {3}}}
6452:{\displaystyle 2-{\sqrt {3}}}
6153:{\displaystyle \tan(\theta )}
6120:{\displaystyle \cos(\theta )}
6087:{\displaystyle \sin(\theta )}
4583:{\displaystyle x^{2}+y^{2}=1}
3939:{\displaystyle \theta >0,}
2049:{\displaystyle \sin(\theta )}
990:{\displaystyle \sin ^{-1}(x)}
787:{\displaystyle \sin(\sin x).}
26703:Elementary special functions
25743:Van Brummelen, Glen (2018).
25241:A course of pure mathematics
25013:(Third ed.). New York.
24628:
20886:{\displaystyle \sec x\tan x}
17627:{\displaystyle z=\pm \pi /2}
17229:{\displaystyle z=\pm \pi /2}
15271:{\displaystyle s\to \infty }
14653:{\displaystyle \sin \theta }
14627:{\displaystyle \cos \theta }
14448:{\displaystyle t\to \infty }
13302:for simplifying the result.
12321:{\displaystyle e^{i\theta }}
10122:Continued fraction expansion
8715:. Their coefficients have a
7368:A Course of Pure Mathematics
4399:-coordinate values of point
4383:The trigonometric functions
3965:{\displaystyle \theta <0}
3267:
3077:
2829:
2581:
2389:
2199:
693:{\displaystyle \sin ^{2}(x)}
96:
7:
26522:Encyclopedia of Mathematics
26504:, accessed 21 January 2006.
26056:'s 1116 translation of the
25959:Walter de Gruyter & Co.
25947:Lötzbeyer, Philipp (1950).
25573:Folland, Gerald B. (2009).
25430:Whittaker and Watson, p 137
25393:Theory of complex functions
25163:Encyclopedia of Mathematics
24778:
24763:as an abbreviation for the
24701:
24693:
24347:, and thus appearing to be
24138:function was discovered by
22431:{\textstyle 0<y<\pi }
22175:{\displaystyle y=\arctan x}
22038:{\displaystyle y=\arccos x}
21884:{\displaystyle y=\arcsin x}
21013:{\displaystyle -\csc ^{2}x}
20282:. The values given for the
18974:Sum and difference formulas
18778:Dividing through by either
17597:and vertical asymptotes at
14302:{\displaystyle (t,\theta )}
7859:{\displaystyle \cos ,\sin }
7187:of sine, cosine and tangent
7118:{\displaystyle 90^{\circ }}
6941:{\displaystyle 75^{\circ }}
6878:{\displaystyle {\sqrt {3}}}
6795:{\displaystyle 60^{\circ }}
6656:{\displaystyle 45^{\circ }}
6510:{\displaystyle 30^{\circ }}
6333:{\displaystyle 15^{\circ }}
3946:and clockwise rotation for
948:{\displaystyle \sin ^{-1}x}
470:
361:which relate an angle of a
10:
26734:
26456:"Madhava of Sangamagramma"
26443:"Madhava of Sangamagramma"
26413:(2nd ed.), New York:
26101:Katx, Victor (July 2008).
25843:Bourbaki, Nicolás (1994).
25390:Remmert, Reinhold (1991).
25324:(3rd ed.), New York:
25284:Whitaker and Watson, p 584
25273:Bartle & Sherbert 1999
25007:Rudin, Walter, 1921–2010.
24992:Protter & Morrey (1970
24956:Protter & Morrey (1970
24816:List of periodic functions
24759:(1620), which defines the
24740:—"cutting"—since the line
24711:
24632:
24212:Habash al-Hasib al-Marwazi
24127:
24123:
23433:
23313:
23115:
22860:
22817:
22791:, etc. are often used for
21809:
20643:{\displaystyle \sec ^{2}x}
20396:{\textstyle \int f(x)\,dx}
18837:{\displaystyle \sin ^{2}x}
18804:{\displaystyle \cos ^{2}x}
18374:
17336:There zeros of cosine are
16643:It therefore follows that
16353:Periodicity and asymptotes
13829:Definition via integration
13468:For a nonzero real number
11891:Infinite product expansion
11067:partial fraction expansion
11061:Partial fraction expansion
11055:proof that π is irrational
7799:{\displaystyle y''+y=0\,.}
6212:{\displaystyle 0^{\circ }}
6043:
5973:exact trigonometric values
5960:Writing the numerators as
5425:. That is, the equalities
5188:. That is, the equalities
3630:)°, so that, for example,
654:{\displaystyle \sin ^{2}x}
607:{\displaystyle \sin(x+y).}
566:{\displaystyle \sin(x)+y,}
486:function concept developed
428:, and an analog among the
194:Trigonometric substitution
36:
29:
26644:
26606:
26517:"Trigonometric functions"
26479:(2nd ed.), Reading:
26430:"Trigonometric functions"
26158:Oxford English Dictionary
26109:. p. 210 (sidebar).
25924:10.1007/978-3-662-36585-4
25895:J. B. Metzlerscher Verlag
25821:"Trigonometric functions"
25761:10.1007/s00407-018-0214-2
25719:"Madhava of Sangamagrama"
25469:Elements of real analysis
25396:. Springer. p. 327.
25159:"Trigonometric Functions"
24791:Small-angle approximation
24680:
20488:{\displaystyle -\cos x+C}
19895:product-to-sum identities
18091:
15900:The sine and cosine of a
12837:This proves the formula.
8726:More precisely, defining
8704:is an arbitrary integer.
6128:
6095:
6062:
6055:
6031:. This is a corollary of
5111:{\displaystyle \pm 2\pi }
5061:Trigonometric functions:
4544:. And since the equation
4380:in the following manner.
4142:and the line of equation
3907:the positive half of the
3698:of points related to the
2175:
2172:
2169:
26409:Nielsen, Kaj L. (1966),
26309:A History of Mathematics
26277:McGraw-Hill Book Company
26105:(3rd ed.). Boston:
26103:A history of mathematics
25831:University of St Andrews
25631:Gingerich, Owen (1986).
25215:15 February 2018 at the
25078:10.4169/math.mag.87.1.37
25060:Diamond, Harvey (2014).
24958:, pp. APP-2, APP-3)
24918:Dover Publications, Inc.
24828:
24801:Generalized trigonometry
24353:transcendental functions
24246:, and Rheticus' student
22664:{\displaystyle \csc y=x}
22499:{\displaystyle \sec y=x}
22362:{\displaystyle \cot y=x}
22208:{\displaystyle \tan y=x}
22071:{\displaystyle \cos y=x}
21917:{\displaystyle \sin y=x}
21855:Set of principal values
21176:and for the integral of
20580:{\displaystyle \sin x+C}
16339:{\displaystyle \csc z\,}
16304:{\displaystyle \sec z\,}
16267:{\displaystyle \cot z\,}
16232:{\displaystyle \tan z\,}
16195:{\displaystyle \cos z\,}
16160:{\displaystyle \sin z\,}
15814:and the added condition
14879:, for all real numbers.
13244:trigonometric identities
8721:alternating permutations
8607:complex-valued functions
3819:that are related to the
3795:alculus" indicates when
3465:. For this purpose, any
525:{\displaystyle \sin x+y}
107:Generalized trigonometry
26688:Trigonometric functions
26394:Visual Complex Analysis
26290:(3rd ed.). Wiley.
24969:"Sine, Cosine, Tangent"
24254:Madhava of Sangamagrama
24146:(180–125 BCE) and
24130:History of trigonometry
23833:uniform circular motion
23819:is large is called the
21316:inverse hyperbolic sine
21132:can also be written as
20547:{\displaystyle -\sin x}
20292:constant of integration
19960:{\displaystyle \theta }
17825:The cotangent function
17192:{\displaystyle \cos(z)}
17090:{\displaystyle \sin(z)}
16093:By taking advantage of
15200:after the substitution
13713:{\displaystyle a=2\pi }
13655:{\displaystyle a=2\pi }
7562:Differentiating again,
7365:noted in his 1908 work
7239:{\displaystyle \cos(x)}
7176:Definitions in analysis
6040:Simple algebraic values
3654:Unit-circle definitions
3574:(360°) is an angle of 2
2111:Trigonometric function
1178:are not in common use.
343:trigonometric functions
26373:Trigonometric Delights
26069:'s translation of the
26046:
26004:Université de Lorraine
25874:, pp. xxiii–xxiv)
25302:Abramowitz; Weisstein.
24908:Klein, Christian Felix
24886:Klein, Christian Felix
24864:
24641:
24189:Aryabhata's sine table
24120:are shown underneath.
24108:
24039:
23977:can be written as the
23965:
23928:
23829:simple harmonic motion
23824:
23801:
23785:
23762:
23612:
23551:
23419:
23282:
23197:
23084:
22991:
22772:
22709:
22665:
22632:
22597:
22544:
22500:
22467:
22432:
22399:
22363:
22330:
22295:
22245:
22209:
22176:
22141:
22108:
22072:
22039:
22004:
21954:
21918:
21885:
21796:
21308:
21288:
21247:
21196:
21195:{\displaystyle \sec x}
21170:
21126:
21125:{\displaystyle \csc x}
21100:
21063:
21014:
20977:
20976:{\displaystyle \cot x}
20948:
20887:
20851:
20850:{\displaystyle \sec x}
20822:
20761:
20722:
20721:{\displaystyle \csc x}
20693:
20644:
20610:
20609:{\displaystyle \tan x}
20581:
20548:
20518:
20517:{\displaystyle \cos x}
20489:
20453:
20452:{\displaystyle \cos x}
20426:
20425:{\displaystyle \sin x}
20397:
20357:
20322:
20245:
20178:
19985:
19961:
19941:
19884:
19521:
19252:
18963:
18899:
18838:
18805:
18768:
18695:
18365:
18055:
17937:
17936:{\displaystyle \pi /2}
17909:
17889:
17817:
17699:
17698:{\displaystyle \pi /2}
17671:
17651:
17628:
17591:
17565:
17498:at each of the zeros.
17492:
17469:
17330:
17230:
17199:has the pair of zeros
17193:
17161:
17117:
17097:has a unique zero (at
17091:
17059:
16938:
16736:
16637:
16529:
16509:
16486:
16378:
16340:
16317:
16305:
16282:
16268:
16245:
16233:
16210:
16196:
16173:
16161:
16138:
16111:
16084:
15932:
15931:{\displaystyle z=x+iy}
15886:
15805:
15702:
15701:{\displaystyle \pi /2}
15674:
15531:
15371:
15272:
15246:
15194:
15033:
14958:
14873:
14850:
14791:
14738:. Now the conditions
14732:
14654:
14628:
14602:
14524:
14449:
14423:
14389:
14335:
14303:
14271:
14124:
14092:
14034:
13956:
13936:
13847:
13819:
13796:
13766:
13714:
13682:
13656:
13627:
13607:
13606:{\displaystyle e(t/a)}
13570:
13529:
13482:
13459:
13403:
13356:
13323:
13296:
13233:
13121:
12960:
12831:
12763:
12703:
12605:
12530:
12475:
12396:
12329:
12322:
12292:
12260:
12217:
12122:
12074:
12073:{\displaystyle \sin z}
12048:
12047:{\displaystyle \cot z}
12019:
11944:
11881:
11820:
11762:
11665:
11613:
11574:
11505:
11447:
11368:
11308:
11268:
11170:
11144:
11069:where just translated
11044:
10634:
10379:
10112:
9888:
9832:
9629:
9558:
9502:
9247:
9191:
8932:
8834:
8694:
8671:
8588:
8521:
8319:
8171:Power series expansion
8151:
8097:
7962:
7903:
7883:
7860:
7800:
7753:
7653:
7553:
7359:
7353:
7295:
7240:
7208:
7200:
7188:
7161:
7140:
7119:
7091:
7061:
7030:
6986:
6942:
6914:
6879:
6854:
6826:
6796:
6768:
6738:
6717:
6687:
6657:
6629:
6599:
6569:
6539:
6511:
6483:
6453:
6422:
6378:
6334:
6306:
6276:
6255:
6234:
6213:
6185:
6154:
6121:
6088:
6029:transcendental numbers
6018:cyclotomic polynomials
5946:
5872:
5804:
5721:
5635:
5564:
5530:
5476:
5419:
5391:
5367:
5344:
5307:
5246:
5182:
5155:
5112:
5089:
5048:
4873:
4828:
4782:
4737:
4689:
4624:
4584:
4530:
4487:
4442:
4366:
4314:
4253:
4214:
4162:
4136:
4084:
4054:
4027:
3966:
3940:
3893:
3860:
3808:
3769:
3681:
3673:
3510:differential equations
3475:elementary mathematics
3457:Radians versus degrees
3445:
3362:
3255:
3172:
3065:
2953:
2817:
2705:
2569:
2485:
2377:
2294:
2161:
2155:in the four quadrants.
2095:
2050:
1980:
1887:
1792:
1693:
1592:
1493:
1355:
1350:are lengths along the
1286:. The points labelled
1272:
1172:
1139:
1101:
1066:
1024:
1001:alternatively written
991:
949:
905:
886:However, the exponent
878:
788:
747:
694:
655:
608:
567:
526:
461:differential equations
334:
26718:Dimensionless numbers
26626:Inverse trigonometric
26362:IEEE Trans. Computers
26091:in Ceccarelli (ed.),
25966:10.1515/9783111507545
25418:20 March 2015 at the
24922:The Macmillan Company
24865:
24109:
24019:
23966:
23908:
23807:
23791:
23779:
23763:
23613:
23552:
23420:
23283:
23198:
23085:
22992:
22773:
22710:
22666:
22633:
22598:
22545:
22501:
22468:
22433:
22400:
22364:
22331:
22296:
22246:
22210:
22177:
22142:
22109:
22073:
22040:
22005:
21955:
21919:
21886:
21830:multivalued functions
21797:
21309:
21289:
21248:
21197:
21171:
21127:
21101:
21064:
21015:
20978:
20949:
20888:
20852:
20823:
20762:
20723:
20694:
20645:
20611:
20582:
20549:
20519:
20490:
20454:
20427:
20398:
20358:
20356:{\displaystyle f'(x)}
20323:
20246:
20179:
19986:
19962:
19942:
19885:
19532:double-angle formulae
19522:
19253:
18964:
18900:
18839:
18806:
18769:
18696:
18366:
18056:
17938:
17910:
17890:
17818:
17700:
17672:
17652:
17629:
17592:
17571:has a simple zero at
17566:
17501:The tangent function
17493:
17491:{\displaystyle \pm 1}
17470:
17331:
17231:
17194:
17162:
17118:
17092:
17060:
16939:
16737:
16638:
16530:
16510:
16508:{\displaystyle 2\pi }
16487:
16379:
16377:{\displaystyle 2\pi }
16341:
16316:
16306:
16281:
16269:
16244:
16234:
16209:
16197:
16172:
16162:
16137:
16112:
16085:
15933:
15887:
15806:
15703:
15675:
15532:
15372:
15273:
15247:
15195:
15034:
14959:
14874:
14872:{\displaystyle 2\pi }
14851:
14792:
14733:
14655:
14629:
14603:
14525:
14450:
14424:
14390:
14336:
14304:
14272:
14125:
14093:
14035:
13957:
13937:
13848:
13820:
13818:{\displaystyle 2\pi }
13797:
13767:
13715:
13683:
13681:{\displaystyle a=360}
13657:
13628:
13608:
13571:
13530:
13483:
13460:
13404:
13362:, via an isomorphism
13357:
13324:
13297:
13234:
13122:
12961:
12832:
12764:
12704:
12606:
12531:
12476:
12397:
12323:
12293:
12261:
12234:
12218:
12102:
12075:
12049:
12020:
11924:
11882:
11800:
11763:
11645:
11614:
11551:
11506:
11427:
11345:
11309:
11248:
11199:absolutely convergent
11183:trick. Combining the
11171:
11121:
11045:
10635:
10380:
10113:
9868:
9833:
9609:
9538:
9503:
9227:
9192:
8912:
8814:
8713:radius of convergence
8695:
8693:{\displaystyle k\pi }
8672:
8626:meromorphic functions
8609:that are defined and
8599:radius of convergence
8589:
8501:
8299:
8152:
8098:
7963:
7904:
7884:
7882:{\displaystyle 2\pi }
7861:
7801:
7754:
7654:
7554:
7402:initial value problem
7354:
7275:
7241:
7214:
7206:
7194:
7183:
7162:
7141:
7120:
7092:
7062:
7031:
6987:
6943:
6915:
6880:
6855:
6827:
6797:
6769:
6739:
6718:
6688:
6658:
6630:
6600:
6570:
6540:
6512:
6484:
6454:
6423:
6379:
6335:
6307:
6277:
6256:
6235:
6214:
6186:
6155:
6122:
6089:
5947:
5873:
5805:
5722:
5636:
5569:algebraic expressions
5558:
5531:
5477:
5420:
5392:
5368:
5366:{\displaystyle 2\pi }
5345:
5343:{\displaystyle 2\pi }
5308:
5247:
5183:
5181:{\displaystyle 2\pi }
5156:
5154:{\displaystyle 2\pi }
5113:
5060:
5049:
4874:
4829:
4783:
4738:
4690:
4625:
4590:holds for all points
4585:
4531:
4488:
4443:
4367:
4315:
4254:
4215:
4163:
4137:
4085:
4055:
4028:
3967:
3941:
3894:
3861:
3775:
3696:Cartesian coordinates
3687:
3679:
3661:
3486:mathematical analysis
3446:
3363:
3256:
3173:
3066:
2954:
2818:
2706:
2570:
2486:
2378:
2295:
2107:
2096:
2051:
1981:
1888:
1793:
1694:
1593:
1494:
1398:and the right angle.
1278:
1189:
1173:
1140:
1102:
1067:
1025:
992:
950:
906:
879:
789:
748:
695:
656:
609:
568:
527:
451:(i.e., a circle with
412:are respectively the
363:right-angled triangle
355:goniometric functions
316:
27:Functions of an angle
26494:Weisstein, Eric W.,
26367:(3), 328–339 (1996).
26132:Mathematics in India
25914:Heß, Adolf (1926) .
25817:Robertson, Edmund F.
25669:D'Ambrosio, Ubiratan
25456:. Springer. §VIII.2.
25358:Proofs from THE BOOK
25328:, pp. 129–140,
25314:Borwein, Jonathan M.
25239:Hardy, G.H. (1950),
25187:Larson, Ron (2013).
25066:Mathematics Magazine
24845:
24270:Madhava's sine table
24228:Nasir al-Din al-Tusi
23984:
23890:
23792:An animation of the
23628:
23568:
23479:
23327:
23207:
23130:
23005:
22895:
22820:Uses of trigonometry
22720:
22676:
22643:
22610:
22555:
22511:
22478:
22445:
22410:
22374:
22341:
22308:
22256:
22220:
22187:
22154:
22119:
22083:
22050:
22017:
21965:
21929:
21896:
21863:
21328:
21298:
21257:
21206:
21180:
21136:
21110:
21078:
21025:
20988:
20961:
20898:
20862:
20835:
20772:
20733:
20706:
20655:
20621:
20594:
20559:
20529:
20502:
20464:
20437:
20410:
20368:
20333:
20321:{\displaystyle f(x)}
20303:
20194:
19998:
19975:
19967:can be expressed as
19951:
19904:
19541:
19266:
18997:
18915:
18851:
18815:
18782:
18720:
18705:Pythagorean identity
18405:
18111:
17947:
17919:
17908:{\displaystyle \pi }
17899:
17829:
17709:
17681:
17670:{\displaystyle \pi }
17661:
17638:
17601:
17575:
17505:
17479:
17340:
17240:
17203:
17171:
17127:
17101:
17069:
16948:
16746:
16647:
16539:
16528:{\displaystyle \pi }
16519:
16496:
16388:
16365:
16323:
16288:
16251:
16216:
16179:
16144:
16101:
15949:
15940:hyperbolic functions
15907:
15896:In the complex plane
15821:
15734:
15725:continuous functions
15718:functional equations
15684:
15541:
15381:
15282:
15256:
15204:
15043:
14968:
14886:
14860:
14801:
14742:
14664:
14638:
14612:
14534:
14459:
14433:
14399:
14348:
14313:
14281:
14134:
14102:
14051:
13966:
13955:{\displaystyle \pi }
13946:
13857:
13837:
13806:
13780:
13724:
13695:
13666:
13637:
13617:
13580:
13539:
13496:
13472:
13413:
13409:In pedestrian terms
13366:
13334:
13313:
13250:
13141:
12980:
12847:
12777:
12713:
12626:
12540:
12485:
12419:
12346:
12337:exponential function
12302:
12270:
12238:
12087:
12058:
12032:
11906:
11773:
11624:
11516:
11324:
11208:
11084:
11071:reciprocal functions
10647:
10392:
10137:
9845:
9515:
9204:
8789:
8681:
8636:
8182:
8113:
7975:
7920:
7902:{\displaystyle \pi }
7893:
7870:
7844:
7769:
7663:
7566:
7411:
7250:
7218:
7151:
7130:
7102:
7074:
7041:
6997:
6953:
6925:
6892:
6865:
6837:
6807:
6779:
6751:
6728:
6698:
6668:
6640:
6612:
6580:
6550:
6522:
6494:
6466:
6433:
6389:
6345:
6317:
6289:
6266:
6245:
6224:
6196:
6175:
6132:
6099:
6066:
5883:
5815:
5732:
5651:
5578:
5486:
5432:
5418:{\displaystyle \pi }
5409:
5390:{\displaystyle \pi }
5381:
5354:
5331:
5256:
5195:
5169:
5142:
5096:
4890:
4838:
4793:
4747:
4705:
4641:
4632:Pythagorean identity
4594:
4548:
4500:
4452:
4410:
4324:
4275:
4236:
4172:
4146:
4094:
4068:
4037:
3976:
3950:
3921:
3879:
3843:
3553:indefinite integrals
3506:exponential function
3373:
3290:
3183:
3100:
2964:
2852:
2716:
2604:
2496:
2412:
2305:
2222:
2118:for selected angles
2060:
2028:
1905:
1812:
1711:
1612:
1511:
1412:
1171:{\displaystyle {-1}}
1157:
1111:
1076:
1034:
1005:
959:
923:
904:{\displaystyle {-1}}
890:
798:
757:
704:
665:
632:
626:function composition
577:
536:
504:
430:hyperbolic functions
214:Trigonometric series
26341:Joseph, George G.,
25978:. Archive ID 541650
25815:O'Connor, John J.;
25638:Scientific American
25633:"Islamic Astronomy"
25491:Mathematische Werke
25413:Extract of page 327
25361:(Second ed.).
25210:Extract of page 153
24769:complementary angle
24196:Islamic mathematics
23300:Pythagorean theorem
23124:Pythagorean theorem
18711:Pythagorean theorem
17590:{\displaystyle z=0}
17116:{\displaystyle z=0}
16129:
15159:
15087:
14309:is on the graph of
13999:
13889:
13795:{\displaystyle t=0}
11003:
10991:
10983:
10971:
10952:
10940:
10932:
10920:
10901:
10889:
10881:
10869:
10850:
10838:
10830:
10818:
10778:
10759:
10745:
10726:
10712:
10693:
10679:
10667:
10579:
10551:
10518:
10490:
10457:
10438:
10424:
10412:
10324:
10296:
10263:
10235:
10202:
10183:
10169:
10157:
10128:continued fractions
5539:hold for any angle
5316:hold for any angle
4259:and intersects the
4161:{\displaystyle y=1}
4083:{\displaystyle x=1}
3702:. The ordinates of
3694:are represented as
2166:
1358:If the acute angle
490:functional notation
459:or as solutions of
379:celestial mechanics
176:Pythagorean theorem
39:log cosine function
26708:Analytic functions
26636:Inverse hyperbolic
26560:Article about the
26547:Article about the
26461:2006-05-05 at the
26415:Barnes & Noble
26387:Needham, Tristan,
26349:, London. (2000).
26275:, second edition,
26218:Abramowitz, Milton
26173:Canon triangulorum
26140:"Clark University"
25791:"Fincke biography"
25539:Farlow, Stanley J.
25471:, pp. 315–316
25454:Topologie generale
25353:Ziegler, Günter M.
24973:www.mathsisfun.com
24860:
24757:Canon triangulorum
24367:exponential series
24363:alternating series
24349:rational functions
24337:algebraic function
24277:Giovanni Bianchini
24168:functions used in
24104:
23961:
23840:periodic functions
23825:
23802:
23794:additive synthesis
23786:
23772:Periodic functions
23758:
23726:
23688:
23650:
23608:
23547:
23415:
23278:
23193:
23094:is the triangle's
23080:
22987:
22808:complex logarithms
22768:
22705:
22661:
22628:
22593:
22540:
22496:
22463:
22428:
22395:
22359:
22326:
22291:
22241:
22205:
22172:
22137:
22104:
22068:
22035:
22000:
21950:
21914:
21881:
21792:
21790:
21304:
21284:
21243:
21192:
21166:
21122:
21096:
21059:
21010:
20973:
20944:
20883:
20847:
20818:
20757:
20718:
20689:
20640:
20606:
20577:
20544:
20514:
20485:
20449:
20422:
20393:
20353:
20318:
20241:
20174:
20172:
19981:
19969:rational fractions
19957:
19937:
19929:
19880:
19878:
19517:
19515:
19248:
19246:
18959:
18895:
18834:
18801:
18764:
18691:
18689:
18381:periodic functions
18361:
18359:
18051:
18023:
17972:
17933:
17905:
17885:
17813:
17785:
17734:
17695:
17667:
17650:{\displaystyle -1}
17647:
17624:
17587:
17561:
17488:
17465:
17326:
17226:
17189:
17157:
17113:
17087:
17055:
16934:
16732:
16633:
16525:
16505:
16482:
16374:
16336:
16318:
16301:
16283:
16264:
16246:
16229:
16211:
16192:
16174:
16157:
16139:
16125:
16107:
16080:
16078:
15928:
15882:
15801:
15698:
15670:
15527:
15367:
15268:
15242:
15190:
15121:
15070:
15029:
14954:
14869:
14846:
14787:
14728:
14650:
14624:
14598:
14520:
14445:
14419:
14385:
14331:
14299:
14267:
14120:
14088:
14030:
13985:
13952:
13932:
13875:
13843:
13815:
13792:
13762:
13710:
13678:
13652:
13623:
13603:
13566:
13525:
13478:
13455:
13399:
13352:
13319:
13307:topological groups
13292:
13229:
13117:
13115:
12956:
12954:
12827:
12759:
12699:
12622:implies thus that
12601:
12526:
12471:
12392:
12330:
12318:
12288:
12256:
12213:
12070:
12044:
12015:
11877:
11846:
11758:
11727:
11609:
11501:
11304:
11166:
11120:
11040:
11036:
11031:
11026:
11021:
11010:
10998:
10978:
10959:
10947:
10927:
10908:
10896:
10876:
10857:
10845:
10825:
10806:
10801:
10796:
10791:
10773:
10740:
10707:
10674:
10630:
10626:
10621:
10616:
10611:
10574:
10513:
10452:
10419:
10375:
10371:
10366:
10361:
10356:
10319:
10258:
10197:
10164:
10108:
10106:
9828:
9826:
9498:
9496:
9187:
9185:
8690:
8667:
8584:
8582:
8147:
8093:
7958:
7899:
7879:
7856:
7796:
7749:
7649:
7549:
7360:
7349:
7236:
7209:
7201:
7189:
7157:
7136:
7115:
7087:
7057:
7026:
6982:
6938:
6910:
6875:
6850:
6822:
6792:
6764:
6734:
6713:
6683:
6653:
6625:
6595:
6565:
6535:
6507:
6479:
6449:
6418:
6374:
6330:
6302:
6272:
6251:
6230:
6209:
6181:
6150:
6117:
6084:
5942:
5868:
5800:
5717:
5631:
5565:
5526:
5472:
5415:
5387:
5375:fundamental period
5363:
5340:
5303:
5242:
5178:
5163:periodic functions
5151:
5108:
5090:
5083:Cotangent (dotted)
5044:
4869:
4824:
4778:
4733:
4685:
4620:
4580:
4526:
4483:
4438:
4362:
4310:
4249:
4210:
4158:
4132:
4080:
4050:
4023:
3962:
3936:
3889:
3856:
3809:
3770:
3682:
3674:
3441:
3358:
3251:
3168:
3061:
2949:
2813:
2701:
2565:
2481:
2373:
2290:
2164:
2162:
2091:
2046:
1976:
1883:
1788:
1689:
1588:
1489:
1356:
1273:
1168:
1135:
1097:
1062:
1020:
987:
945:
901:
874:
784:
743:
690:
651:
604:
563:
522:
387:periodic functions
347:circular functions
335:
26675:
26674:
26284:Bartle, Robert G.
26279:, New York, 1966.
26237:978-0-486-61272-0
26222:Stegun, Irene Ann
26080:Robert of Chester
26067:Gerard of Cremona
25933:978-3-662-35755-2
25686:978-1-4020-0260-1
25588:978-0-8218-4790-9
25571:See for example,
25554:978-0-486-67620-3
25483:Weierstrass, Karl
25450:Bourbaki, Nicolas
25403:978-0-387-97195-7
25372:978-3-642-00855-9
25322:Pi, a source book
25318:Borwein, Peter B.
25200:978-1-285-60718-4
25143:978-0-88385-525-6
24858:
24857:
24765:sinus complementi
24736:stems from Latin
24724:comes from Latin
24326:Gottfried Leibniz
24296:Geometria rotundi
24204:solving triangles
24099:
24017:
23994:
23973:For example, the
23753:
23740:
23725:
23702:
23687:
23664:
23649:
23606:
23585:
23545:
23498:
23436:Law of cotangents
23430:Law of cotangents
23413:
23384:
23381:
23355:
23273:
23203:or equivalently,
23066:
23045:
23024:
22982:
22956:
22935:
22914:
22781:
22780:
22753:
22734:
22694:
22591:
22529:
22289:
22270:
21998:
21979:
21806:Inverse functions
21689:
21667:
21516:
21494:
21381:
21359:
21072:
21071:
20229:
20165:
20112:
20049:
19984:{\displaystyle t}
19928:
19871:
19803:
19631:
19508:
19239:
18001:
17950:
17763:
17712:
17441:
17423:
17410:
17394:
17351:
16350:
16349:
16110:{\displaystyle z}
15864:
15647:
15646:
15619:
15618:
15569:
15504:
15503:
15471:
15468:
15409:
15305:
15240:
15188:
15157:
15116:
14949:
14028:
13977:
13918:
13846:{\displaystyle t}
13740:
13626:{\displaystyle a}
13481:{\displaystyle a}
13322:{\displaystyle U}
13108:
13047:
12188:
11990:
11872:
11845:
11753:
11726:
11604:
11496:
11416:
11403:
11299:
11237:
11161:
11105:
11038:
11033:
11028:
11023:
11012:
11002:
10990:
10982:
10970:
10961:
10951:
10939:
10931:
10919:
10910:
10900:
10888:
10880:
10868:
10859:
10849:
10837:
10829:
10817:
10808:
10803:
10798:
10793:
10777:
10758:
10744:
10725:
10711:
10692:
10678:
10666:
10628:
10623:
10618:
10613:
10578:
10550:
10517:
10489:
10456:
10437:
10423:
10411:
10373:
10368:
10363:
10358:
10323:
10295:
10262:
10234:
10201:
10182:
10168:
10156:
10074:
10049:
10026:
10010:
9955:
9819:
9793:
9768:
9745:
9722:
9683:
9591:
9464:
9439:
9416:
9400:
9345:
9178:
9152:
9127:
9104:
9081:
9034:
8879:
8665:
8562:
8483:
8458:
8433:
8366:
8281:
8256:
8231:
8062:
7991:
7723:
7693:
7623:
7596:
7527:
7500:
7469:
7455:
7427:
7347:
7197:Taylor polynomial
7173:
7172:
7160:{\displaystyle 0}
7139:{\displaystyle 1}
7085:
7055:
7024:
7018:
7008:
6980:
6974:
6964:
6908:
6873:
6848:
6820:
6816:
6762:
6737:{\displaystyle 1}
6711:
6707:
6681:
6677:
6623:
6593:
6589:
6563:
6559:
6533:
6477:
6447:
6416:
6410:
6400:
6372:
6366:
6356:
6300:
6275:{\displaystyle 0}
6254:{\displaystyle 1}
6233:{\displaystyle 0}
6184:{\displaystyle 0}
6035:, proved in 1966.
6003:algebraic numbers
5977:ruler and compass
5934:
5930:
5900:
5866:
5862:
5832:
5798:
5797:
5783:
5779:
5749:
5715:
5702:
5698:
5668:
5623:
5619:
5075:Cosecant (dotted)
5039:
5005:
4971:
4929:
4807:
3854:
3815:of points on the
3813:coordinate values
3454:
3453:
3439:
3356:
3324:
3249:
3166:
3134:
3059:
3003:
2947:
2915:
2891:
2811:
2755:
2699:
2667:
2643:
2562:
2478:
2446:
2371:
2288:
2256:
1994:Various mnemonics
1991:
1990:
1974:
1881:
1786:
1687:
1586:
1487:
1151:iterated function
408:functions. Their
311:
310:
203:inverse functions
146:Laws and theorems
32:log sine function
16:(Redirected from
26725:
26595:
26588:
26581:
26572:
26571:
26530:
26491:
26454:Pearce, Ian G.,
26425:
26335:
26301:
26265:
26224:, eds. (1983) .
26203:
26202:
26200:
26199:
26183:
26177:
26176:
26165:
26159:
26156:
26150:
26147:
26127:
26121:
26120:
26095:, Springer, 2004
26054:Plato Tiburtinus
26049:
26042:
26036:
26025:
26019:
26018:
26016:
26015:
26006:. hal-01357842.
25993:
25987:
25986:
25984:
25983:
25975:978-3-11114038-4
25944:
25938:
25937:
25911:
25905:
25904:
25902:
25901:
25881:
25875:
25869:
25863:
25862:
25850:
25840:
25834:
25833:
25812:
25806:
25805:
25803:
25802:
25787:
25781:
25780:
25740:
25734:
25733:
25731:
25730:
25714:
25708:
25707:
25700:
25691:
25690:
25660:
25654:
25653:
25651:
25650:
25628:
25619:
25608:
25597:
25596:
25569:
25563:
25562:
25535:
25529:
25526:
25520:
25519:
25501:
25495:
25494:
25479:
25473:
25472:
25464:
25458:
25457:
25446:
25440:
25437:
25431:
25428:
25422:
25411:
25387:
25381:
25380:
25345:
25339:
25338:
25309:
25303:
25300:
25294:
25291:
25285:
25282:
25276:
25270:
25264:
25261:
25255:
25251:
25245:
25244:
25236:
25219:
25208:
25184:
25178:
25177:
25175:
25174:
25154:
25148:
25147:
25127:
25121:
25120:
25104:
25098:
25097:
25057:
25051:
25050:
25040:
25032:
25004:
24995:
24994:, p. APP-7)
24989:
24983:
24982:
24980:
24979:
24965:
24959:
24953:
24947:
24946:
24944:
24943:
24931:978-0-48643480-3
24904:
24898:
24897:
24882:
24870:
24869:
24867:
24866:
24861:
24859:
24853:
24849:
24839:
24716:
24715:
24704:
24696:
24685:
24684:
24644:
24617:
24607:
24605:
24604:
24601:
24598:
24597:
24582:
24569:
24567:
24565:
24564:
24561:
24558:
24545:
24543:
24542:
24539:
24536:
24522:
24516:
24514:
24513:
24510:
24507:
24506:
24487:
24485:
24483:
24482:
24479:
24476:
24456:
24454:
24452:
24451:
24448:
24445:
24369:. He presented "
24342:
24334:
24238:(14th century),
24234:(14th century),
24232:Jamshīd al-Kāshī
24173:Indian astronomy
24113:
24111:
24110:
24105:
24100:
24098:
24084:
24083:
24082:
24055:
24054:
24041:
24038:
24033:
24018:
24010:
23996:
23995:
23992:
23970:
23968:
23967:
23962:
23948:
23947:
23938:
23937:
23927:
23922:
23886:takes the form:
23885:
23874:
23859:
23821:Gibbs phenomenon
23818:
23814:
23767:
23765:
23764:
23759:
23754:
23746:
23741:
23739:
23728:
23727:
23718:
23708:
23703:
23701:
23690:
23689:
23680:
23670:
23665:
23663:
23652:
23651:
23642:
23632:
23621:It follows that
23617:
23615:
23614:
23609:
23607:
23602:
23591:
23586:
23578:
23556:
23554:
23553:
23548:
23546:
23499:
23491:
23489:
23424:
23422:
23421:
23416:
23414:
23412:
23401:
23390:
23385:
23383:
23382:
23377:
23366:
23357:
23356:
23351:
23340:
23331:
23297:
23293:
23287:
23285:
23284:
23279:
23274:
23272:
23261:
23260:
23259:
23247:
23246:
23234:
23233:
23223:
23202:
23200:
23199:
23194:
23168:
23167:
23155:
23154:
23142:
23141:
23093:
23089:
23087:
23086:
23081:
23067:
23065:
23051:
23046:
23044:
23030:
23025:
23023:
23009:
23000:
22996:
22994:
22993:
22988:
22983:
22981:
22970:
22962:
22957:
22952:
22941:
22936:
22931:
22920:
22915:
22910:
22899:
22890:
22886:
22882:
22878:
22874:
22870:
22852:
22848:
22844:
22840:
22836:
22832:
22829:In this section
22798:
22794:
22790:
22786:
22777:
22775:
22774:
22769:
22754:
22746:
22735:
22727:
22714:
22712:
22711:
22706:
22695:
22692:
22670:
22668:
22667:
22662:
22637:
22635:
22634:
22629:
22602:
22600:
22599:
22594:
22592:
22584:
22549:
22547:
22546:
22541:
22530:
22527:
22505:
22503:
22502:
22497:
22472:
22470:
22469:
22464:
22437:
22435:
22434:
22429:
22404:
22402:
22401:
22396:
22368:
22366:
22365:
22360:
22335:
22333:
22332:
22327:
22300:
22298:
22297:
22292:
22290:
22282:
22271:
22263:
22250:
22248:
22247:
22242:
22214:
22212:
22211:
22206:
22181:
22179:
22178:
22173:
22146:
22144:
22143:
22138:
22113:
22111:
22110:
22105:
22077:
22075:
22074:
22069:
22044:
22042:
22041:
22036:
22009:
22007:
22006:
22001:
21999:
21991:
21980:
21972:
21959:
21957:
21956:
21951:
21923:
21921:
21920:
21915:
21890:
21888:
21887:
21882:
21843:
21842:
21838:principal values
21822:inverse function
21801:
21799:
21798:
21793:
21791:
21777:
21776:
21749:
21735:
21734:
21707:
21690:
21688:
21677:
21668:
21666:
21658:
21644:
21598:
21569:
21534:
21517:
21515:
21504:
21495:
21493:
21485:
21471:
21434:
21399:
21382:
21380:
21369:
21360:
21358:
21350:
21336:
21313:
21311:
21310:
21305:
21293:
21291:
21290:
21285:
21252:
21250:
21249:
21244:
21239:
21219:
21201:
21199:
21198:
21193:
21175:
21173:
21172:
21167:
21131:
21129:
21128:
21123:
21106:the integral of
21105:
21103:
21102:
21097:
21068:
21066:
21065:
21060:
21052:
21048:
21019:
21017:
21016:
21011:
21003:
21002:
20982:
20980:
20979:
20974:
20953:
20951:
20950:
20945:
20937:
20933:
20892:
20890:
20889:
20884:
20856:
20854:
20853:
20848:
20827:
20825:
20824:
20819:
20811:
20807:
20766:
20764:
20763:
20758:
20727:
20725:
20724:
20719:
20698:
20696:
20695:
20690:
20682:
20678:
20649:
20647:
20646:
20641:
20633:
20632:
20615:
20613:
20612:
20607:
20586:
20584:
20583:
20578:
20553:
20551:
20550:
20545:
20523:
20521:
20520:
20515:
20494:
20492:
20491:
20486:
20458:
20456:
20455:
20450:
20431:
20429:
20428:
20423:
20402:
20400:
20399:
20394:
20362:
20360:
20359:
20354:
20343:
20327:
20325:
20324:
20319:
20297:
20296:
20289:
20250:
20248:
20247:
20242:
20230:
20228:
20227:
20226:
20207:
20183:
20181:
20180:
20175:
20173:
20166:
20164:
20163:
20162:
20146:
20138:
20113:
20111:
20110:
20109:
20093:
20092:
20091:
20075:
20050:
20048:
20047:
20046:
20030:
20022:
19990:
19988:
19987:
19982:
19966:
19964:
19963:
19958:
19946:
19944:
19943:
19938:
19930:
19921:
19889:
19887:
19886:
19881:
19879:
19872:
19870:
19863:
19862:
19846:
19832:
19804:
19802:
19795:
19794:
19778:
19771:
19770:
19754:
19743:
19742:
19709:
19708:
19687:
19686:
19668:
19667:
19632:
19630:
19623:
19622:
19606:
19592:
19526:
19524:
19523:
19518:
19516:
19509:
19507:
19481:
19458:
19375:
19371:
19297:
19293:
19257:
19255:
19254:
19249:
19247:
19240:
19238:
19212:
19189:
19106:
19102:
19028:
19024:
18968:
18966:
18965:
18960:
18952:
18951:
18933:
18932:
18904:
18902:
18901:
18896:
18888:
18887:
18863:
18862:
18843:
18841:
18840:
18835:
18827:
18826:
18810:
18808:
18807:
18802:
18794:
18793:
18773:
18771:
18770:
18765:
18751:
18750:
18732:
18731:
18700:
18698:
18697:
18692:
18690:
18397:
18393:
18389:
18388:
18370:
18368:
18367:
18362:
18360:
18064:Basic identities
18060:
18058:
18057:
18052:
18022:
18021:
18020:
17971:
17970:
17969:
17942:
17940:
17939:
17934:
17929:
17914:
17912:
17911:
17906:
17894:
17892:
17891:
17886:
17869:
17822:
17820:
17819:
17814:
17784:
17783:
17782:
17733:
17732:
17731:
17704:
17702:
17701:
17696:
17691:
17676:
17674:
17673:
17668:
17656:
17654:
17653:
17648:
17633:
17631:
17630:
17625:
17620:
17596:
17594:
17593:
17588:
17570:
17568:
17567:
17562:
17545:
17497:
17495:
17494:
17489:
17474:
17472:
17471:
17466:
17461:
17453:
17449:
17442:
17437:
17429:
17424:
17416:
17411:
17403:
17395:
17390:
17382:
17363:
17352:
17344:
17335:
17333:
17332:
17327:
17322:
17314:
17310:
17250:
17235:
17233:
17232:
17227:
17222:
17198:
17196:
17195:
17190:
17167:. The function
17166:
17164:
17163:
17158:
17122:
17120:
17119:
17114:
17096:
17094:
17093:
17088:
17064:
17062:
17061:
17056:
17027:
16973:
16943:
16941:
16940:
16935:
16815:
16814:
16789:
16788:
16758:
16757:
16741:
16739:
16738:
16733:
16642:
16640:
16639:
16634:
16534:
16532:
16531:
16526:
16514:
16512:
16511:
16506:
16491:
16489:
16488:
16483:
16383:
16381:
16380:
16375:
16345:
16343:
16342:
16337:
16310:
16308:
16307:
16302:
16273:
16271:
16270:
16265:
16238:
16236:
16235:
16230:
16201:
16199:
16198:
16193:
16166:
16164:
16163:
16158:
16130:
16124:
16116:
16114:
16113:
16108:
16089:
16087:
16086:
16081:
16079:
15937:
15935:
15934:
15929:
15891:
15889:
15888:
15883:
15865:
15862:
15810:
15808:
15807:
15802:
15707:
15705:
15704:
15699:
15694:
15679:
15677:
15676:
15671:
15648:
15645:
15644:
15629:
15625:
15620:
15617:
15616:
15604:
15584:
15580:
15575:
15571:
15570:
15562:
15536:
15534:
15533:
15528:
15505:
15502:
15501:
15486:
15485:
15477:
15472:
15470:
15469:
15467:
15466:
15454:
15434:
15428:
15420:
15415:
15411:
15410:
15402:
15376:
15374:
15373:
15368:
15329:
15306:
15298:
15277:
15275:
15274:
15269:
15251:
15249:
15248:
15243:
15241:
15239:
15225:
15214:
15199:
15197:
15196:
15191:
15189:
15187:
15186:
15185:
15169:
15161:
15158:
15156:
15142:
15131:
15129:
15117:
15115:
15114:
15113:
15097:
15089:
15086:
15081:
15038:
15036:
15035:
15030:
15022:
15008:
14964:holds, provided
14963:
14961:
14960:
14955:
14950:
14948:
14934:
14923:
14878:
14876:
14875:
14870:
14855:
14853:
14852:
14847:
14796:
14794:
14793:
14788:
14737:
14735:
14734:
14729:
14715:
14683:
14659:
14657:
14656:
14651:
14633:
14631:
14630:
14625:
14607:
14605:
14604:
14599:
14591:
14590:
14586:
14570:
14569:
14529:
14527:
14526:
14521:
14513:
14512:
14508:
14492:
14491:
14454:
14452:
14451:
14446:
14428:
14426:
14425:
14420:
14415:
14394:
14392:
14391:
14386:
14378:
14364:
14340:
14338:
14337:
14332:
14308:
14306:
14305:
14300:
14277:where the point
14276:
14274:
14273:
14268:
14266:
14265:
14261:
14245:
14244:
14207:
14206:
14202:
14186:
14185:
14129:
14127:
14126:
14121:
14097:
14095:
14094:
14089:
14084:
14064:
14047:On the interval
14042:Karl Weierstrass
14039:
14037:
14036:
14031:
14029:
14027:
14026:
14025:
14009:
14001:
13998:
13993:
13978:
13970:
13961:
13959:
13958:
13953:
13941:
13939:
13938:
13933:
13919:
13917:
13916:
13915:
13899:
13891:
13888:
13883:
13852:
13850:
13849:
13844:
13824:
13822:
13821:
13816:
13801:
13799:
13798:
13793:
13771:
13769:
13768:
13763:
13755:
13741:
13739:
13728:
13719:
13717:
13716:
13711:
13687:
13685:
13684:
13679:
13661:
13659:
13658:
13653:
13632:
13630:
13629:
13624:
13612:
13610:
13609:
13604:
13596:
13575:
13573:
13572:
13567:
13559:
13551:
13546:
13534:
13532:
13531:
13526:
13518:
13492:), the function
13487:
13485:
13484:
13479:
13464:
13462:
13461:
13456:
13408:
13406:
13405:
13400:
13389:
13384:
13379:
13361:
13359:
13358:
13353:
13351:
13346:
13341:
13328:
13326:
13325:
13320:
13301:
13299:
13298:
13293:
13291:
13290:
13281:
13280:
13268:
13267:
13238:
13236:
13235:
13230:
13225:
13221:
13220:
13182:
13178:
13177:
13133:
13126:
13124:
13123:
13118:
13116:
13109:
13104:
13103:
13102:
13084:
13083:
13070:
13048:
13046:
13038:
13037:
13036:
13018:
13017:
13004:
12965:
12963:
12962:
12957:
12955:
12917:
12916:
12866:
12865:
12836:
12834:
12833:
12828:
12811:
12810:
12789:
12788:
12772:
12768:
12766:
12765:
12760:
12749:
12748:
12739:
12725:
12724:
12708:
12706:
12705:
12700:
12680:
12679:
12670:
12656:
12655:
12636:
12617:
12610:
12608:
12607:
12602:
12591:
12590:
12569:
12555:
12554:
12535:
12533:
12532:
12527:
12522:
12521:
12497:
12496:
12480:
12478:
12477:
12472:
12431:
12430:
12408:
12401:
12399:
12398:
12393:
12361:
12360:
12327:
12325:
12324:
12319:
12317:
12316:
12297:
12295:
12294:
12289:
12265:
12263:
12262:
12257:
12222:
12220:
12219:
12214:
12209:
12194:
12190:
12189:
12187:
12186:
12185:
12176:
12175:
12163:
12145:
12144:
12135:
12121:
12116:
12079:
12077:
12076:
12071:
12053:
12051:
12050:
12045:
12024:
12022:
12021:
12016:
12011:
11996:
11992:
11991:
11989:
11988:
11987:
11978:
11977:
11967:
11966:
11957:
11943:
11938:
11886:
11884:
11883:
11878:
11873:
11871:
11870:
11869:
11857:
11856:
11847:
11838:
11822:
11819:
11814:
11767:
11765:
11764:
11759:
11754:
11752:
11751:
11750:
11738:
11737:
11728:
11719:
11706:
11686:
11684:
11683:
11664:
11659:
11618:
11616:
11615:
11610:
11605:
11603:
11602:
11601:
11576:
11573:
11568:
11538:
11537:
11528:
11527:
11510:
11508:
11507:
11502:
11497:
11495:
11494:
11493:
11481:
11480:
11470:
11469:
11468:
11449:
11446:
11441:
11417:
11409:
11404:
11402:
11391:
11390:
11389:
11370:
11367:
11362:
11313:
11311:
11310:
11305:
11300:
11298:
11297:
11296:
11284:
11283:
11270:
11267:
11262:
11238:
11230:
11197:th term lead to
11196:
11190:
11175:
11173:
11172:
11167:
11162:
11160:
11146:
11143:
11138:
11119:
11049:
11047:
11046:
11041:
11039:
11037:
11035:
11034:
11032:
11030:
11029:
11027:
11025:
11024:
11022:
11020:
11013:
11011:
11009:
10999:
10997:
10987:
10979:
10977:
10967:
10962:
10960:
10958:
10948:
10946:
10936:
10928:
10926:
10916:
10911:
10909:
10907:
10897:
10895:
10885:
10877:
10875:
10865:
10860:
10858:
10856:
10846:
10844:
10834:
10826:
10824:
10814:
10809:
10807:
10805:
10804:
10802:
10800:
10799:
10797:
10795:
10794:
10792:
10790:
10774:
10772:
10771:
10770:
10755:
10741:
10739:
10738:
10737:
10722:
10708:
10706:
10705:
10704:
10689:
10675:
10673:
10663:
10639:
10637:
10636:
10631:
10629:
10627:
10625:
10624:
10622:
10620:
10619:
10617:
10615:
10614:
10612:
10610:
10603:
10602:
10575:
10573:
10572:
10571:
10547:
10542:
10541:
10514:
10512:
10511:
10510:
10486:
10481:
10480:
10453:
10451:
10450:
10449:
10434:
10420:
10418:
10408:
10384:
10382:
10381:
10376:
10374:
10372:
10370:
10369:
10367:
10365:
10364:
10362:
10360:
10359:
10357:
10355:
10348:
10347:
10320:
10318:
10317:
10316:
10292:
10287:
10286:
10259:
10257:
10256:
10255:
10231:
10226:
10225:
10198:
10196:
10195:
10194:
10179:
10165:
10163:
10153:
10117:
10115:
10114:
10109:
10107:
10094:
10086:
10075:
10072:
10060:
10059:
10050:
10042:
10037:
10036:
10027:
10019:
10011:
10003:
9998:
9997:
9979:
9975:
9974:
9956:
9954:
9937:
9936:
9935:
9923:
9922:
9910:
9909:
9890:
9887:
9882:
9837:
9835:
9834:
9829:
9827:
9820:
9812:
9807:
9799:
9794:
9791:
9779:
9778:
9769:
9761:
9756:
9755:
9746:
9738:
9733:
9732:
9723:
9715:
9701:
9697:
9696:
9684:
9682:
9665:
9664:
9663:
9651:
9650:
9631:
9628:
9623:
9605:
9604:
9592:
9590:
9573:
9572:
9560:
9557:
9552:
9507:
9505:
9504:
9499:
9497:
9484:
9476:
9465:
9462:
9450:
9449:
9440:
9432:
9427:
9426:
9417:
9409:
9401:
9393:
9388:
9387:
9369:
9365:
9364:
9346:
9344:
9327:
9326:
9325:
9313:
9309:
9302:
9301:
9275:
9274:
9249:
9246:
9241:
9196:
9194:
9193:
9188:
9186:
9179:
9171:
9166:
9158:
9153:
9150:
9138:
9137:
9128:
9120:
9115:
9114:
9105:
9097:
9092:
9091:
9082:
9074:
9063:
9058:
9054:
9053:
9035:
9033:
9016:
9015:
9014:
9002:
8998:
8991:
8990:
8973:
8972:
8960:
8959:
8934:
8931:
8926:
8908:
8903:
8899:
8898:
8880:
8878:
8855:
8854:
8836:
8833:
8828:
8810:
8773:
8769:
8759:Bernoulli number
8756:
8752:
8739:
8735:
8723:of finite sets.
8703:
8699:
8697:
8696:
8691:
8676:
8674:
8673:
8668:
8666:
8658:
8603:entire functions
8593:
8591:
8590:
8585:
8583:
8576:
8575:
8563:
8561:
8544:
8543:
8542:
8523:
8520:
8515:
8494:
8484:
8482:
8474:
8473:
8464:
8459:
8457:
8449:
8448:
8439:
8434:
8432:
8424:
8423:
8414:
8386:
8385:
8367:
8365:
8342:
8341:
8340:
8321:
8318:
8313:
8292:
8282:
8280:
8272:
8271:
8262:
8257:
8255:
8247:
8246:
8237:
8232:
8230:
8222:
8221:
8212:
8166:
8156:
8154:
8153:
8148:
8142:
8141:
8123:
8102:
8100:
8099:
8094:
8082:
8081:
8063:
8061:
8054:
8053:
8043:
8036:
8035:
8017:
8016:
8006:
7992:
7990:
7979:
7967:
7965:
7964:
7959:
7948:
7908:
7906:
7905:
7900:
7888:
7886:
7885:
7880:
7865:
7863:
7862:
7857:
7836:
7829:
7822:
7815:
7805:
7803:
7802:
7797:
7779:
7758:
7756:
7755:
7750:
7724:
7722:
7711:
7694:
7692:
7691:
7690:
7677:
7676:
7667:
7658:
7656:
7655:
7650:
7624:
7622:
7611:
7597:
7595:
7594:
7593:
7580:
7579:
7570:
7558:
7556:
7555:
7550:
7525:
7498:
7470:
7468:
7457:
7453:
7428:
7426:
7415:
7358:
7356:
7355:
7350:
7348:
7346:
7329:
7328:
7316:
7314:
7313:
7294:
7289:
7262:
7261:
7245:
7243:
7242:
7237:
7166:
7164:
7163:
7158:
7145:
7143:
7142:
7137:
7124:
7122:
7121:
7116:
7114:
7113:
7096:
7094:
7093:
7088:
7086:
7078:
7066:
7064:
7063:
7058:
7056:
7051:
7035:
7033:
7032:
7027:
7025:
7020:
7019:
7014:
7009:
7004:
7001:
6991:
6989:
6988:
6983:
6981:
6976:
6975:
6970:
6965:
6960:
6957:
6947:
6945:
6944:
6939:
6937:
6936:
6919:
6917:
6916:
6911:
6909:
6904:
6896:
6884:
6882:
6881:
6876:
6874:
6869:
6859:
6857:
6856:
6851:
6849:
6841:
6831:
6829:
6828:
6823:
6821:
6812:
6811:
6801:
6799:
6798:
6793:
6791:
6790:
6773:
6771:
6770:
6765:
6763:
6755:
6743:
6741:
6740:
6735:
6722:
6720:
6719:
6714:
6712:
6703:
6702:
6692:
6690:
6689:
6684:
6682:
6673:
6672:
6662:
6660:
6659:
6654:
6652:
6651:
6634:
6632:
6631:
6626:
6624:
6616:
6604:
6602:
6601:
6596:
6594:
6585:
6584:
6574:
6572:
6571:
6566:
6564:
6555:
6554:
6544:
6542:
6541:
6536:
6534:
6526:
6516:
6514:
6513:
6508:
6506:
6505:
6488:
6486:
6485:
6480:
6478:
6470:
6458:
6456:
6455:
6450:
6448:
6443:
6427:
6425:
6424:
6419:
6417:
6412:
6411:
6406:
6401:
6396:
6393:
6383:
6381:
6380:
6375:
6373:
6368:
6367:
6362:
6357:
6352:
6349:
6339:
6337:
6336:
6331:
6329:
6328:
6311:
6309:
6308:
6303:
6301:
6293:
6281:
6279:
6278:
6273:
6260:
6258:
6257:
6252:
6239:
6237:
6236:
6231:
6218:
6216:
6215:
6210:
6208:
6207:
6190:
6188:
6187:
6182:
6159:
6157:
6156:
6151:
6126:
6124:
6123:
6118:
6093:
6091:
6090:
6085:
6053:
6052:
6009:
5951:
5949:
5948:
5943:
5935:
5926:
5925:
5920:
5919:
5901:
5893:
5877:
5875:
5874:
5869:
5867:
5858:
5857:
5852:
5851:
5833:
5825:
5809:
5807:
5806:
5801:
5799:
5793:
5789:
5784:
5775:
5774:
5769:
5768:
5750:
5742:
5726:
5724:
5723:
5718:
5716:
5708:
5703:
5694:
5693:
5688:
5687:
5669:
5661:
5640:
5638:
5637:
5632:
5624:
5615:
5614:
5608:
5607:
5551:Algebraic values
5546:
5543:and any integer
5542:
5535:
5533:
5532:
5527:
5481:
5479:
5478:
5473:
5424:
5422:
5421:
5416:
5404:
5400:
5396:
5394:
5393:
5388:
5372:
5370:
5369:
5364:
5349:
5347:
5346:
5341:
5326:
5319:
5312:
5310:
5309:
5304:
5302:
5298:
5251:
5249:
5248:
5243:
5240:
5236:
5187:
5185:
5184:
5179:
5160:
5158:
5157:
5152:
5137:
5133:
5129:
5125:
5121:
5117:
5115:
5114:
5109:
5084:
5080:
5076:
5072:
5068:
5064:
5053:
5051:
5050:
5045:
5040:
5038:
5024:
5006:
5004:
4990:
4972:
4970:
4959:
4948:
4930:
4928:
4917:
4906:
4878:
4876:
4875:
4870:
4865:
4864:
4863:
4833:
4831:
4830:
4825:
4822:
4821:
4820:
4805:
4787:
4785:
4784:
4779:
4774:
4773:
4772:
4742:
4740:
4739:
4734:
4731:
4730:
4729:
4694:
4692:
4691:
4686:
4672:
4671:
4653:
4652:
4629:
4627:
4626:
4621:
4601:
4589:
4587:
4586:
4581:
4573:
4572:
4560:
4559:
4539:
4535:
4533:
4532:
4527:
4522:
4492:
4490:
4489:
4484:
4479:
4478:
4477:
4447:
4445:
4444:
4439:
4436:
4435:
4434:
4402:
4390:
4386:
4379:
4371:
4369:
4368:
4363:
4349:
4348:
4347:
4331:
4319:
4317:
4316:
4311:
4306:
4305:
4304:
4282:
4271:-axes at points
4270:
4264:
4258:
4256:
4255:
4250:
4245:
4244:
4227:
4219:
4217:
4216:
4211:
4197:
4196:
4195:
4179:
4167:
4165:
4164:
4159:
4141:
4139:
4138:
4133:
4125:
4124:
4123:
4101:
4089:
4087:
4086:
4081:
4059:
4057:
4056:
4051:
4046:
4045:
4032:
4030:
4029:
4024:
4016:
4015:
4014:
4001:
4000:
3999:
3983:
3971:
3969:
3968:
3963:
3945:
3943:
3942:
3937:
3915:counterclockwise
3912:
3906:
3898:
3896:
3895:
3890:
3888:
3887:
3871:
3865:
3863:
3862:
3857:
3855:
3847:
3838:
3830:
3767:
3760:
3753:
3746:
3742:
3738:
3734:
3727:
3720:
3713:
3709:
3705:
3693:
3667:
3649:
3645:
3637:
3635:
3629:
3613:
3606:
3595:
3588:
3582:, the notations
3577:
3539:
3535:
3531:
3527:
3523:
3519:
3503:
3499:
3450:
3448:
3447:
3442:
3440:
3438:
3424:
3419:
3415:
3408:
3407:
3367:
3365:
3364:
3359:
3357:
3355:
3341:
3336:
3332:
3325:
3317:
3284:
3283:
3281:
3280:
3277:
3274:
3260:
3258:
3257:
3252:
3250:
3248:
3234:
3229:
3225:
3218:
3217:
3177:
3175:
3174:
3169:
3167:
3165:
3151:
3146:
3142:
3135:
3127:
3094:
3093:
3091:
3090:
3087:
3084:
3070:
3068:
3067:
3062:
3060:
3058:
3044:
3039:
3035:
3028:
3027:
3004:
3002:
2991:
2980:
2958:
2956:
2955:
2950:
2948:
2946:
2932:
2927:
2923:
2916:
2908:
2892:
2890:
2879:
2868:
2846:
2845:
2843:
2842:
2839:
2836:
2822:
2820:
2819:
2814:
2812:
2810:
2796:
2791:
2787:
2780:
2779:
2756:
2754:
2743:
2732:
2710:
2708:
2707:
2702:
2700:
2698:
2684:
2679:
2675:
2668:
2660:
2644:
2642:
2631:
2620:
2598:
2597:
2595:
2594:
2591:
2588:
2574:
2572:
2571:
2566:
2563:
2561:
2547:
2542:
2538:
2531:
2530:
2490:
2488:
2487:
2482:
2479:
2477:
2463:
2458:
2454:
2447:
2439:
2406:
2405:
2403:
2402:
2399:
2396:
2382:
2380:
2379:
2374:
2372:
2370:
2356:
2351:
2347:
2340:
2339:
2299:
2297:
2296:
2291:
2289:
2287:
2273:
2268:
2264:
2257:
2249:
2216:
2215:
2213:
2212:
2209:
2206:
2167:
2163:
2154:
2149:
2143:
2138:
2133:
2128:
2123:
2117:
2100:
2098:
2097:
2092:
2081:
2080:
2055:
2053:
2052:
2047:
2023:
2019:
2017:
2016:
2013:
2010:
2002:
1985:
1983:
1982:
1977:
1975:
1973:
1947:
1921:
1892:
1890:
1889:
1884:
1882:
1880:
1854:
1828:
1797:
1795:
1794:
1789:
1787:
1785:
1759:
1727:
1698:
1696:
1695:
1690:
1688:
1686:
1654:
1628:
1597:
1595:
1594:
1589:
1587:
1585:
1559:
1527:
1498:
1496:
1495:
1490:
1488:
1486:
1454:
1428:
1401:
1400:
1397:
1389:
1377:
1373:
1365:
1361:
1353:
1349:
1341:
1337:
1329:
1325:
1321:
1313:
1305:
1297:
1289:
1285:
1270:
1269:
1267:
1266:
1261:
1258:
1243:
1242:
1240:
1239:
1234:
1231:
1216:
1215:
1213:
1212:
1207:
1204:
1177:
1175:
1174:
1169:
1167:
1144:
1142:
1141:
1136:
1106:
1104:
1103:
1098:
1071:
1069:
1068:
1063:
1055:
1054:
1029:
1027:
1026:
1021:
996:
994:
993:
988:
974:
973:
954:
952:
951:
946:
938:
937:
913:inverse function
910:
908:
907:
902:
900:
883:
881:
880:
875:
810:
809:
793:
791:
790:
785:
752:
750:
749:
744:
699:
697:
696:
691:
677:
676:
660:
658:
657:
652:
644:
643:
618:positive integer
613:
611:
610:
605:
572:
570:
569:
564:
531:
529:
528:
523:
499:
426:inverse function
391:Fourier analysis
303:
296:
289:
60:
46:
45:
21:
26733:
26732:
26728:
26727:
26726:
26724:
26723:
26722:
26678:
26677:
26676:
26671:
26640:
26619:Sine and cosine
26602:
26599:
26515:
26512:
26507:
26463:Wayback Machine
26324:Cajori, Florian
26298:
26238:
26212:
26207:
26206:
26197:
26195:
26184:
26180:
26166:
26162:
26157:
26153:
26148:
26138:
26136:
26128:
26124:
26117:
26098:
26096:
26043:
26039:
26026:
26022:
26013:
26011:
25994:
25990:
25981:
25979:
25976:
25945:
25941:
25934:
25912:
25908:
25899:
25897:
25882:
25878:
25870:
25866:
25859:
25841:
25837:
25813:
25809:
25800:
25798:
25789:
25788:
25784:
25741:
25737:
25728:
25726:
25715:
25711:
25702:
25701:
25694:
25687:
25671:, eds. (2000).
25661:
25657:
25648:
25646:
25629:
25622:
25609:
25600:
25589:
25570:
25566:
25555:
25536:
25532:
25527:
25523:
25516:
25502:
25498:
25480:
25476:
25467:Bartle (1964),
25465:
25461:
25447:
25443:
25438:
25434:
25429:
25425:
25420:Wayback Machine
25404:
25388:
25384:
25373:
25365:. p. 149.
25363:Springer-Verlag
25346:
25342:
25336:
25326:Springer-Verlag
25310:
25306:
25301:
25297:
25292:
25288:
25283:
25279:
25271:
25267:
25262:
25258:
25252:
25248:
25237:
25222:
25217:Wayback Machine
25201:
25185:
25181:
25172:
25170:
25155:
25151:
25144:
25128:
25124:
25105:
25101:
25058:
25054:
25034:
25033:
25021:
25005:
24998:
24990:
24986:
24977:
24975:
24967:
24966:
24962:
24954:
24950:
24941:
24939:
24932:
24905:
24901:
24883:
24879:
24874:
24873:
24848:
24846:
24843:
24842:
24840:
24836:
24831:
24826:
24781:
24688:transliteration
24637:
24631:
24602:
24599:
24595:
24594:
24593:
24591:
24585:
24572:
24562:
24559:
24554:
24553:
24551:
24540:
24537:
24534:
24533:
24531:
24525:
24511:
24508:
24504:
24503:
24502:
24500:
24490:
24480:
24477:
24472:
24471:
24469:
24459:
24449:
24446:
24441:
24440:
24438:
24432:
24371:Euler's formula
24340:
24329:
24262:infinite series
24248:Valentinus Otho
24184:Surya Siddhanta
24132:
24126:
24085:
24078:
24077:
24050:
24049:
24042:
24040:
24034:
24023:
24009:
23991:
23987:
23985:
23982:
23981:
23943:
23939:
23933:
23929:
23923:
23912:
23891:
23888:
23887:
23876:
23873:
23869:
23866:basis functions
23850:
23816:
23809:
23782:Lissajous curve
23774:
23745:
23729:
23716:
23709:
23707:
23691:
23678:
23671:
23669:
23653:
23640:
23633:
23631:
23629:
23626:
23625:
23592:
23590:
23577:
23569:
23566:
23565:
23490:
23488:
23480:
23477:
23476:
23470:Heron's formula
23438:
23432:
23402:
23391:
23389:
23367:
23365:
23358:
23341:
23339:
23332:
23330:
23328:
23325:
23324:
23318:
23316:Law of tangents
23312:
23310:Law of tangents
23295:
23291:
23262:
23255:
23251:
23242:
23238:
23229:
23225:
23224:
23222:
23208:
23205:
23204:
23163:
23159:
23150:
23146:
23137:
23133:
23131:
23128:
23127:
23120:
23114:
23091:
23055:
23050:
23034:
23029:
23013:
23008:
23006:
23003:
23002:
22998:
22971:
22963:
22961:
22942:
22940:
22921:
22919:
22900:
22898:
22896:
22893:
22892:
22888:
22884:
22880:
22876:
22872:
22868:
22865:
22859:
22850:
22846:
22842:
22838:
22834:
22830:
22827:
22822:
22816:
22796:
22792:
22788:
22784:
22745:
22726:
22721:
22718:
22717:
22691:
22677:
22674:
22673:
22644:
22641:
22640:
22611:
22608:
22607:
22583:
22556:
22553:
22552:
22526:
22512:
22509:
22508:
22479:
22476:
22475:
22446:
22443:
22442:
22411:
22408:
22407:
22375:
22372:
22371:
22342:
22339:
22338:
22309:
22306:
22305:
22281:
22262:
22257:
22254:
22253:
22221:
22218:
22217:
22188:
22185:
22184:
22155:
22152:
22151:
22120:
22117:
22116:
22084:
22081:
22080:
22051:
22048:
22047:
22018:
22015:
22014:
21990:
21971:
21966:
21963:
21962:
21930:
21927:
21926:
21897:
21894:
21893:
21864:
21861:
21860:
21814:
21808:
21789:
21788:
21772:
21768:
21745:
21730:
21726:
21703:
21681:
21676:
21669:
21659:
21645:
21643:
21640:
21639:
21594:
21565:
21530:
21508:
21503:
21496:
21486:
21472:
21470:
21467:
21466:
21430:
21395:
21373:
21368:
21361:
21351:
21337:
21335:
21331:
21329:
21326:
21325:
21299:
21296:
21295:
21258:
21255:
21254:
21235:
21215:
21207:
21204:
21203:
21181:
21178:
21177:
21137:
21134:
21133:
21111:
21108:
21107:
21079:
21076:
21075:
21038:
21034:
21026:
21023:
21022:
20998:
20994:
20989:
20986:
20985:
20962:
20959:
20958:
20911:
20907:
20899:
20896:
20895:
20863:
20860:
20859:
20836:
20833:
20832:
20785:
20781:
20773:
20770:
20769:
20734:
20731:
20730:
20707:
20704:
20703:
20668:
20664:
20656:
20653:
20652:
20628:
20624:
20622:
20619:
20618:
20595:
20592:
20591:
20560:
20557:
20556:
20530:
20527:
20526:
20503:
20500:
20499:
20465:
20462:
20461:
20438:
20435:
20434:
20411:
20408:
20407:
20369:
20366:
20365:
20336:
20334:
20331:
20330:
20304:
20301:
20300:
20287:
20284:antiderivatives
20272:
20264:antiderivatives
20222:
20218:
20211:
20206:
20195:
20192:
20191:
20187:Together with
20171:
20170:
20158:
20154:
20147:
20139:
20137:
20130:
20118:
20117:
20105:
20101:
20094:
20087:
20083:
20076:
20074:
20067:
20055:
20054:
20042:
20038:
20031:
20023:
20021:
20014:
20001:
19999:
19996:
19995:
19976:
19973:
19972:
19952:
19949:
19948:
19919:
19905:
19902:
19901:
19877:
19876:
19858:
19854:
19847:
19833:
19831:
19824:
19809:
19808:
19790:
19786:
19779:
19766:
19762:
19755:
19753:
19738:
19734:
19704:
19700:
19682:
19678:
19663:
19659:
19652:
19637:
19636:
19618:
19614:
19607:
19593:
19591:
19560:
19544:
19542:
19539:
19538:
19514:
19513:
19482:
19459:
19457:
19450:
19426:
19425:
19376:
19361:
19357:
19348:
19347:
19298:
19283:
19279:
19269:
19267:
19264:
19263:
19245:
19244:
19213:
19190:
19188:
19181:
19157:
19156:
19107:
19092:
19088:
19079:
19078:
19029:
19014:
19010:
19000:
18998:
18995:
18994:
18985:Euler's formula
18976:
18947:
18943:
18928:
18924:
18916:
18913:
18912:
18883:
18879:
18858:
18854:
18852:
18849:
18848:
18822:
18818:
18816:
18813:
18812:
18789:
18785:
18783:
18780:
18779:
18746:
18742:
18727:
18723:
18721:
18718:
18717:
18707:
18688:
18687:
18670:
18656:
18638:
18637:
18623:
18609:
18591:
18590:
18576:
18565:
18547:
18546:
18532:
18521:
18503:
18502:
18488:
18474:
18456:
18455:
18441:
18427:
18408:
18406:
18403:
18402:
18395:
18391:
18386:
18384:
18377:
18358:
18357:
18338:
18317:
18316:
18297:
18276:
18275:
18256:
18235:
18234:
18215:
18194:
18193:
18177:
18156:
18155:
18136:
18114:
18112:
18109:
18108:
18094:
18066:
18016:
18012:
18005:
17965:
17961:
17954:
17948:
17945:
17944:
17925:
17920:
17917:
17916:
17900:
17897:
17896:
17865:
17830:
17827:
17826:
17778:
17774:
17767:
17727:
17723:
17716:
17710:
17707:
17706:
17687:
17682:
17679:
17678:
17662:
17659:
17658:
17639:
17636:
17635:
17616:
17602:
17599:
17598:
17576:
17573:
17572:
17541:
17506:
17503:
17502:
17480:
17477:
17476:
17457:
17430:
17428:
17415:
17402:
17383:
17381:
17371:
17367:
17359:
17343:
17341:
17338:
17337:
17318:
17258:
17254:
17246:
17241:
17238:
17237:
17218:
17204:
17201:
17200:
17172:
17169:
17168:
17128:
17125:
17124:
17123:) in the strip
17102:
17099:
17098:
17070:
17067:
17066:
17023:
16969:
16949:
16946:
16945:
16810:
16806:
16784:
16780:
16753:
16749:
16747:
16744:
16743:
16648:
16645:
16644:
16540:
16537:
16536:
16520:
16517:
16516:
16497:
16494:
16493:
16389:
16386:
16385:
16366:
16363:
16362:
16355:
16324:
16321:
16320:
16289:
16286:
16285:
16252:
16249:
16248:
16217:
16214:
16213:
16180:
16177:
16176:
16145:
16142:
16141:
16102:
16099:
16098:
16095:domain coloring
16077:
16076:
16027:
16015:
16014:
15965:
15952:
15950:
15947:
15946:
15908:
15905:
15904:
15898:
15863: for
15861:
15822:
15819:
15818:
15735:
15732:
15731:
15714:
15690:
15685:
15682:
15681:
15640:
15636:
15624:
15612:
15608:
15600:
15579:
15561:
15554:
15550:
15542:
15539:
15538:
15497:
15493:
15478:
15476:
15462:
15458:
15450:
15433:
15429:
15421:
15419:
15401:
15394:
15390:
15382:
15379:
15378:
15325:
15297:
15283:
15280:
15279:
15257:
15254:
15253:
15226:
15215:
15213:
15205:
15202:
15201:
15181:
15177:
15170:
15162:
15160:
15143:
15132:
15130:
15125:
15109:
15105:
15098:
15090:
15088:
15082:
15074:
15044:
15041:
15040:
15018:
15004:
14969:
14966:
14965:
14935:
14924:
14922:
14887:
14884:
14883:
14861:
14858:
14857:
14802:
14799:
14798:
14743:
14740:
14739:
14711:
14679:
14665:
14662:
14661:
14639:
14636:
14635:
14613:
14610:
14609:
14582:
14575:
14571:
14565:
14561:
14535:
14532:
14531:
14504:
14497:
14493:
14487:
14483:
14460:
14457:
14456:
14434:
14431:
14430:
14411:
14400:
14397:
14396:
14374:
14360:
14349:
14346:
14345:
14314:
14311:
14310:
14282:
14279:
14278:
14257:
14250:
14246:
14240:
14236:
14198:
14191:
14187:
14181:
14177:
14135:
14132:
14131:
14103:
14100:
14099:
14080:
14060:
14052:
14049:
14048:
14021:
14017:
14010:
14002:
14000:
13994:
13989:
13969:
13967:
13964:
13963:
13947:
13944:
13943:
13911:
13907:
13900:
13892:
13890:
13884:
13879:
13858:
13855:
13854:
13838:
13835:
13834:
13831:
13807:
13804:
13803:
13781:
13778:
13777:
13751:
13732:
13727:
13725:
13722:
13721:
13696:
13693:
13692:
13667:
13664:
13663:
13638:
13635:
13634:
13618:
13615:
13614:
13592:
13581:
13578:
13577:
13555:
13547:
13542:
13540:
13537:
13536:
13514:
13497:
13494:
13493:
13473:
13470:
13469:
13414:
13411:
13410:
13385:
13380:
13375:
13367:
13364:
13363:
13347:
13342:
13337:
13335:
13332:
13331:
13314:
13311:
13310:
13286:
13282:
13276:
13272:
13257:
13253:
13251:
13248:
13247:
13213:
13209:
13205:
13170:
13166:
13162:
13142:
13139:
13138:
13131:
13114:
13113:
13092:
13088:
13076:
13072:
13071:
13069:
13062:
13050:
13049:
13039:
13026:
13022:
13010:
13006:
13005:
13003:
12996:
12983:
12981:
12978:
12977:
12953:
12952:
12918:
12906:
12902:
12899:
12898:
12867:
12858:
12854:
12850:
12848:
12845:
12844:
12806:
12802:
12784:
12780:
12778:
12775:
12774:
12770:
12744:
12740:
12735:
12720:
12716:
12714:
12711:
12710:
12675:
12671:
12666:
12651:
12647:
12632:
12627:
12624:
12623:
12612:
12586:
12582:
12565:
12550:
12546:
12541:
12538:
12537:
12514:
12510:
12492:
12488:
12486:
12483:
12482:
12426:
12422:
12420:
12417:
12416:
12406:
12353:
12349:
12347:
12344:
12343:
12333:Euler's formula
12309:
12305:
12303:
12300:
12299:
12271:
12268:
12267:
12239:
12236:
12235:
12229:
12205:
12181:
12177:
12171:
12167:
12159:
12146:
12140:
12136:
12134:
12127:
12123:
12117:
12106:
12088:
12085:
12084:
12059:
12056:
12055:
12033:
12030:
12029:
12007:
11983:
11979:
11973:
11969:
11968:
11962:
11958:
11956:
11949:
11945:
11939:
11928:
11907:
11904:
11903:
11893:
11865:
11861:
11852:
11848:
11836:
11826:
11821:
11815:
11804:
11774:
11771:
11770:
11746:
11742:
11733:
11729:
11717:
11707:
11687:
11685:
11679:
11675:
11660:
11649:
11625:
11622:
11621:
11597:
11593:
11580:
11575:
11569:
11555:
11533:
11529:
11523:
11519:
11517:
11514:
11513:
11489:
11485:
11476:
11472:
11471:
11464:
11460:
11450:
11448:
11442:
11431:
11408:
11392:
11385:
11381:
11371:
11369:
11363:
11349:
11325:
11322:
11321:
11292:
11288:
11279:
11275:
11274:
11269:
11263:
11252:
11229:
11209:
11206:
11205:
11192:
11184:
11150:
11145:
11139:
11125:
11109:
11085:
11082:
11081:
11063:
11005:
11000:
10993:
10988:
10986:
10985:
10980:
10973:
10968:
10966:
10954:
10949:
10942:
10937:
10935:
10934:
10929:
10922:
10917:
10915:
10903:
10898:
10891:
10886:
10884:
10883:
10878:
10871:
10866:
10864:
10852:
10847:
10840:
10835:
10833:
10832:
10827:
10820:
10815:
10813:
10780:
10775:
10766:
10762:
10761:
10756:
10754:
10747:
10742:
10733:
10729:
10728:
10723:
10721:
10714:
10709:
10700:
10696:
10695:
10690:
10688:
10681:
10676:
10669:
10664:
10662:
10648:
10645:
10644:
10598:
10594:
10581:
10576:
10567:
10563:
10553:
10548:
10546:
10537:
10533:
10520:
10515:
10506:
10502:
10492:
10487:
10485:
10476:
10472:
10459:
10454:
10445:
10441:
10440:
10435:
10433:
10426:
10421:
10414:
10409:
10407:
10393:
10390:
10389:
10343:
10339:
10326:
10321:
10312:
10308:
10298:
10293:
10291:
10282:
10278:
10265:
10260:
10251:
10247:
10237:
10232:
10230:
10221:
10217:
10204:
10199:
10190:
10186:
10185:
10180:
10178:
10171:
10166:
10159:
10154:
10152:
10138:
10135:
10134:
10124:
10105:
10104:
10090:
10082:
10071:
10055:
10051:
10041:
10032:
10028:
10018:
10002:
9990:
9986:
9977:
9976:
9961:
9957:
9938:
9928:
9924:
9915:
9911:
9905:
9901:
9891:
9889:
9883:
9872:
9861:
9848:
9846:
9843:
9842:
9825:
9824:
9811:
9803:
9795:
9790:
9774:
9770:
9760:
9751:
9747:
9737:
9728:
9724:
9714:
9699:
9698:
9689:
9685:
9666:
9656:
9652:
9646:
9642:
9632:
9630:
9624:
9613:
9597:
9593:
9574:
9565:
9561:
9559:
9553:
9542:
9531:
9518:
9516:
9513:
9512:
9495:
9494:
9480:
9472:
9461:
9445:
9441:
9431:
9422:
9418:
9408:
9392:
9380:
9376:
9367:
9366:
9351:
9347:
9328:
9318:
9314:
9288:
9284:
9283:
9279:
9264:
9260:
9250:
9248:
9242:
9231:
9220:
9207:
9205:
9202:
9201:
9184:
9183:
9170:
9162:
9154:
9149:
9133:
9129:
9119:
9110:
9106:
9096:
9087:
9083:
9073:
9062:
9056:
9055:
9040:
9036:
9017:
9007:
9003:
8983:
8979:
8978:
8974:
8965:
8961:
8949:
8945:
8935:
8933:
8927:
8916:
8907:
8901:
8900:
8885:
8881:
8856:
8841:
8837:
8835:
8829:
8818:
8809:
8805:
8792:
8790:
8787:
8786:
8771:
8768:
8764:
8754:
8751:
8747:
8737:
8734:
8730:
8701:
8682:
8679:
8678:
8657:
8637:
8634:
8633:
8581:
8580:
8568:
8564:
8545:
8538:
8534:
8524:
8522:
8516:
8505:
8492:
8491:
8475:
8469:
8465:
8463:
8450:
8444:
8440:
8438:
8425:
8419:
8415:
8413:
8400:
8388:
8387:
8372:
8368:
8343:
8336:
8332:
8322:
8320:
8314:
8303:
8290:
8289:
8273:
8267:
8263:
8261:
8248:
8242:
8238:
8236:
8223:
8217:
8213:
8211:
8198:
8185:
8183:
8180:
8179:
8173:
8161:
8137:
8133:
8116:
8114:
8111:
8110:
8077:
8073:
8049:
8045:
8044:
8031:
8027:
8012:
8008:
8007:
8005:
7983:
7978:
7976:
7973:
7972:
7944:
7921:
7918:
7917:
7916:to the tangent
7894:
7891:
7890:
7871:
7868:
7867:
7845:
7842:
7841:
7831:
7824:
7817:
7810:
7772:
7770:
7767:
7766:
7715:
7710:
7686:
7682:
7678:
7672:
7668:
7666:
7664:
7661:
7660:
7615:
7610:
7589:
7585:
7581:
7575:
7571:
7569:
7567:
7564:
7563:
7461:
7456:
7419:
7414:
7412:
7409:
7408:
7398:
7330:
7321:
7317:
7315:
7309:
7305:
7290:
7279:
7257:
7253:
7251:
7248:
7247:
7219:
7216:
7215:
7178:
7152:
7149:
7148:
7131:
7128:
7127:
7109:
7105:
7103:
7100:
7099:
7077:
7075:
7072:
7071:
7050:
7042:
7039:
7038:
7013:
7003:
7002:
7000:
6998:
6995:
6994:
6969:
6959:
6958:
6956:
6954:
6951:
6950:
6932:
6928:
6926:
6923:
6922:
6897:
6895:
6893:
6890:
6889:
6868:
6866:
6863:
6862:
6840:
6838:
6835:
6834:
6810:
6808:
6805:
6804:
6786:
6782:
6780:
6777:
6776:
6754:
6752:
6749:
6748:
6729:
6726:
6725:
6701:
6699:
6696:
6695:
6671:
6669:
6666:
6665:
6647:
6643:
6641:
6638:
6637:
6615:
6613:
6610:
6609:
6583:
6581:
6578:
6577:
6553:
6551:
6548:
6547:
6525:
6523:
6520:
6519:
6501:
6497:
6495:
6492:
6491:
6469:
6467:
6464:
6463:
6442:
6434:
6431:
6430:
6405:
6395:
6394:
6392:
6390:
6387:
6386:
6361:
6351:
6350:
6348:
6346:
6343:
6342:
6324:
6320:
6318:
6315:
6314:
6292:
6290:
6287:
6286:
6267:
6264:
6263:
6246:
6243:
6242:
6225:
6222:
6221:
6203:
6199:
6197:
6194:
6193:
6176:
6173:
6172:
6133:
6130:
6129:
6100:
6097:
6096:
6067:
6064:
6063:
6048:
6042:
6033:Baker's theorem
6007:
5999:rational number
5924:
5915:
5911:
5892:
5884:
5881:
5880:
5856:
5847:
5843:
5824:
5816:
5813:
5812:
5788:
5773:
5764:
5760:
5741:
5733:
5730:
5729:
5707:
5692:
5683:
5679:
5660:
5652:
5649:
5648:
5613:
5603:
5599:
5579:
5576:
5575:
5553:
5544:
5540:
5487:
5484:
5483:
5433:
5430:
5429:
5410:
5407:
5406:
5402:
5398:
5382:
5379:
5378:
5355:
5352:
5351:
5332:
5329:
5328:
5324:
5317:
5282:
5278:
5257:
5254:
5253:
5220:
5216:
5196:
5193:
5192:
5170:
5167:
5166:
5143:
5140:
5139:
5135:
5131:
5127:
5123:
5119:
5097:
5094:
5093:
5082:
5079:Secant (dotted)
5078:
5074:
5070:
5066:
5062:
5028:
5023:
4994:
4989:
4960:
4949:
4947:
4918:
4907:
4905:
4891:
4888:
4887:
4859:
4858:
4854:
4839:
4836:
4835:
4816:
4815:
4811:
4794:
4791:
4790:
4768:
4767:
4763:
4748:
4745:
4744:
4725:
4724:
4720:
4706:
4703:
4702:
4667:
4663:
4648:
4644:
4642:
4639:
4638:
4597:
4595:
4592:
4591:
4568:
4564:
4555:
4551:
4549:
4546:
4545:
4537:
4518:
4501:
4498:
4497:
4473:
4472:
4468:
4453:
4450:
4449:
4430:
4429:
4425:
4411:
4408:
4407:
4400:
4388:
4384:
4377:
4343:
4342:
4338:
4327:
4325:
4322:
4321:
4300:
4299:
4295:
4278:
4276:
4273:
4272:
4266:
4260:
4240:
4239:
4237:
4234:
4233:
4225:
4191:
4190:
4186:
4175:
4173:
4170:
4169:
4147:
4144:
4143:
4119:
4118:
4114:
4097:
4095:
4092:
4091:
4069:
4066:
4065:
4041:
4040:
4038:
4035:
4034:
4010:
4009:
4005:
3995:
3994:
3990:
3979:
3977:
3974:
3973:
3951:
3948:
3947:
3922:
3919:
3918:
3908:
3904:
3883:
3882:
3880:
3877:
3876:
3869:
3846:
3844:
3841:
3840:
3836:
3828:
3823:, which is the
3817:Euclidean plane
3768:, respectively.
3762:
3755:
3748:
3744:
3740:
3736:
3729:
3722:
3715:
3711:
3707:
3703:
3689:
3663:
3656:
3650:/180 ≈ 0.0175.
3647:
3643:
3633:
3631:
3627:
3608:
3601:
3590:
3583:
3575:
3537:
3533:
3529:
3525:
3521:
3517:
3501:
3497:
3494:complex numbers
3459:
3428:
3423:
3403:
3399:
3398:
3394:
3374:
3371:
3370:
3345:
3340:
3316:
3315:
3311:
3291:
3288:
3287:
3278:
3275:
3272:
3271:
3269:
3268:
3238:
3233:
3213:
3209:
3208:
3204:
3184:
3181:
3180:
3155:
3150:
3126:
3125:
3121:
3101:
3098:
3097:
3088:
3085:
3082:
3081:
3079:
3078:
3048:
3043:
3023:
3019:
3018:
3014:
2992:
2981:
2979:
2965:
2962:
2961:
2936:
2931:
2907:
2906:
2902:
2880:
2869:
2867:
2853:
2850:
2849:
2840:
2837:
2834:
2833:
2831:
2830:
2800:
2795:
2775:
2771:
2770:
2766:
2744:
2733:
2731:
2717:
2714:
2713:
2688:
2683:
2659:
2658:
2654:
2632:
2621:
2619:
2605:
2602:
2601:
2592:
2589:
2586:
2585:
2583:
2582:
2551:
2546:
2526:
2522:
2521:
2517:
2497:
2494:
2493:
2467:
2462:
2438:
2437:
2433:
2413:
2410:
2409:
2400:
2397:
2394:
2393:
2391:
2390:
2360:
2355:
2335:
2331:
2330:
2326:
2306:
2303:
2302:
2277:
2272:
2248:
2247:
2243:
2223:
2220:
2219:
2210:
2207:
2204:
2203:
2201:
2200:
2156:
2147:
2145:
2136:
2135:
2126:
2125:
2119:
2112:
2076:
2072:
2061:
2058:
2057:
2029:
2026:
2025:
2014:
2011:
2008:
2007:
2005:
2004:
2000:
1948:
1922:
1920:
1906:
1903:
1902:
1855:
1829:
1827:
1813:
1810:
1809:
1760:
1728:
1726:
1712:
1709:
1708:
1655:
1629:
1627:
1613:
1610:
1609:
1560:
1528:
1526:
1512:
1509:
1508:
1455:
1429:
1427:
1413:
1410:
1409:
1395:
1387:
1375:
1371:
1363:
1359:
1351:
1343:
1339:
1331:
1327:
1323:
1315:
1307:
1299:
1291:
1287:
1280:
1262:
1259:
1254:
1253:
1251:
1245:
1235:
1232:
1227:
1226:
1224:
1218:
1208:
1205:
1200:
1199:
1197:
1191:
1184:
1160:
1158:
1155:
1154:
1112:
1109:
1108:
1077:
1074:
1073:
1047:
1043:
1035:
1032:
1031:
1006:
1003:
1002:
966:
962:
960:
957:
956:
930:
926:
924:
921:
920:
893:
891:
888:
887:
805:
801:
799:
796:
795:
758:
755:
754:
705:
702:
701:
672:
668:
666:
663:
662:
639:
635:
633:
630:
629:
578:
575:
574:
537:
534:
533:
505:
502:
501:
493:
473:
457:infinite series
375:solid mechanics
351:angle functions
319:right triangles
307:
128:Exact constants
42:
35:
28:
23:
22:
15:
12:
11:
5:
26731:
26721:
26720:
26715:
26710:
26705:
26700:
26695:
26690:
26673:
26672:
26670:
26669:
26664:
26659:
26654:
26648:
26646:
26642:
26641:
26639:
26638:
26633:
26628:
26623:
26622:
26621:
26610:
26608:
26604:
26603:
26598:
26597:
26590:
26583:
26575:
26569:
26568:
26555:
26542:
26536:
26531:
26511:
26510:External links
26508:
26506:
26505:
26492:
26481:Addison-Wesley
26472:
26452:
26439:
26426:
26406:
26385:
26368:
26358:
26339:
26336:
26320:
26305:Boyer, Carl B.
26302:
26296:
26280:
26266:
26236:
26213:
26211:
26208:
26205:
26204:
26178:
26169:Gunter, Edmund
26160:
26151:
26122:
26116:978-0321387004
26115:
26084:
26083:
26077:
26064:
26037:
26020:
25988:
25974:
25939:
25932:
25906:
25887:, ed. (1897).
25876:
25864:
25857:
25835:
25807:
25782:
25755:(5): 547–563.
25735:
25709:
25704:"trigonometry"
25692:
25685:
25665:Selin, Helaine
25655:
25620:
25598:
25587:
25564:
25553:
25530:
25521:
25515:978-0387894911
25514:
25496:
25474:
25459:
25441:
25439:Ahlfors, p 197
25432:
25423:
25402:
25382:
25371:
25349:Aigner, Martin
25340:
25334:
25304:
25295:
25286:
25277:
25275:, p. 247.
25265:
25256:
25246:
25220:
25199:
25179:
25149:
25142:
25122:
25099:
25052:
25019:
24996:
24984:
24960:
24948:
24930:
24899:
24876:
24875:
24872:
24871:
24856:
24852:
24841:Also equal to
24833:
24832:
24830:
24827:
24825:
24824:
24818:
24813:
24808:
24803:
24798:
24793:
24788:
24782:
24780:
24777:
24690:from Sanskrit
24672:Medieval Latin
24633:Main article:
24630:
24627:
24619:
24618:
24583:
24570:
24523:
24488:
24457:
24345:right triangle
24266:Madhava series
24128:Main article:
24125:
24122:
24103:
24097:
24094:
24091:
24088:
24081:
24076:
24073:
24070:
24067:
24064:
24061:
24058:
24053:
24048:
24045:
24037:
24032:
24029:
24026:
24022:
24016:
24013:
24008:
24005:
24002:
23999:
23990:
23979:Fourier series
23960:
23957:
23954:
23951:
23946:
23942:
23936:
23932:
23926:
23921:
23918:
23915:
23911:
23907:
23904:
23901:
23898:
23895:
23871:
23862:Fourier series
23773:
23770:
23769:
23768:
23757:
23752:
23749:
23744:
23738:
23735:
23732:
23724:
23721:
23715:
23712:
23706:
23700:
23697:
23694:
23686:
23683:
23677:
23674:
23668:
23662:
23659:
23656:
23648:
23645:
23639:
23636:
23619:
23618:
23605:
23601:
23598:
23595:
23589:
23584:
23581:
23576:
23573:
23559:
23558:
23544:
23541:
23538:
23535:
23532:
23529:
23526:
23523:
23520:
23517:
23514:
23511:
23508:
23505:
23502:
23497:
23494:
23487:
23484:
23472:implies that:
23434:Main article:
23431:
23428:
23427:
23426:
23411:
23408:
23405:
23400:
23397:
23394:
23388:
23380:
23376:
23373:
23370:
23364:
23361:
23354:
23350:
23347:
23344:
23338:
23335:
23314:Main article:
23311:
23308:
23277:
23271:
23268:
23265:
23258:
23254:
23250:
23245:
23241:
23237:
23232:
23228:
23221:
23218:
23215:
23212:
23192:
23189:
23186:
23183:
23180:
23177:
23174:
23171:
23166:
23162:
23158:
23153:
23149:
23145:
23140:
23136:
23118:Law of cosines
23116:Main article:
23113:
23112:Law of cosines
23110:
23079:
23076:
23073:
23070:
23064:
23061:
23058:
23054:
23049:
23043:
23040:
23037:
23033:
23028:
23022:
23019:
23016:
23012:
22986:
22980:
22977:
22974:
22969:
22966:
22960:
22955:
22951:
22948:
22945:
22939:
22934:
22930:
22927:
22924:
22918:
22913:
22909:
22906:
22903:
22861:Main article:
22858:
22855:
22826:
22823:
22818:Main article:
22815:
22812:
22783:The notations
22779:
22778:
22767:
22764:
22761:
22757:
22752:
22749:
22744:
22741:
22738:
22733:
22730:
22725:
22715:
22704:
22701:
22698:
22693: or
22690:
22687:
22684:
22681:
22671:
22660:
22657:
22654:
22651:
22648:
22638:
22627:
22624:
22621:
22618:
22615:
22604:
22603:
22590:
22587:
22582:
22579:
22575:
22572:
22569:
22566:
22563:
22560:
22550:
22539:
22536:
22533:
22528: or
22525:
22522:
22519:
22516:
22506:
22495:
22492:
22489:
22486:
22483:
22473:
22462:
22459:
22456:
22453:
22450:
22439:
22438:
22427:
22424:
22421:
22418:
22415:
22405:
22394:
22391:
22388:
22385:
22382:
22379:
22369:
22358:
22355:
22352:
22349:
22346:
22336:
22325:
22322:
22319:
22316:
22313:
22302:
22301:
22288:
22285:
22280:
22277:
22274:
22269:
22266:
22261:
22251:
22240:
22237:
22234:
22231:
22228:
22225:
22215:
22204:
22201:
22198:
22195:
22192:
22182:
22171:
22168:
22165:
22162:
22159:
22148:
22147:
22136:
22133:
22130:
22127:
22124:
22114:
22103:
22100:
22097:
22094:
22091:
22088:
22078:
22067:
22064:
22061:
22058:
22055:
22045:
22034:
22031:
22028:
22025:
22022:
22011:
22010:
21997:
21994:
21989:
21986:
21983:
21978:
21975:
21970:
21960:
21949:
21946:
21943:
21940:
21937:
21934:
21924:
21913:
21910:
21907:
21904:
21901:
21891:
21880:
21877:
21874:
21871:
21868:
21857:
21856:
21853:
21850:
21847:
21810:Main article:
21807:
21804:
21803:
21802:
21787:
21783:
21780:
21775:
21771:
21767:
21764:
21761:
21758:
21755:
21752:
21748:
21744:
21741:
21738:
21733:
21729:
21725:
21722:
21719:
21716:
21713:
21710:
21706:
21702:
21699:
21696:
21693:
21687:
21684:
21680:
21675:
21672:
21670:
21665:
21662:
21657:
21654:
21651:
21648:
21642:
21641:
21638:
21634:
21631:
21628:
21625:
21622:
21619:
21616:
21613:
21610:
21607:
21604:
21601:
21597:
21593:
21590:
21587:
21584:
21581:
21578:
21575:
21572:
21568:
21564:
21561:
21558:
21555:
21552:
21549:
21546:
21543:
21540:
21537:
21533:
21529:
21526:
21523:
21520:
21514:
21511:
21507:
21502:
21499:
21497:
21492:
21489:
21484:
21481:
21478:
21475:
21469:
21468:
21465:
21461:
21458:
21455:
21452:
21449:
21446:
21443:
21440:
21437:
21433:
21429:
21426:
21423:
21420:
21417:
21414:
21411:
21408:
21405:
21402:
21398:
21394:
21391:
21388:
21385:
21379:
21376:
21372:
21367:
21364:
21362:
21357:
21354:
21349:
21346:
21343:
21340:
21334:
21333:
21303:
21283:
21280:
21277:
21274:
21271:
21268:
21265:
21262:
21242:
21238:
21234:
21231:
21228:
21225:
21222:
21218:
21214:
21211:
21191:
21188:
21185:
21165:
21162:
21159:
21156:
21153:
21150:
21147:
21144:
21141:
21121:
21118:
21115:
21095:
21092:
21089:
21086:
21083:
21070:
21069:
21058:
21055:
21051:
21047:
21044:
21041:
21037:
21033:
21030:
21020:
21009:
21006:
21001:
20997:
20993:
20983:
20972:
20969:
20966:
20955:
20954:
20943:
20940:
20936:
20932:
20929:
20926:
20923:
20920:
20917:
20914:
20910:
20906:
20903:
20893:
20882:
20879:
20876:
20873:
20870:
20867:
20857:
20846:
20843:
20840:
20829:
20828:
20817:
20814:
20810:
20806:
20803:
20800:
20797:
20794:
20791:
20788:
20784:
20780:
20777:
20767:
20756:
20753:
20750:
20747:
20744:
20741:
20738:
20728:
20717:
20714:
20711:
20700:
20699:
20688:
20685:
20681:
20677:
20674:
20671:
20667:
20663:
20660:
20650:
20639:
20636:
20631:
20627:
20616:
20605:
20602:
20599:
20588:
20587:
20576:
20573:
20570:
20567:
20564:
20554:
20543:
20540:
20537:
20534:
20524:
20513:
20510:
20507:
20496:
20495:
20484:
20481:
20478:
20475:
20472:
20469:
20459:
20448:
20445:
20442:
20432:
20421:
20418:
20415:
20404:
20403:
20392:
20389:
20385:
20382:
20379:
20376:
20373:
20363:
20352:
20349:
20346:
20342:
20339:
20328:
20317:
20314:
20311:
20308:
20271:
20268:
20252:
20251:
20240:
20237:
20234:
20225:
20221:
20217:
20214:
20210:
20205:
20202:
20199:
20185:
20184:
20169:
20161:
20157:
20153:
20150:
20145:
20142:
20136:
20133:
20131:
20129:
20126:
20123:
20120:
20119:
20116:
20108:
20104:
20100:
20097:
20090:
20086:
20082:
20079:
20073:
20070:
20068:
20066:
20063:
20060:
20057:
20056:
20053:
20045:
20041:
20037:
20034:
20029:
20026:
20020:
20017:
20015:
20013:
20010:
20007:
20004:
20003:
19980:
19956:
19936:
19933:
19927:
19924:
19918:
19915:
19912:
19909:
19891:
19890:
19875:
19869:
19866:
19861:
19857:
19853:
19850:
19845:
19842:
19839:
19836:
19830:
19827:
19825:
19823:
19820:
19817:
19814:
19811:
19810:
19807:
19801:
19798:
19793:
19789:
19785:
19782:
19777:
19774:
19769:
19765:
19761:
19758:
19752:
19749:
19746:
19741:
19737:
19733:
19730:
19727:
19724:
19721:
19718:
19715:
19712:
19707:
19703:
19699:
19696:
19693:
19690:
19685:
19681:
19677:
19674:
19671:
19666:
19662:
19658:
19655:
19653:
19651:
19648:
19645:
19642:
19639:
19638:
19635:
19629:
19626:
19621:
19617:
19613:
19610:
19605:
19602:
19599:
19596:
19590:
19587:
19584:
19581:
19578:
19575:
19572:
19569:
19566:
19563:
19561:
19559:
19556:
19553:
19550:
19547:
19546:
19528:
19527:
19512:
19506:
19503:
19500:
19497:
19494:
19491:
19488:
19485:
19480:
19477:
19474:
19471:
19468:
19465:
19462:
19456:
19453:
19451:
19449:
19446:
19443:
19440:
19437:
19434:
19431:
19428:
19427:
19424:
19421:
19418:
19415:
19412:
19409:
19406:
19403:
19400:
19397:
19394:
19391:
19388:
19385:
19382:
19379:
19377:
19374:
19370:
19367:
19364:
19360:
19356:
19353:
19350:
19349:
19346:
19343:
19340:
19337:
19334:
19331:
19328:
19325:
19322:
19319:
19316:
19313:
19310:
19307:
19304:
19301:
19299:
19296:
19292:
19289:
19286:
19282:
19278:
19275:
19272:
19271:
19261:
19258:
19243:
19237:
19234:
19231:
19228:
19225:
19222:
19219:
19216:
19211:
19208:
19205:
19202:
19199:
19196:
19193:
19187:
19184:
19182:
19180:
19177:
19174:
19171:
19168:
19165:
19162:
19159:
19158:
19155:
19152:
19149:
19146:
19143:
19140:
19137:
19134:
19131:
19128:
19125:
19122:
19119:
19116:
19113:
19110:
19108:
19105:
19101:
19098:
19095:
19091:
19087:
19084:
19081:
19080:
19077:
19074:
19071:
19068:
19065:
19062:
19059:
19056:
19053:
19050:
19047:
19044:
19041:
19038:
19035:
19032:
19030:
19027:
19023:
19020:
19017:
19013:
19009:
19006:
19003:
19002:
18992:
18975:
18972:
18971:
18970:
18958:
18955:
18950:
18946:
18942:
18939:
18936:
18931:
18927:
18923:
18920:
18906:
18905:
18894:
18891:
18886:
18882:
18878:
18875:
18872:
18869:
18866:
18861:
18857:
18833:
18830:
18825:
18821:
18800:
18797:
18792:
18788:
18776:
18775:
18763:
18760:
18757:
18754:
18749:
18745:
18741:
18738:
18735:
18730:
18726:
18706:
18703:
18702:
18701:
18686:
18683:
18680:
18677:
18674:
18671:
18669:
18666:
18663:
18660:
18657:
18655:
18652:
18649:
18646:
18643:
18640:
18639:
18636:
18633:
18630:
18627:
18624:
18622:
18619:
18616:
18613:
18610:
18608:
18605:
18602:
18599:
18596:
18593:
18592:
18589:
18586:
18583:
18580:
18577:
18575:
18572:
18569:
18566:
18564:
18561:
18558:
18555:
18552:
18549:
18548:
18545:
18542:
18539:
18536:
18533:
18531:
18528:
18525:
18522:
18520:
18517:
18514:
18511:
18508:
18505:
18504:
18501:
18498:
18495:
18492:
18489:
18487:
18484:
18481:
18478:
18475:
18473:
18470:
18467:
18464:
18461:
18458:
18457:
18454:
18451:
18448:
18445:
18442:
18440:
18437:
18434:
18431:
18428:
18426:
18423:
18420:
18417:
18414:
18411:
18410:
18376:
18373:
18372:
18371:
18356:
18353:
18350:
18347:
18344:
18341:
18339:
18337:
18334:
18331:
18328:
18325:
18322:
18319:
18318:
18315:
18312:
18309:
18306:
18303:
18300:
18298:
18296:
18293:
18290:
18287:
18284:
18281:
18278:
18277:
18274:
18271:
18268:
18265:
18262:
18259:
18257:
18255:
18252:
18249:
18246:
18243:
18240:
18237:
18236:
18233:
18230:
18227:
18224:
18221:
18218:
18216:
18214:
18211:
18208:
18205:
18202:
18199:
18196:
18195:
18192:
18189:
18186:
18183:
18180:
18178:
18176:
18173:
18170:
18167:
18164:
18161:
18158:
18157:
18154:
18151:
18148:
18145:
18142:
18139:
18137:
18135:
18132:
18129:
18126:
18123:
18120:
18117:
18116:
18098:even functions
18093:
18090:
18065:
18062:
18050:
18047:
18044:
18041:
18038:
18035:
18032:
18029:
18026:
18019:
18015:
18011:
18008:
18004:
17999:
17996:
17993:
17990:
17987:
17984:
17981:
17978:
17975:
17968:
17964:
17960:
17957:
17953:
17932:
17928:
17924:
17904:
17884:
17881:
17878:
17875:
17872:
17868:
17864:
17861:
17858:
17855:
17852:
17849:
17846:
17843:
17840:
17837:
17834:
17812:
17809:
17806:
17803:
17800:
17797:
17794:
17791:
17788:
17781:
17777:
17773:
17770:
17766:
17761:
17758:
17755:
17752:
17749:
17746:
17743:
17740:
17737:
17730:
17726:
17722:
17719:
17715:
17694:
17690:
17686:
17666:
17646:
17643:
17623:
17619:
17615:
17612:
17609:
17606:
17586:
17583:
17580:
17560:
17557:
17554:
17551:
17548:
17544:
17540:
17537:
17534:
17531:
17528:
17525:
17522:
17519:
17516:
17513:
17510:
17487:
17484:
17464:
17460:
17456:
17452:
17448:
17445:
17440:
17436:
17433:
17427:
17422:
17419:
17414:
17409:
17406:
17401:
17398:
17393:
17389:
17386:
17380:
17377:
17374:
17370:
17366:
17362:
17358:
17355:
17350:
17347:
17325:
17321:
17317:
17313:
17309:
17306:
17303:
17300:
17297:
17294:
17291:
17288:
17285:
17282:
17279:
17276:
17273:
17270:
17267:
17264:
17261:
17257:
17253:
17249:
17245:
17225:
17221:
17217:
17214:
17211:
17208:
17188:
17185:
17182:
17179:
17176:
17156:
17153:
17150:
17147:
17144:
17141:
17138:
17135:
17132:
17112:
17109:
17106:
17086:
17083:
17080:
17077:
17074:
17054:
17051:
17048:
17045:
17042:
17039:
17036:
17033:
17030:
17026:
17022:
17019:
17016:
17013:
17010:
17007:
17003:
17000:
16997:
16994:
16991:
16988:
16985:
16982:
16979:
16976:
16972:
16968:
16965:
16962:
16959:
16956:
16953:
16933:
16930:
16927:
16924:
16921:
16918:
16915:
16912:
16909:
16906:
16903:
16900:
16897:
16894:
16891:
16888:
16885:
16882:
16879:
16876:
16873:
16870:
16867:
16864:
16861:
16858:
16854:
16851:
16848:
16845:
16842:
16839:
16836:
16833:
16830:
16827:
16824:
16821:
16818:
16813:
16809:
16804:
16801:
16798:
16795:
16792:
16787:
16783:
16779:
16776:
16773:
16770:
16767:
16764:
16761:
16756:
16752:
16731:
16728:
16725:
16722:
16719:
16716:
16713:
16710:
16707:
16704:
16701:
16698:
16695:
16691:
16688:
16685:
16682:
16679:
16676:
16673:
16670:
16667:
16664:
16661:
16658:
16655:
16652:
16632:
16629:
16626:
16623:
16620:
16617:
16614:
16611:
16608:
16605:
16602:
16599:
16596:
16593:
16590:
16586:
16583:
16580:
16577:
16574:
16571:
16568:
16565:
16562:
16559:
16556:
16553:
16550:
16547:
16544:
16524:
16504:
16501:
16481:
16478:
16475:
16472:
16469:
16466:
16463:
16460:
16457:
16454:
16451:
16448:
16445:
16442:
16439:
16435:
16432:
16429:
16426:
16423:
16420:
16417:
16414:
16411:
16408:
16405:
16402:
16399:
16396:
16393:
16373:
16370:
16361:, with period
16354:
16351:
16348:
16347:
16334:
16331:
16328:
16299:
16296:
16293:
16275:
16262:
16259:
16256:
16227:
16224:
16221:
16203:
16190:
16187:
16184:
16155:
16152:
16149:
16119:zeros or poles
16106:
16091:
16090:
16075:
16072:
16069:
16066:
16063:
16060:
16057:
16054:
16051:
16048:
16045:
16042:
16039:
16036:
16033:
16030:
16028:
16026:
16023:
16020:
16017:
16016:
16013:
16010:
16007:
16004:
16001:
15998:
15995:
15992:
15989:
15986:
15983:
15980:
15977:
15974:
15971:
15968:
15966:
15964:
15961:
15958:
15955:
15954:
15927:
15924:
15921:
15918:
15915:
15912:
15902:complex number
15897:
15894:
15893:
15892:
15881:
15878:
15875:
15872:
15869:
15859:
15856:
15853:
15850:
15847:
15844:
15841:
15838:
15835:
15832:
15829:
15826:
15812:
15811:
15799:
15796:
15793:
15790:
15787:
15784:
15781:
15778:
15775:
15772:
15769:
15766:
15763:
15760:
15757:
15754:
15751:
15748:
15745:
15742:
15739:
15713:
15710:
15697:
15693:
15689:
15669:
15666:
15663:
15660:
15657:
15654:
15651:
15643:
15639:
15635:
15632:
15628:
15623:
15615:
15611:
15607:
15603:
15599:
15596:
15593:
15590:
15587:
15583:
15578:
15574:
15568:
15565:
15560:
15557:
15553:
15549:
15546:
15526:
15523:
15520:
15517:
15514:
15511:
15508:
15500:
15496:
15492:
15489:
15484:
15481:
15475:
15465:
15461:
15457:
15453:
15449:
15446:
15443:
15440:
15437:
15432:
15427:
15424:
15418:
15414:
15408:
15405:
15400:
15397:
15393:
15389:
15386:
15366:
15363:
15360:
15357:
15354:
15351:
15348:
15345:
15342:
15338:
15335:
15332:
15328:
15324:
15321:
15318:
15315:
15312:
15309:
15304:
15301:
15296:
15293:
15290:
15287:
15267:
15264:
15261:
15238:
15235:
15232:
15229:
15224:
15221:
15218:
15212:
15209:
15184:
15180:
15176:
15173:
15168:
15165:
15155:
15152:
15149:
15146:
15141:
15138:
15135:
15128:
15124:
15120:
15112:
15108:
15104:
15101:
15096:
15093:
15085:
15080:
15077:
15073:
15069:
15066:
15063:
15060:
15057:
15054:
15051:
15048:
15028:
15025:
15021:
15017:
15014:
15011:
15007:
15003:
15000:
14997:
14994:
14991:
14988:
14985:
14982:
14979:
14976:
14973:
14953:
14947:
14944:
14941:
14938:
14933:
14930:
14927:
14921:
14918:
14915:
14912:
14909:
14906:
14903:
14900:
14897:
14894:
14891:
14868:
14865:
14845:
14842:
14839:
14836:
14833:
14830:
14827:
14824:
14821:
14818:
14815:
14812:
14809:
14806:
14786:
14783:
14780:
14777:
14774:
14771:
14768:
14765:
14762:
14759:
14756:
14753:
14750:
14747:
14727:
14724:
14721:
14718:
14714:
14710:
14707:
14704:
14701:
14698:
14695:
14692:
14689:
14686:
14682:
14678:
14675:
14672:
14669:
14649:
14646:
14643:
14623:
14620:
14617:
14597:
14594:
14589:
14585:
14581:
14578:
14574:
14568:
14564:
14560:
14557:
14554:
14551:
14548:
14545:
14542:
14539:
14519:
14516:
14511:
14507:
14503:
14500:
14496:
14490:
14486:
14482:
14479:
14476:
14473:
14470:
14467:
14464:
14444:
14441:
14438:
14418:
14414:
14410:
14407:
14404:
14384:
14381:
14377:
14373:
14370:
14367:
14363:
14359:
14356:
14353:
14330:
14327:
14324:
14321:
14318:
14298:
14295:
14292:
14289:
14286:
14264:
14260:
14256:
14253:
14249:
14243:
14239:
14235:
14232:
14229:
14226:
14223:
14220:
14217:
14214:
14210:
14205:
14201:
14197:
14194:
14190:
14184:
14180:
14176:
14173:
14170:
14167:
14164:
14161:
14158:
14154:
14151:
14148:
14145:
14142:
14139:
14119:
14116:
14113:
14110:
14107:
14087:
14083:
14079:
14076:
14073:
14070:
14067:
14063:
14059:
14056:
14024:
14020:
14016:
14013:
14008:
14005:
13997:
13992:
13988:
13984:
13981:
13976:
13973:
13962:is defined by
13951:
13931:
13928:
13925:
13922:
13914:
13910:
13906:
13903:
13898:
13895:
13887:
13882:
13878:
13874:
13871:
13868:
13865:
13862:
13842:
13830:
13827:
13814:
13811:
13791:
13788:
13785:
13761:
13758:
13754:
13750:
13747:
13744:
13738:
13735:
13731:
13709:
13706:
13703:
13700:
13677:
13674:
13671:
13651:
13648:
13645:
13642:
13622:
13602:
13599:
13595:
13591:
13588:
13585:
13565:
13562:
13558:
13554:
13550:
13545:
13524:
13521:
13517:
13513:
13510:
13507:
13504:
13501:
13477:
13454:
13451:
13448:
13445:
13442:
13439:
13436:
13433:
13430:
13427:
13424:
13421:
13418:
13398:
13395:
13392:
13388:
13383:
13378:
13374:
13371:
13350:
13345:
13340:
13318:
13289:
13285:
13279:
13275:
13271:
13266:
13263:
13260:
13256:
13240:
13239:
13228:
13224:
13219:
13216:
13212:
13208:
13204:
13201:
13198:
13195:
13192:
13189:
13185:
13181:
13176:
13173:
13169:
13165:
13161:
13158:
13155:
13152:
13149:
13146:
13128:
13127:
13112:
13107:
13101:
13098:
13095:
13091:
13087:
13082:
13079:
13075:
13068:
13065:
13063:
13061:
13058:
13055:
13052:
13051:
13045:
13042:
13035:
13032:
13029:
13025:
13021:
13016:
13013:
13009:
13002:
12999:
12997:
12995:
12992:
12989:
12986:
12985:
12967:
12966:
12951:
12948:
12945:
12942:
12939:
12936:
12933:
12930:
12927:
12924:
12921:
12919:
12915:
12912:
12909:
12905:
12901:
12900:
12897:
12894:
12891:
12888:
12885:
12882:
12879:
12876:
12873:
12870:
12868:
12864:
12861:
12857:
12853:
12852:
12826:
12823:
12820:
12817:
12814:
12809:
12805:
12801:
12798:
12795:
12792:
12787:
12783:
12758:
12755:
12752:
12747:
12743:
12738:
12734:
12731:
12728:
12723:
12719:
12698:
12695:
12692:
12689:
12686:
12683:
12678:
12674:
12669:
12665:
12662:
12659:
12654:
12650:
12646:
12642:
12639:
12635:
12631:
12600:
12597:
12594:
12589:
12585:
12581:
12578:
12575:
12572:
12568:
12564:
12561:
12558:
12553:
12549:
12545:
12525:
12520:
12517:
12513:
12509:
12506:
12503:
12500:
12495:
12491:
12470:
12467:
12464:
12461:
12458:
12455:
12452:
12449:
12446:
12443:
12440:
12437:
12434:
12429:
12425:
12403:
12402:
12391:
12388:
12385:
12382:
12379:
12376:
12373:
12370:
12367:
12364:
12359:
12356:
12352:
12315:
12312:
12308:
12287:
12284:
12281:
12278:
12275:
12255:
12252:
12249:
12246:
12243:
12228:
12225:
12224:
12223:
12212:
12208:
12204:
12201:
12197:
12193:
12184:
12180:
12174:
12170:
12166:
12162:
12158:
12155:
12152:
12149:
12143:
12139:
12133:
12130:
12126:
12120:
12115:
12112:
12109:
12105:
12101:
12098:
12095:
12092:
12069:
12066:
12063:
12043:
12040:
12037:
12026:
12025:
12014:
12010:
12006:
12003:
11999:
11995:
11986:
11982:
11976:
11972:
11965:
11961:
11955:
11952:
11948:
11942:
11937:
11934:
11931:
11927:
11923:
11920:
11917:
11914:
11911:
11897:Leonhard Euler
11892:
11889:
11888:
11887:
11876:
11868:
11864:
11860:
11855:
11851:
11844:
11841:
11835:
11832:
11829:
11825:
11818:
11813:
11810:
11807:
11803:
11799:
11796:
11793:
11790:
11787:
11784:
11781:
11778:
11768:
11757:
11749:
11745:
11741:
11736:
11732:
11725:
11722:
11716:
11713:
11710:
11705:
11702:
11699:
11696:
11693:
11690:
11682:
11678:
11674:
11671:
11668:
11663:
11658:
11655:
11652:
11648:
11644:
11641:
11638:
11635:
11632:
11629:
11619:
11608:
11600:
11596:
11592:
11589:
11586:
11583:
11579:
11572:
11567:
11564:
11561:
11558:
11554:
11550:
11547:
11544:
11541:
11536:
11532:
11526:
11522:
11511:
11500:
11492:
11488:
11484:
11479:
11475:
11467:
11463:
11459:
11456:
11453:
11445:
11440:
11437:
11434:
11430:
11426:
11423:
11420:
11415:
11412:
11407:
11401:
11398:
11395:
11388:
11384:
11380:
11377:
11374:
11366:
11361:
11358:
11355:
11352:
11348:
11344:
11341:
11338:
11335:
11332:
11329:
11315:
11314:
11303:
11295:
11291:
11287:
11282:
11278:
11273:
11266:
11261:
11258:
11255:
11251:
11247:
11244:
11241:
11236:
11233:
11228:
11225:
11222:
11219:
11216:
11213:
11177:
11176:
11165:
11159:
11156:
11153:
11149:
11142:
11137:
11134:
11131:
11128:
11124:
11118:
11115:
11112:
11108:
11104:
11101:
11098:
11095:
11092:
11089:
11062:
11059:
11051:
11050:
11019:
11016:
11008:
10996:
10976:
10965:
10957:
10945:
10925:
10914:
10906:
10894:
10874:
10863:
10855:
10843:
10823:
10812:
10789:
10786:
10783:
10769:
10765:
10753:
10750:
10736:
10732:
10720:
10717:
10703:
10699:
10687:
10684:
10672:
10661:
10658:
10655:
10652:
10641:
10640:
10609:
10606:
10601:
10597:
10593:
10590:
10587:
10584:
10570:
10566:
10562:
10559:
10556:
10545:
10540:
10536:
10532:
10529:
10526:
10523:
10509:
10505:
10501:
10498:
10495:
10484:
10479:
10475:
10471:
10468:
10465:
10462:
10448:
10444:
10432:
10429:
10417:
10406:
10403:
10400:
10397:
10386:
10385:
10354:
10351:
10346:
10342:
10338:
10335:
10332:
10329:
10315:
10311:
10307:
10304:
10301:
10290:
10285:
10281:
10277:
10274:
10271:
10268:
10254:
10250:
10246:
10243:
10240:
10229:
10224:
10220:
10216:
10213:
10210:
10207:
10193:
10189:
10177:
10174:
10162:
10151:
10148:
10145:
10142:
10126:The following
10123:
10120:
10119:
10118:
10103:
10100:
10097:
10093:
10089:
10085:
10081:
10078:
10069:
10066:
10063:
10058:
10054:
10048:
10045:
10040:
10035:
10031:
10025:
10022:
10017:
10014:
10009:
10006:
10001:
9996:
9993:
9989:
9985:
9982:
9980:
9978:
9973:
9970:
9967:
9964:
9960:
9953:
9950:
9947:
9944:
9941:
9934:
9931:
9927:
9921:
9918:
9914:
9908:
9904:
9900:
9897:
9894:
9886:
9881:
9878:
9875:
9871:
9867:
9864:
9862:
9860:
9857:
9854:
9851:
9850:
9839:
9838:
9823:
9818:
9815:
9810:
9806:
9802:
9798:
9788:
9785:
9782:
9777:
9773:
9767:
9764:
9759:
9754:
9750:
9744:
9741:
9736:
9731:
9727:
9721:
9718:
9713:
9710:
9707:
9704:
9702:
9700:
9695:
9692:
9688:
9681:
9678:
9675:
9672:
9669:
9662:
9659:
9655:
9649:
9645:
9641:
9638:
9635:
9627:
9622:
9619:
9616:
9612:
9608:
9603:
9600:
9596:
9589:
9586:
9583:
9580:
9577:
9571:
9568:
9564:
9556:
9551:
9548:
9545:
9541:
9537:
9534:
9532:
9530:
9527:
9524:
9521:
9520:
9509:
9508:
9493:
9490:
9487:
9483:
9479:
9475:
9471:
9468:
9459:
9456:
9453:
9448:
9444:
9438:
9435:
9430:
9425:
9421:
9415:
9412:
9407:
9404:
9399:
9396:
9391:
9386:
9383:
9379:
9375:
9372:
9370:
9368:
9363:
9360:
9357:
9354:
9350:
9343:
9340:
9337:
9334:
9331:
9324:
9321:
9317:
9312:
9308:
9305:
9300:
9297:
9294:
9291:
9287:
9282:
9278:
9273:
9270:
9267:
9263:
9259:
9256:
9253:
9245:
9240:
9237:
9234:
9230:
9226:
9223:
9221:
9219:
9216:
9213:
9210:
9209:
9198:
9197:
9182:
9177:
9174:
9169:
9165:
9161:
9157:
9147:
9144:
9141:
9136:
9132:
9126:
9123:
9118:
9113:
9109:
9103:
9100:
9095:
9090:
9086:
9080:
9077:
9072:
9069:
9066:
9061:
9059:
9057:
9052:
9049:
9046:
9043:
9039:
9032:
9029:
9026:
9023:
9020:
9013:
9010:
9006:
9001:
8997:
8994:
8989:
8986:
8982:
8977:
8971:
8968:
8964:
8958:
8955:
8952:
8948:
8944:
8941:
8938:
8930:
8925:
8922:
8919:
8915:
8911:
8906:
8904:
8902:
8897:
8894:
8891:
8888:
8884:
8877:
8874:
8871:
8868:
8865:
8862:
8859:
8853:
8850:
8847:
8844:
8840:
8832:
8827:
8824:
8821:
8817:
8813:
8808:
8806:
8804:
8801:
8798:
8795:
8794:
8780:
8779:
8766:
8762:
8749:
8745:
8742:up/down number
8732:
8689:
8686:
8664:
8661:
8656:
8653:
8650:
8647:
8644:
8641:
8595:
8594:
8579:
8574:
8571:
8567:
8560:
8557:
8554:
8551:
8548:
8541:
8537:
8533:
8530:
8527:
8519:
8514:
8511:
8508:
8504:
8500:
8497:
8495:
8493:
8490:
8487:
8481:
8478:
8472:
8468:
8462:
8456:
8453:
8447:
8443:
8437:
8431:
8428:
8422:
8418:
8412:
8409:
8406:
8403:
8401:
8399:
8396:
8393:
8390:
8389:
8384:
8381:
8378:
8375:
8371:
8364:
8361:
8358:
8355:
8352:
8349:
8346:
8339:
8335:
8331:
8328:
8325:
8317:
8312:
8309:
8306:
8302:
8298:
8295:
8293:
8291:
8288:
8285:
8279:
8276:
8270:
8266:
8260:
8254:
8251:
8245:
8241:
8235:
8229:
8226:
8220:
8216:
8210:
8207:
8204:
8201:
8199:
8197:
8194:
8191:
8188:
8187:
8172:
8169:
8158:
8157:
8146:
8140:
8136:
8132:
8129:
8126:
8122:
8119:
8104:
8103:
8092:
8088:
8085:
8080:
8076:
8072:
8069:
8066:
8060:
8057:
8052:
8048:
8042:
8039:
8034:
8030:
8026:
8023:
8020:
8015:
8011:
8004:
8001:
7998:
7995:
7989:
7986:
7982:
7957:
7954:
7951:
7947:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7898:
7878:
7875:
7855:
7852:
7849:
7807:
7806:
7795:
7791:
7788:
7785:
7782:
7778:
7775:
7748:
7745:
7742:
7739:
7736:
7733:
7730:
7727:
7721:
7718:
7714:
7709:
7706:
7703:
7700:
7697:
7689:
7685:
7681:
7675:
7671:
7648:
7645:
7642:
7639:
7636:
7633:
7630:
7627:
7621:
7618:
7614:
7609:
7606:
7603:
7600:
7592:
7588:
7584:
7578:
7574:
7560:
7559:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7524:
7521:
7518:
7515:
7512:
7509:
7506:
7503:
7497:
7494:
7491:
7488:
7485:
7482:
7479:
7476:
7473:
7467:
7464:
7460:
7452:
7449:
7446:
7443:
7440:
7437:
7434:
7431:
7425:
7422:
7418:
7397:
7394:
7393:
7392:
7389:
7386:
7383:
7380:
7345:
7342:
7339:
7336:
7333:
7327:
7324:
7320:
7312:
7308:
7304:
7301:
7298:
7293:
7288:
7285:
7282:
7278:
7274:
7271:
7268:
7265:
7260:
7256:
7235:
7232:
7229:
7226:
7223:
7177:
7174:
7171:
7170:
7167:
7156:
7146:
7135:
7125:
7112:
7108:
7097:
7084:
7081:
7068:
7067:
7054:
7049:
7046:
7036:
7023:
7017:
7012:
7007:
6992:
6979:
6973:
6968:
6963:
6948:
6935:
6931:
6920:
6907:
6903:
6900:
6886:
6885:
6872:
6860:
6847:
6844:
6832:
6819:
6815:
6802:
6789:
6785:
6774:
6761:
6758:
6745:
6744:
6733:
6723:
6710:
6706:
6693:
6680:
6676:
6663:
6650:
6646:
6635:
6622:
6619:
6606:
6605:
6592:
6588:
6575:
6562:
6558:
6545:
6532:
6529:
6517:
6504:
6500:
6489:
6476:
6473:
6460:
6459:
6446:
6441:
6438:
6428:
6415:
6409:
6404:
6399:
6384:
6371:
6365:
6360:
6355:
6340:
6327:
6323:
6312:
6299:
6296:
6283:
6282:
6271:
6261:
6250:
6240:
6229:
6219:
6206:
6202:
6191:
6180:
6169:
6168:
6165:
6161:
6160:
6149:
6146:
6143:
6140:
6137:
6127:
6116:
6113:
6110:
6107:
6104:
6094:
6083:
6080:
6077:
6074:
6071:
6061:
6044:Main article:
6041:
6038:
6037:
6036:
6025:
5995:
5988:complex number
5986:of a non-real
5980:
5958:
5957:
5941:
5938:
5933:
5929:
5923:
5918:
5914:
5910:
5907:
5904:
5899:
5896:
5891:
5888:
5878:
5865:
5861:
5855:
5850:
5846:
5842:
5839:
5836:
5831:
5828:
5823:
5820:
5810:
5796:
5792:
5787:
5782:
5778:
5772:
5767:
5763:
5759:
5756:
5753:
5748:
5745:
5740:
5737:
5727:
5714:
5711:
5706:
5701:
5697:
5691:
5686:
5682:
5678:
5675:
5672:
5667:
5664:
5659:
5656:
5646:
5630:
5627:
5622:
5618:
5612:
5606:
5602:
5598:
5595:
5592:
5589:
5586:
5583:
5552:
5549:
5537:
5536:
5525:
5522:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5470:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5414:
5386:
5362:
5359:
5339:
5336:
5314:
5313:
5301:
5297:
5294:
5291:
5288:
5285:
5281:
5277:
5274:
5271:
5268:
5265:
5262:
5239:
5235:
5232:
5229:
5226:
5223:
5219:
5215:
5212:
5209:
5206:
5203:
5200:
5177:
5174:
5150:
5147:
5107:
5104:
5101:
5055:
5054:
5043:
5037:
5034:
5031:
5027:
5022:
5019:
5016:
5013:
5009:
5003:
5000:
4997:
4993:
4988:
4985:
4982:
4979:
4975:
4969:
4966:
4963:
4958:
4955:
4952:
4946:
4943:
4940:
4937:
4933:
4927:
4924:
4921:
4916:
4913:
4910:
4904:
4901:
4898:
4895:
4880:
4879:
4868:
4862:
4857:
4853:
4850:
4847:
4844:
4819:
4814:
4810:
4804:
4801:
4798:
4788:
4777:
4771:
4766:
4762:
4759:
4756:
4753:
4728:
4723:
4719:
4716:
4713:
4710:
4696:
4695:
4684:
4681:
4678:
4675:
4670:
4666:
4662:
4659:
4656:
4651:
4647:
4619:
4616:
4613:
4610:
4607:
4604:
4600:
4579:
4576:
4571:
4567:
4563:
4558:
4554:
4525:
4521:
4517:
4514:
4511:
4508:
4505:
4494:
4493:
4482:
4476:
4471:
4467:
4464:
4461:
4458:
4433:
4428:
4424:
4421:
4418:
4415:
4361:
4358:
4355:
4352:
4346:
4341:
4337:
4334:
4330:
4309:
4303:
4298:
4294:
4291:
4288:
4285:
4281:
4248:
4243:
4209:
4206:
4203:
4200:
4194:
4189:
4185:
4182:
4178:
4157:
4154:
4151:
4131:
4128:
4122:
4117:
4113:
4110:
4107:
4104:
4100:
4079:
4076:
4073:
4060:extended to a
4049:
4044:
4022:
4019:
4013:
4008:
4004:
3998:
3993:
3989:
3986:
3982:
3961:
3958:
3955:
3935:
3932:
3929:
3926:
3886:
3853:
3850:
3655:
3652:
3458:
3455:
3452:
3451:
3437:
3434:
3431:
3427:
3422:
3418:
3414:
3411:
3406:
3402:
3397:
3393:
3390:
3387:
3384:
3381:
3378:
3368:
3354:
3351:
3348:
3344:
3339:
3335:
3331:
3328:
3323:
3320:
3314:
3310:
3307:
3304:
3301:
3298:
3295:
3285:
3266:
3262:
3261:
3247:
3244:
3241:
3237:
3232:
3228:
3224:
3221:
3216:
3212:
3207:
3203:
3200:
3197:
3194:
3191:
3188:
3178:
3164:
3161:
3158:
3154:
3149:
3145:
3141:
3138:
3133:
3130:
3124:
3120:
3117:
3114:
3111:
3108:
3105:
3095:
3076:
3072:
3071:
3057:
3054:
3051:
3047:
3042:
3038:
3034:
3031:
3026:
3022:
3017:
3013:
3010:
3007:
3001:
2998:
2995:
2990:
2987:
2984:
2978:
2975:
2972:
2969:
2959:
2945:
2942:
2939:
2935:
2930:
2926:
2922:
2919:
2914:
2911:
2905:
2901:
2898:
2895:
2889:
2886:
2883:
2878:
2875:
2872:
2866:
2863:
2860:
2857:
2847:
2828:
2824:
2823:
2809:
2806:
2803:
2799:
2794:
2790:
2786:
2783:
2778:
2774:
2769:
2765:
2762:
2759:
2753:
2750:
2747:
2742:
2739:
2736:
2730:
2727:
2724:
2721:
2711:
2697:
2694:
2691:
2687:
2682:
2678:
2674:
2671:
2666:
2663:
2657:
2653:
2650:
2647:
2641:
2638:
2635:
2630:
2627:
2624:
2618:
2615:
2612:
2609:
2599:
2580:
2576:
2575:
2560:
2557:
2554:
2550:
2545:
2541:
2537:
2534:
2529:
2525:
2520:
2516:
2513:
2510:
2507:
2504:
2501:
2491:
2476:
2473:
2470:
2466:
2461:
2457:
2453:
2450:
2445:
2442:
2436:
2432:
2429:
2426:
2423:
2420:
2417:
2407:
2388:
2384:
2383:
2369:
2366:
2363:
2359:
2354:
2350:
2346:
2343:
2338:
2334:
2329:
2325:
2322:
2319:
2316:
2313:
2310:
2300:
2286:
2283:
2280:
2276:
2271:
2267:
2263:
2260:
2255:
2252:
2246:
2242:
2239:
2236:
2233:
2230:
2227:
2217:
2198:
2194:
2193:
2187:
2180:
2179:
2174:
2171:
2090:
2087:
2084:
2079:
2075:
2071:
2068:
2065:
2045:
2042:
2039:
2036:
2033:
1989:
1988:
1987:
1986:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1919:
1916:
1913:
1910:
1900:
1895:
1894:
1893:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1826:
1823:
1820:
1817:
1807:
1801:
1800:
1799:
1798:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1725:
1722:
1719:
1716:
1706:
1701:
1700:
1699:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1626:
1623:
1620:
1617:
1607:
1601:
1600:
1599:
1598:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1525:
1522:
1519:
1516:
1506:
1501:
1500:
1499:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1426:
1423:
1420:
1417:
1407:
1183:
1180:
1166:
1163:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1096:
1093:
1090:
1087:
1084:
1081:
1061:
1058:
1053:
1050:
1046:
1042:
1039:
1019:
1016:
1013:
1010:
986:
983:
980:
977:
972:
969:
965:
944:
941:
936:
933:
929:
919:. For example
899:
896:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
808:
804:
783:
780:
777:
774:
771:
768:
765:
762:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
689:
686:
683:
680:
675:
671:
650:
647:
642:
638:
628:. For example
622:exponentiation
603:
600:
597:
594:
591:
588:
585:
582:
562:
559:
556:
553:
550:
547:
544:
541:
521:
518:
515:
512:
509:
492:, for example
472:
469:
359:real functions
309:
308:
306:
305:
298:
291:
283:
280:
279:
278:
277:
272:
267:
262:
257:
252:
247:
242:
237:
232:
224:
223:
222:Mathematicians
219:
218:
217:
216:
211:
206:
196:
188:
187:
181:
180:
179:
178:
172:
171:
166:
161:
156:
148:
147:
143:
142:
141:
140:
135:
130:
125:
117:
116:
112:
111:
110:
109:
104:
81:
80:
75:
70:
62:
61:
53:
52:
26:
9:
6:
4:
3:
2:
26730:
26719:
26716:
26714:
26711:
26709:
26706:
26704:
26701:
26699:
26696:
26694:
26691:
26689:
26686:
26685:
26683:
26668:
26665:
26663:
26660:
26658:
26655:
26653:
26650:
26649:
26647:
26643:
26637:
26634:
26632:
26629:
26627:
26624:
26620:
26617:
26616:
26615:
26614:Trigonometric
26612:
26611:
26609:
26605:
26596:
26591:
26589:
26584:
26582:
26577:
26576:
26573:
26567:
26563:
26559:
26556:
26554:
26550:
26546:
26543:
26540:
26537:
26535:
26532:
26528:
26524:
26523:
26518:
26514:
26513:
26503:
26502:
26497:
26493:
26490:
26486:
26482:
26478:
26473:
26470:
26469:
26464:
26460:
26457:
26453:
26450:
26449:
26444:
26440:
26437:
26436:
26431:
26427:
26424:
26420:
26416:
26412:
26407:
26404:
26403:0-19-853446-9
26400:
26396:
26395:
26390:
26386:
26383:
26382:0-691-09541-8
26379:
26375:
26374:
26369:
26366:
26363:
26359:
26356:
26355:0-691-00659-8
26352:
26348:
26347:Penguin Books
26344:
26340:
26337:
26333:
26329:
26325:
26321:
26318:
26317:0-471-54397-7
26314:
26310:
26306:
26303:
26299:
26297:9780471321484
26293:
26289:
26285:
26281:
26278:
26274:
26270:
26267:
26263:
26259:
26255:
26251:
26247:
26243:
26239:
26233:
26229:
26228:
26223:
26219:
26215:
26214:
26193:
26189:
26182:
26174:
26170:
26164:
26155:
26145:
26141:
26134:
26133:
26129:See Plofker,
26126:
26118:
26112:
26108:
26104:
26094:
26090:
26089:
26081:
26078:
26076:
26072:
26068:
26065:
26063:
26059:
26055:
26052:
26051:
26048:
26041:
26034:
26030:
26024:
26009:
26005:
26001:
26000:
25992:
25977:
25971:
25967:
25963:
25960:
25956:
25955:
25950:
25943:
25935:
25929:
25925:
25921:
25917:
25910:
25896:
25892:
25891:
25886:
25880:
25873:
25872:Nielsen (1966
25868:
25860:
25858:9783540647676
25854:
25849:
25848:
25839:
25832:
25828:
25827:
25822:
25818:
25811:
25796:
25792:
25786:
25778:
25774:
25770:
25766:
25762:
25758:
25754:
25750:
25746:
25739:
25725:on 2006-05-14
25724:
25720:
25713:
25705:
25699:
25697:
25688:
25682:
25678:
25674:
25670:
25666:
25659:
25645:on 2013-10-19
25644:
25640:
25639:
25634:
25627:
25625:
25617:
25616:0-471-54397-7
25613:
25607:
25605:
25603:
25594:
25590:
25584:
25580:
25576:
25568:
25560:
25556:
25550:
25546:
25545:
25540:
25534:
25525:
25517:
25511:
25507:
25500:
25492:
25488:
25484:
25478:
25470:
25463:
25455:
25451:
25445:
25436:
25427:
25421:
25417:
25414:
25409:
25405:
25399:
25395:
25394:
25386:
25378:
25374:
25368:
25364:
25360:
25359:
25354:
25350:
25344:
25337:
25335:0-387-20571-3
25331:
25327:
25323:
25319:
25315:
25308:
25299:
25290:
25281:
25274:
25269:
25260:
25250:
25242:
25235:
25233:
25231:
25229:
25227:
25225:
25218:
25214:
25211:
25206:
25202:
25196:
25192:
25191:
25183:
25168:
25164:
25160:
25153:
25145:
25139:
25135:
25134:
25126:
25118:
25114:
25110:
25103:
25095:
25091:
25087:
25083:
25079:
25075:
25071:
25067:
25063:
25056:
25048:
25044:
25038:
25030:
25026:
25022:
25020:0-07-054235-X
25016:
25012:
25011:
25003:
25001:
24993:
24988:
24974:
24970:
24964:
24957:
24952:
24937:
24933:
24927:
24923:
24919:
24915:
24914:
24909:
24903:
24895:
24891:
24887:
24881:
24877:
24854:
24850:
24838:
24834:
24822:
24819:
24817:
24814:
24812:
24809:
24807:
24804:
24802:
24799:
24797:
24794:
24792:
24789:
24787:
24784:
24783:
24776:
24774:
24770:
24767:(sine of the
24766:
24762:
24758:
24754:
24753:Edmund Gunter
24750:
24745:
24743:
24739:
24735:
24731:
24727:
24723:
24718:
24714:
24709:
24708:Ancient Greek
24705:
24703:
24697:
24695:
24689:
24683:
24677:
24673:
24669:
24665:
24661:
24657:
24653:
24652:
24648:
24645:derives from
24643:
24636:
24626:
24624:
24615:
24611:
24589:
24584:
24580:
24576:
24571:
24557:
24549:
24529:
24524:
24520:
24498:
24494:
24489:
24475:
24467:
24463:
24458:
24444:
24436:
24431:
24430:
24429:
24427:
24423:
24419:
24415:
24411:
24407:
24403:
24398:
24396:
24392:
24388:
24384:
24380:
24376:
24372:
24368:
24364:
24360:
24359:
24354:
24350:
24346:
24338:
24333:
24327:
24322:
24320:
24319:Trigonométrie
24316:
24312:
24308:
24304:
24303:Albert Girard
24299:
24297:
24293:
24292:Thomas Fincke
24289:
24285:
24280:
24278:
24273:
24271:
24267:
24263:
24259:
24255:
24251:
24249:
24245:
24241:
24240:Regiomontanus
24237:
24233:
24229:
24225:
24221:
24217:
24213:
24209:
24205:
24201:
24197:
24192:
24190:
24186:
24185:
24180:
24179:
24174:
24171:
24167:
24166:
24162:
24157:
24153:
24149:
24145:
24141:
24137:
24131:
24121:
24119:
24118:sawtooth wave
24114:
24101:
24095:
24092:
24089:
24086:
24074:
24068:
24065:
24062:
24059:
24046:
24043:
24030:
24027:
24024:
24020:
24014:
24011:
24006:
24000:
23988:
23980:
23976:
23971:
23958:
23952:
23944:
23940:
23934:
23930:
23919:
23916:
23913:
23909:
23905:
23899:
23893:
23883:
23879:
23867:
23863:
23857:
23853:
23847:
23845:
23841:
23836:
23834:
23830:
23822:
23812:
23806:
23799:
23795:
23790:
23783:
23778:
23755:
23750:
23747:
23742:
23736:
23733:
23730:
23722:
23719:
23713:
23710:
23704:
23698:
23695:
23692:
23684:
23681:
23675:
23672:
23666:
23660:
23657:
23654:
23646:
23643:
23637:
23634:
23624:
23623:
23622:
23603:
23599:
23596:
23593:
23587:
23582:
23579:
23574:
23571:
23564:
23563:
23562:
23539:
23536:
23533:
23524:
23521:
23518:
23509:
23506:
23503:
23495:
23492:
23485:
23482:
23475:
23474:
23473:
23471:
23467:
23463:
23459:
23455:
23451:
23447:
23443:
23437:
23409:
23406:
23403:
23398:
23395:
23392:
23386:
23378:
23374:
23371:
23368:
23362:
23359:
23352:
23348:
23345:
23342:
23336:
23333:
23323:
23322:
23321:
23317:
23307:
23303:
23301:
23288:
23275:
23269:
23266:
23263:
23256:
23252:
23248:
23243:
23239:
23235:
23230:
23226:
23219:
23216:
23213:
23210:
23190:
23187:
23184:
23181:
23178:
23175:
23172:
23169:
23164:
23160:
23156:
23151:
23147:
23143:
23138:
23134:
23125:
23119:
23109:
23107:
23106:
23105:triangulation
23099:
23097:
23077:
23074:
23071:
23068:
23062:
23059:
23056:
23052:
23047:
23041:
23038:
23035:
23031:
23026:
23020:
23017:
23014:
23010:
22984:
22978:
22975:
22972:
22964:
22958:
22953:
22949:
22946:
22943:
22937:
22932:
22928:
22925:
22922:
22916:
22911:
22907:
22904:
22901:
22864:
22854:
22821:
22811:
22809:
22804:
22802:
22765:
22762:
22759:
22755:
22750:
22747:
22742:
22739:
22736:
22731:
22728:
22723:
22716:
22702:
22699:
22696:
22688:
22685:
22682:
22679:
22672:
22658:
22655:
22652:
22649:
22646:
22639:
22625:
22622:
22619:
22616:
22613:
22606:
22605:
22588:
22585:
22580:
22577:
22573:
22570:
22567:
22564:
22561:
22558:
22551:
22537:
22534:
22531:
22523:
22520:
22517:
22514:
22507:
22493:
22490:
22487:
22484:
22481:
22474:
22460:
22457:
22454:
22451:
22448:
22441:
22440:
22425:
22422:
22419:
22416:
22413:
22406:
22389:
22386:
22383:
22377:
22370:
22356:
22353:
22350:
22347:
22344:
22337:
22323:
22320:
22317:
22314:
22311:
22304:
22303:
22286:
22283:
22278:
22275:
22272:
22267:
22264:
22259:
22252:
22235:
22232:
22229:
22223:
22216:
22202:
22199:
22196:
22193:
22190:
22183:
22169:
22166:
22163:
22160:
22157:
22150:
22149:
22134:
22131:
22128:
22125:
22122:
22115:
22101:
22098:
22095:
22092:
22089:
22086:
22079:
22065:
22062:
22059:
22056:
22053:
22046:
22032:
22029:
22026:
22023:
22020:
22013:
22012:
21995:
21992:
21987:
21984:
21981:
21976:
21973:
21968:
21961:
21947:
21944:
21941:
21938:
21935:
21932:
21925:
21911:
21908:
21905:
21902:
21899:
21892:
21878:
21875:
21872:
21869:
21866:
21859:
21858:
21854:
21851:
21848:
21845:
21844:
21841:
21839:
21835:
21831:
21827:
21823:
21819:
21813:
21785:
21781:
21778:
21773:
21769:
21765:
21762:
21756:
21753:
21750:
21746:
21742:
21736:
21731:
21727:
21723:
21720:
21714:
21711:
21708:
21704:
21700:
21694:
21691:
21685:
21682:
21678:
21673:
21671:
21663:
21660:
21655:
21652:
21649:
21646:
21636:
21632:
21629:
21626:
21623:
21620:
21617:
21614:
21611:
21605:
21602:
21599:
21595:
21591:
21585:
21582:
21576:
21573:
21570:
21566:
21562:
21556:
21553:
21550:
21547:
21541:
21538:
21535:
21531:
21527:
21521:
21518:
21512:
21509:
21505:
21500:
21498:
21490:
21487:
21482:
21479:
21476:
21473:
21463:
21459:
21456:
21453:
21450:
21447:
21441:
21438:
21435:
21431:
21427:
21421:
21418:
21415:
21412:
21406:
21403:
21400:
21396:
21392:
21386:
21383:
21377:
21374:
21370:
21365:
21363:
21355:
21352:
21347:
21344:
21341:
21338:
21324:
21323:
21322:
21319:
21317:
21301:
21281:
21275:
21272:
21269:
21263:
21260:
21240:
21236:
21232:
21229:
21226:
21223:
21220:
21216:
21212:
21209:
21189:
21186:
21183:
21163:
21157:
21154:
21151:
21145:
21142:
21139:
21119:
21116:
21113:
21093:
21090:
21087:
21084:
21081:
21056:
21053:
21049:
21045:
21042:
21039:
21035:
21031:
21028:
21021:
21007:
21004:
20999:
20995:
20991:
20984:
20970:
20967:
20964:
20957:
20956:
20941:
20938:
20934:
20930:
20927:
20924:
20921:
20918:
20915:
20912:
20908:
20904:
20901:
20894:
20880:
20877:
20874:
20871:
20868:
20865:
20858:
20844:
20841:
20838:
20831:
20830:
20815:
20812:
20808:
20804:
20801:
20798:
20795:
20792:
20789:
20786:
20782:
20778:
20775:
20768:
20754:
20751:
20748:
20745:
20742:
20739:
20736:
20729:
20715:
20712:
20709:
20702:
20701:
20686:
20683:
20679:
20675:
20672:
20669:
20665:
20661:
20658:
20651:
20637:
20634:
20629:
20625:
20617:
20603:
20600:
20597:
20590:
20589:
20574:
20571:
20568:
20565:
20562:
20555:
20541:
20538:
20535:
20532:
20525:
20511:
20508:
20505:
20498:
20497:
20482:
20479:
20476:
20473:
20470:
20467:
20460:
20446:
20443:
20440:
20433:
20419:
20416:
20413:
20406:
20405:
20390:
20387:
20380:
20374:
20371:
20364:
20347:
20340:
20337:
20329:
20312:
20306:
20299:
20298:
20295:
20293:
20285:
20281:
20280:quotient rule
20277:
20267:
20265:
20261:
20257:
20238:
20235:
20232:
20223:
20219:
20215:
20212:
20208:
20203:
20200:
20197:
20190:
20189:
20188:
20167:
20159:
20155:
20151:
20148:
20143:
20140:
20134:
20132:
20127:
20124:
20121:
20114:
20106:
20102:
20098:
20095:
20088:
20084:
20080:
20077:
20071:
20069:
20064:
20061:
20058:
20051:
20043:
20039:
20035:
20032:
20027:
20024:
20018:
20016:
20011:
20008:
20005:
19994:
19993:
19992:
19978:
19970:
19954:
19934:
19931:
19925:
19922:
19916:
19913:
19910:
19907:
19898:
19896:
19873:
19867:
19864:
19859:
19855:
19851:
19848:
19843:
19840:
19837:
19834:
19828:
19826:
19821:
19818:
19815:
19812:
19805:
19799:
19796:
19791:
19787:
19783:
19780:
19775:
19772:
19767:
19763:
19759:
19756:
19750:
19747:
19744:
19739:
19735:
19731:
19728:
19725:
19722:
19719:
19716:
19713:
19710:
19705:
19701:
19697:
19694:
19691:
19688:
19683:
19679:
19675:
19672:
19669:
19664:
19660:
19656:
19654:
19649:
19646:
19643:
19640:
19633:
19627:
19624:
19619:
19615:
19611:
19608:
19603:
19600:
19597:
19594:
19588:
19585:
19582:
19579:
19576:
19573:
19570:
19567:
19564:
19562:
19557:
19554:
19551:
19548:
19537:
19536:
19535:
19533:
19510:
19504:
19501:
19498:
19495:
19492:
19489:
19486:
19483:
19478:
19475:
19472:
19469:
19466:
19463:
19460:
19454:
19452:
19444:
19441:
19438:
19432:
19429:
19422:
19419:
19416:
19413:
19410:
19407:
19404:
19401:
19398:
19395:
19392:
19389:
19386:
19383:
19380:
19378:
19372:
19368:
19365:
19362:
19358:
19354:
19351:
19344:
19341:
19338:
19335:
19332:
19329:
19326:
19323:
19320:
19317:
19314:
19311:
19308:
19305:
19302:
19300:
19294:
19290:
19287:
19284:
19280:
19276:
19273:
19262:
19259:
19241:
19235:
19232:
19229:
19226:
19223:
19220:
19217:
19214:
19209:
19206:
19203:
19200:
19197:
19194:
19191:
19185:
19183:
19175:
19172:
19169:
19163:
19160:
19153:
19150:
19147:
19144:
19141:
19138:
19135:
19132:
19129:
19126:
19123:
19120:
19117:
19114:
19111:
19109:
19103:
19099:
19096:
19093:
19089:
19085:
19082:
19075:
19072:
19069:
19066:
19063:
19060:
19057:
19054:
19051:
19048:
19045:
19042:
19039:
19036:
19033:
19031:
19025:
19021:
19018:
19015:
19011:
19007:
19004:
18993:
18990:
18989:
18988:
18986:
18982:
18956:
18953:
18948:
18944:
18940:
18937:
18934:
18929:
18925:
18921:
18918:
18911:
18910:
18909:
18892:
18889:
18884:
18880:
18876:
18873:
18870:
18867:
18864:
18859:
18855:
18847:
18846:
18845:
18831:
18828:
18823:
18819:
18798:
18795:
18790:
18786:
18761:
18758:
18755:
18752:
18747:
18743:
18739:
18736:
18733:
18728:
18724:
18716:
18715:
18714:
18712:
18684:
18681:
18678:
18675:
18672:
18664:
18661:
18658:
18653:
18650:
18644:
18641:
18634:
18631:
18628:
18625:
18617:
18614:
18611:
18606:
18603:
18597:
18594:
18587:
18584:
18581:
18578:
18570:
18567:
18562:
18559:
18553:
18550:
18543:
18540:
18537:
18534:
18526:
18523:
18518:
18515:
18509:
18506:
18499:
18496:
18493:
18490:
18482:
18479:
18476:
18471:
18468:
18462:
18459:
18452:
18449:
18446:
18443:
18435:
18432:
18429:
18424:
18421:
18415:
18412:
18401:
18400:
18399:
18382:
18354:
18351:
18348:
18345:
18342:
18340:
18332:
18329:
18323:
18320:
18313:
18310:
18307:
18304:
18301:
18299:
18291:
18288:
18282:
18279:
18272:
18269:
18266:
18263:
18260:
18258:
18250:
18247:
18241:
18238:
18231:
18228:
18225:
18222:
18219:
18217:
18209:
18206:
18200:
18197:
18190:
18187:
18184:
18181:
18179:
18171:
18168:
18162:
18159:
18152:
18149:
18146:
18143:
18140:
18138:
18130:
18127:
18121:
18118:
18107:
18106:
18105:
18103:
18102:odd functions
18099:
18089:
18087:
18083:
18079:
18075:
18071:
18061:
18048:
18042:
18039:
18033:
18027:
18024:
18017:
18013:
18006:
17997:
17991:
17988:
17982:
17976:
17973:
17966:
17962:
17955:
17930:
17926:
17922:
17902:
17879:
17873:
17870:
17866:
17859:
17853:
17850:
17847:
17841:
17835:
17832:
17823:
17810:
17804:
17801:
17795:
17789:
17786:
17779:
17775:
17768:
17759:
17753:
17750:
17744:
17738:
17735:
17728:
17724:
17717:
17692:
17688:
17684:
17664:
17644:
17641:
17621:
17617:
17613:
17610:
17607:
17604:
17584:
17581:
17578:
17555:
17549:
17546:
17542:
17535:
17529:
17526:
17523:
17517:
17511:
17508:
17499:
17485:
17482:
17462:
17454:
17450:
17446:
17443:
17438:
17434:
17431:
17425:
17420:
17417:
17412:
17407:
17404:
17399:
17396:
17391:
17387:
17384:
17378:
17375:
17372:
17368:
17364:
17356:
17353:
17348:
17345:
17323:
17315:
17311:
17307:
17304:
17301:
17298:
17295:
17292:
17289:
17286:
17283:
17280:
17277:
17274:
17271:
17268:
17265:
17262:
17259:
17255:
17251:
17243:
17223:
17219:
17215:
17212:
17209:
17206:
17183:
17177:
17174:
17154:
17151:
17145:
17136:
17133:
17130:
17110:
17107:
17104:
17081:
17075:
17072:
17065:The function
17052:
17046:
17040:
17037:
17034:
17028:
17024:
17020:
17017:
17014:
17008:
17005:
17001:
16995:
16989:
16986:
16983:
16980:
16974:
16970:
16966:
16963:
16960:
16954:
16951:
16944:We also have
16931:
16925:
16919:
16916:
16910:
16904:
16901:
16898:
16892:
16889:
16886:
16880:
16877:
16871:
16868:
16865:
16859:
16856:
16852:
16846:
16840:
16837:
16834:
16828:
16825:
16822:
16816:
16811:
16807:
16802:
16796:
16790:
16785:
16781:
16777:
16771:
16768:
16765:
16759:
16754:
16750:
16726:
16720:
16717:
16714:
16708:
16705:
16702:
16696:
16693:
16689:
16683:
16677:
16674:
16671:
16665:
16662:
16659:
16653:
16650:
16630:
16624:
16618:
16615:
16612:
16609:
16603:
16600:
16597:
16591:
16588:
16584:
16578:
16572:
16569:
16566:
16563:
16557:
16554:
16551:
16545:
16542:
16522:
16502:
16499:
16479:
16473:
16467:
16464:
16461:
16455:
16452:
16449:
16446:
16440:
16437:
16433:
16427:
16421:
16418:
16415:
16409:
16406:
16403:
16400:
16394:
16391:
16371:
16368:
16360:
16346:
16332:
16329:
16326:
16315:
16311:
16297:
16294:
16291:
16280:
16276:
16274:
16260:
16257:
16254:
16243:
16239:
16225:
16222:
16219:
16208:
16204:
16202:
16188:
16185:
16182:
16171:
16167:
16153:
16150:
16147:
16136:
16132:
16131:
16128:
16123:
16120:
16104:
16096:
16073:
16070:
16067:
16064:
16061:
16058:
16055:
16052:
16049:
16046:
16043:
16040:
16037:
16034:
16031:
16029:
16024:
16021:
16018:
16011:
16008:
16005:
16002:
15999:
15996:
15993:
15990:
15987:
15984:
15981:
15978:
15975:
15972:
15969:
15967:
15962:
15959:
15956:
15945:
15944:
15943:
15941:
15925:
15922:
15919:
15916:
15913:
15910:
15903:
15879:
15876:
15873:
15870:
15867:
15857:
15854:
15851:
15848:
15845:
15842:
15839:
15836:
15833:
15830:
15827:
15824:
15817:
15816:
15815:
15797:
15794:
15791:
15788:
15785:
15782:
15779:
15776:
15773:
15770:
15767:
15764:
15761:
15758:
15752:
15749:
15746:
15740:
15737:
15730:
15729:
15728:
15726:
15721:
15719:
15709:
15695:
15691:
15687:
15667:
15661:
15655:
15652:
15649:
15641:
15637:
15633:
15630:
15626:
15621:
15613:
15605:
15601:
15597:
15594:
15588:
15585:
15581:
15576:
15572:
15566:
15563:
15558:
15555:
15551:
15547:
15544:
15521:
15515:
15512:
15509:
15506:
15498:
15494:
15490:
15487:
15482:
15479:
15473:
15463:
15455:
15451:
15447:
15444:
15438:
15435:
15430:
15425:
15422:
15416:
15412:
15406:
15403:
15398:
15395:
15391:
15387:
15384:
15377:Thus we have
15364:
15358:
15355:
15349:
15343:
15340:
15336:
15330:
15326:
15322:
15319:
15313:
15310:
15307:
15302:
15299:
15294:
15291:
15288:
15285:
15259:
15236:
15233:
15230:
15227:
15222:
15219:
15216:
15207:
15182:
15178:
15174:
15171:
15166:
15163:
15153:
15150:
15147:
15144:
15139:
15136:
15133:
15126:
15122:
15118:
15110:
15106:
15102:
15099:
15094:
15091:
15083:
15078:
15075:
15071:
15067:
15064:
15061:
15058:
15055:
15052:
15049:
15046:
15023:
15019:
15015:
15012:
15009:
15005:
15001:
14998:
14992:
14989:
14986:
14983:
14980:
14977:
14974:
14971:
14945:
14942:
14939:
14936:
14931:
14928:
14925:
14916:
14913:
14910:
14907:
14904:
14901:
14898:
14895:
14892:
14889:
14880:
14866:
14863:
14840:
14834:
14831:
14828:
14825:
14819:
14816:
14813:
14807:
14804:
14781:
14775:
14772:
14769:
14766:
14760:
14757:
14754:
14748:
14745:
14725:
14722:
14716:
14712:
14708:
14702:
14699:
14696:
14693:
14690:
14684:
14680:
14676:
14670:
14667:
14647:
14644:
14641:
14621:
14618:
14615:
14595:
14587:
14583:
14579:
14576:
14566:
14562:
14558:
14555:
14549:
14546:
14543:
14540:
14537:
14517:
14509:
14505:
14501:
14498:
14488:
14484:
14480:
14477:
14471:
14468:
14465:
14462:
14436:
14416:
14412:
14408:
14402:
14379:
14375:
14371:
14368:
14365:
14361:
14357:
14354:
14342:
14328:
14325:
14322:
14319:
14316:
14293:
14290:
14287:
14262:
14258:
14254:
14251:
14241:
14237:
14233:
14230:
14224:
14221:
14218:
14215:
14212:
14208:
14203:
14199:
14195:
14192:
14182:
14178:
14174:
14171:
14165:
14162:
14159:
14156:
14152:
14149:
14146:
14143:
14140:
14137:
14117:
14114:
14111:
14108:
14105:
14085:
14081:
14077:
14074:
14071:
14068:
14065:
14061:
14057:
14054:
14045:
14043:
14022:
14018:
14014:
14011:
14006:
14003:
13990:
13986:
13982:
13979:
13974:
13971:
13949:
13929:
13926:
13923:
13920:
13912:
13908:
13904:
13901:
13896:
13893:
13885:
13880:
13876:
13872:
13866:
13860:
13840:
13826:
13812:
13809:
13789:
13786:
13783:
13775:
13756:
13752:
13748:
13742:
13736:
13733:
13729:
13707:
13704:
13701:
13698:
13689:
13675:
13672:
13669:
13649:
13646:
13643:
13640:
13620:
13597:
13593:
13589:
13583:
13563:
13552:
13548:
13519:
13515:
13511:
13505:
13499:
13491:
13475:
13466:
13449:
13446:
13443:
13440:
13434:
13431:
13428:
13422:
13416:
13396:
13393:
13381:
13372:
13369:
13343:
13316:
13308:
13303:
13287:
13283:
13277:
13273:
13269:
13264:
13261:
13258:
13254:
13245:
13226:
13222:
13217:
13214:
13210:
13206:
13202:
13199:
13196:
13193:
13190:
13187:
13183:
13179:
13174:
13171:
13167:
13163:
13159:
13156:
13153:
13150:
13147:
13144:
13137:
13136:
13135:
13110:
13105:
13099:
13096:
13093:
13089:
13085:
13080:
13077:
13073:
13066:
13064:
13059:
13056:
13053:
13043:
13040:
13033:
13030:
13027:
13023:
13019:
13014:
13011:
13007:
13000:
12998:
12993:
12990:
12987:
12976:
12975:
12974:
12972:
12971:linear system
12969:Solving this
12949:
12946:
12943:
12940:
12937:
12934:
12931:
12928:
12925:
12922:
12920:
12913:
12910:
12907:
12903:
12895:
12892:
12889:
12886:
12883:
12880:
12877:
12874:
12871:
12869:
12862:
12859:
12855:
12843:
12842:
12841:
12838:
12824:
12821:
12815:
12807:
12803:
12799:
12793:
12785:
12781:
12753:
12745:
12741:
12736:
12729:
12721:
12717:
12709:. Therefore,
12696:
12693:
12684:
12676:
12672:
12667:
12660:
12652:
12648:
12640:
12637:
12633:
12629:
12621:
12620:quotient rule
12615:
12595:
12587:
12583:
12579:
12576:
12573:
12570:
12566:
12559:
12551:
12547:
12543:
12523:
12518:
12515:
12511:
12507:
12501:
12493:
12489:
12468:
12465:
12462:
12459:
12456:
12453:
12450:
12447:
12444:
12441:
12435:
12427:
12423:
12414:
12410:
12389:
12386:
12383:
12380:
12377:
12374:
12371:
12368:
12365:
12362:
12357:
12354:
12350:
12342:
12341:
12340:
12338:
12334:
12328:respectively.
12313:
12310:
12306:
12282:
12276:
12273:
12250:
12244:
12241:
12233:
12210:
12202:
12199:
12195:
12191:
12182:
12178:
12172:
12164:
12160:
12156:
12153:
12150:
12141:
12137:
12131:
12128:
12124:
12113:
12110:
12107:
12103:
12099:
12096:
12093:
12090:
12083:
12082:
12081:
12067:
12064:
12061:
12041:
12038:
12035:
12012:
12004:
12001:
11997:
11993:
11984:
11980:
11974:
11970:
11963:
11959:
11953:
11950:
11946:
11935:
11932:
11929:
11925:
11921:
11918:
11915:
11912:
11909:
11902:
11901:
11900:
11898:
11874:
11866:
11862:
11858:
11853:
11842:
11839:
11833:
11830:
11823:
11811:
11808:
11805:
11801:
11797:
11794:
11791:
11788:
11785:
11782:
11779:
11776:
11769:
11755:
11747:
11743:
11739:
11734:
11723:
11720:
11714:
11711:
11700:
11697:
11694:
11691:
11680:
11672:
11669:
11656:
11653:
11650:
11646:
11642:
11639:
11636:
11633:
11630:
11627:
11620:
11606:
11598:
11590:
11587:
11584:
11577:
11562:
11559:
11556:
11552:
11548:
11545:
11542:
11539:
11534:
11530:
11524:
11520:
11512:
11498:
11490:
11486:
11482:
11477:
11473:
11465:
11457:
11454:
11438:
11435:
11432:
11428:
11424:
11421:
11418:
11413:
11410:
11405:
11399:
11396:
11393:
11386:
11378:
11375:
11356:
11353:
11350:
11346:
11342:
11339:
11336:
11333:
11330:
11327:
11320:
11319:
11318:
11301:
11293:
11289:
11285:
11280:
11276:
11271:
11259:
11256:
11253:
11249:
11245:
11242:
11239:
11234:
11231:
11226:
11223:
11220:
11217:
11214:
11211:
11204:
11203:
11202:
11200:
11195:
11188:
11182:
11163:
11157:
11154:
11151:
11147:
11140:
11135:
11132:
11129:
11126:
11122:
11110:
11102:
11099:
11096:
11093:
11090:
11087:
11080:
11079:
11078:
11076:
11072:
11068:
11058:
11056:
11017:
11014:
11006:
10994:
10974:
10963:
10955:
10943:
10923:
10912:
10904:
10892:
10872:
10861:
10853:
10841:
10821:
10810:
10787:
10784:
10781:
10767:
10763:
10751:
10748:
10734:
10730:
10718:
10715:
10701:
10697:
10685:
10682:
10670:
10659:
10656:
10653:
10650:
10643:
10642:
10607:
10604:
10599:
10595:
10591:
10588:
10585:
10582:
10568:
10564:
10560:
10557:
10554:
10543:
10538:
10534:
10530:
10527:
10524:
10521:
10507:
10503:
10499:
10496:
10493:
10482:
10477:
10473:
10469:
10466:
10463:
10460:
10446:
10442:
10430:
10427:
10415:
10404:
10401:
10398:
10395:
10388:
10387:
10352:
10349:
10344:
10340:
10336:
10333:
10330:
10327:
10313:
10309:
10305:
10302:
10299:
10288:
10283:
10279:
10275:
10272:
10269:
10266:
10252:
10248:
10244:
10241:
10238:
10227:
10222:
10218:
10214:
10211:
10208:
10205:
10191:
10187:
10175:
10172:
10160:
10149:
10146:
10143:
10140:
10133:
10132:
10131:
10129:
10101:
10098:
10095:
10087:
10079:
10076:
10067:
10064:
10061:
10056:
10052:
10046:
10043:
10038:
10033:
10029:
10023:
10020:
10015:
10012:
10007:
10004:
9999:
9994:
9991:
9987:
9983:
9981:
9971:
9968:
9965:
9962:
9958:
9951:
9945:
9942:
9932:
9929:
9925:
9919:
9916:
9912:
9906:
9898:
9895:
9879:
9876:
9873:
9869:
9865:
9863:
9858:
9855:
9852:
9841:
9840:
9821:
9816:
9813:
9808:
9800:
9786:
9783:
9780:
9775:
9771:
9765:
9762:
9757:
9752:
9748:
9742:
9739:
9734:
9729:
9725:
9719:
9716:
9711:
9708:
9705:
9703:
9693:
9690:
9686:
9679:
9673:
9670:
9660:
9657:
9653:
9647:
9639:
9636:
9620:
9617:
9614:
9610:
9606:
9601:
9598:
9594:
9587:
9581:
9578:
9569:
9566:
9562:
9549:
9546:
9543:
9539:
9535:
9533:
9528:
9525:
9522:
9511:
9510:
9491:
9488:
9485:
9477:
9469:
9466:
9457:
9454:
9451:
9446:
9442:
9436:
9433:
9428:
9423:
9419:
9413:
9410:
9405:
9402:
9397:
9394:
9389:
9384:
9381:
9377:
9373:
9371:
9361:
9358:
9355:
9352:
9348:
9341:
9335:
9332:
9322:
9319:
9315:
9310:
9306:
9303:
9298:
9295:
9292:
9289:
9285:
9280:
9276:
9271:
9268:
9265:
9257:
9254:
9238:
9235:
9232:
9228:
9224:
9222:
9217:
9214:
9211:
9200:
9199:
9180:
9175:
9172:
9167:
9159:
9145:
9142:
9139:
9134:
9130:
9124:
9121:
9116:
9111:
9107:
9101:
9098:
9093:
9088:
9084:
9078:
9075:
9070:
9067:
9064:
9060:
9050:
9047:
9044:
9041:
9037:
9030:
9024:
9021:
9011:
9008:
9004:
8999:
8995:
8992:
8987:
8984:
8980:
8975:
8969:
8966:
8962:
8956:
8953:
8950:
8942:
8939:
8923:
8920:
8917:
8913:
8909:
8905:
8895:
8892:
8889:
8886:
8882:
8875:
8869:
8866:
8863:
8860:
8851:
8848:
8845:
8842:
8838:
8825:
8822:
8819:
8815:
8811:
8807:
8802:
8799:
8796:
8785:
8784:
8783:
8777:
8763:
8760:
8746:
8743:
8729:
8728:
8727:
8724:
8722:
8718:
8717:combinatorial
8714:
8710:
8709:Taylor series
8705:
8687:
8684:
8662:
8659:
8651:
8648:
8645:
8642:
8631:
8627:
8622:
8618:
8616:
8615:complex plane
8613:on the whole
8612:
8608:
8604:
8600:
8577:
8572:
8569:
8565:
8558:
8552:
8549:
8539:
8531:
8528:
8512:
8509:
8506:
8502:
8498:
8496:
8488:
8485:
8479:
8476:
8470:
8466:
8460:
8454:
8451:
8445:
8441:
8435:
8429:
8426:
8420:
8416:
8410:
8407:
8404:
8402:
8397:
8394:
8391:
8382:
8379:
8376:
8373:
8369:
8362:
8356:
8353:
8350:
8347:
8337:
8329:
8326:
8310:
8307:
8304:
8300:
8296:
8294:
8286:
8283:
8277:
8274:
8268:
8264:
8258:
8252:
8249:
8243:
8239:
8233:
8227:
8224:
8218:
8214:
8208:
8205:
8202:
8200:
8195:
8192:
8189:
8178:
8177:
8176:
8168:
8164:
8144:
8138:
8134:
8130:
8127:
8124:
8120:
8117:
8109:
8108:
8107:
8090:
8086:
8083:
8078:
8074:
8070:
8067:
8064:
8058:
8055:
8050:
8046:
8040:
8037:
8032:
8028:
8024:
8021:
8018:
8013:
8009:
8002:
7999:
7996:
7993:
7987:
7984:
7980:
7971:
7970:
7969:
7955:
7952:
7949:
7945:
7941:
7938:
7935:
7932:
7929:
7926:
7923:
7915:
7914:quotient rule
7912:Applying the
7910:
7896:
7876:
7873:
7853:
7850:
7847:
7838:
7834:
7827:
7820:
7813:
7793:
7789:
7786:
7783:
7780:
7776:
7773:
7765:
7764:
7763:
7762:
7746:
7743:
7740:
7737:
7734:
7731:
7728:
7725:
7719:
7716:
7712:
7707:
7704:
7701:
7698:
7695:
7687:
7683:
7679:
7673:
7669:
7646:
7643:
7640:
7637:
7634:
7631:
7628:
7625:
7619:
7616:
7612:
7607:
7604:
7601:
7598:
7590:
7586:
7582:
7576:
7572:
7546:
7543:
7537:
7531:
7528:
7522:
7519:
7516:
7510:
7504:
7501:
7495:
7492:
7489:
7486:
7483:
7480:
7477:
7474:
7471:
7465:
7462:
7458:
7450:
7447:
7444:
7441:
7438:
7435:
7432:
7429:
7423:
7420:
7416:
7407:
7406:
7405:
7403:
7390:
7387:
7384:
7381:
7378:
7377:
7376:
7373:
7370:
7369:
7364:
7343:
7337:
7334:
7325:
7322:
7318:
7310:
7302:
7299:
7291:
7286:
7283:
7280:
7276:
7272:
7266:
7258:
7254:
7230:
7224:
7221:
7213:
7205:
7198:
7193:
7186:
7182:
7154:
7147:
7133:
7126:
7110:
7106:
7098:
7082:
7079:
7070:
7069:
7052:
7047:
7044:
7037:
7021:
7015:
7010:
7005:
6993:
6977:
6971:
6966:
6961:
6949:
6933:
6929:
6921:
6905:
6901:
6898:
6888:
6887:
6870:
6861:
6845:
6842:
6833:
6817:
6813:
6803:
6787:
6783:
6775:
6759:
6756:
6747:
6746:
6731:
6724:
6708:
6704:
6694:
6678:
6674:
6664:
6648:
6644:
6636:
6620:
6617:
6608:
6607:
6590:
6586:
6576:
6560:
6556:
6546:
6530:
6527:
6518:
6502:
6498:
6490:
6474:
6471:
6462:
6461:
6444:
6439:
6436:
6429:
6413:
6407:
6402:
6397:
6385:
6369:
6363:
6358:
6353:
6341:
6325:
6321:
6313:
6297:
6294:
6285:
6284:
6269:
6262:
6248:
6241:
6227:
6220:
6204:
6200:
6192:
6178:
6171:
6170:
6166:
6163:
6162:
6144:
6138:
6135:
6111:
6105:
6102:
6078:
6072:
6069:
6059:
6054:
6051:
6047:
6034:
6030:
6026:
6023:
6019:
6015:
6014:Galois groups
6011:
6004:
6000:
5996:
5993:
5992:Galois theory
5989:
5985:
5981:
5978:
5974:
5970:
5969:
5968:
5965:
5963:
5955:
5939:
5936:
5931:
5927:
5921:
5916:
5912:
5908:
5905:
5902:
5897:
5894:
5889:
5886:
5879:
5863:
5859:
5853:
5848:
5844:
5840:
5837:
5834:
5829:
5826:
5821:
5818:
5811:
5794:
5790:
5785:
5780:
5776:
5770:
5765:
5761:
5757:
5754:
5751:
5746:
5743:
5738:
5735:
5728:
5712:
5709:
5704:
5699:
5695:
5689:
5684:
5680:
5676:
5673:
5670:
5665:
5662:
5657:
5654:
5647:
5644:
5628:
5625:
5620:
5616:
5610:
5604:
5600:
5596:
5593:
5590:
5587:
5584:
5581:
5574:
5573:
5572:
5570:
5562:
5557:
5548:
5520:
5517:
5514:
5511:
5505:
5502:
5499:
5496:
5493:
5490:
5465:
5462:
5459:
5456:
5450:
5447:
5444:
5441:
5438:
5435:
5428:
5427:
5426:
5412:
5397:, the points
5384:
5376:
5360:
5357:
5337:
5334:
5323:
5299:
5295:
5292:
5289:
5286:
5283:
5279:
5275:
5272:
5269:
5266:
5263:
5260:
5237:
5233:
5230:
5227:
5224:
5221:
5217:
5213:
5210:
5207:
5204:
5201:
5198:
5191:
5190:
5189:
5175:
5172:
5164:
5148:
5145:
5105:
5102:
5099:
5088:
5059:
5041:
5035:
5032:
5029:
5025:
5020:
5017:
5014:
5011:
5007:
5001:
4998:
4995:
4991:
4986:
4983:
4980:
4977:
4973:
4967:
4964:
4961:
4956:
4953:
4950:
4944:
4941:
4938:
4935:
4931:
4925:
4922:
4919:
4914:
4911:
4908:
4902:
4899:
4896:
4893:
4886:
4885:
4884:
4866:
4855:
4851:
4848:
4845:
4842:
4812:
4808:
4802:
4799:
4796:
4789:
4775:
4764:
4760:
4757:
4754:
4751:
4721:
4717:
4714:
4711:
4708:
4701:
4700:
4699:
4682:
4679:
4676:
4673:
4668:
4664:
4660:
4657:
4654:
4649:
4645:
4637:
4636:
4635:
4633:
4614:
4611:
4608:
4602:
4577:
4574:
4569:
4565:
4561:
4556:
4552:
4543:
4523:
4519:
4515:
4512:
4509:
4506:
4503:
4496:In the range
4480:
4469:
4465:
4462:
4459:
4456:
4426:
4422:
4419:
4416:
4413:
4406:
4405:
4404:
4398:
4394:
4381:
4375:
4359:
4353:
4350:
4339:
4332:
4296:
4292:
4289:
4283:
4269:
4263:
4246:
4231:
4230:perpendicular
4223:
4207:
4201:
4198:
4187:
4180:
4155:
4152:
4149:
4129:
4115:
4111:
4108:
4102:
4077:
4074:
4071:
4063:
4047:
4020:
4006:
4002:
3991:
3984:
3959:
3956:
3953:
3933:
3930:
3927:
3924:
3917:rotation for
3916:
3911:
3902:
3873:
3868:
3851:
3848:
3834:
3826:
3822:
3818:
3814:
3806:
3802:
3798:
3794:
3790:
3786:
3783:
3779:
3774:
3766:
3759:
3752:
3733:
3726:
3719:
3701:
3697:
3692:
3686:
3678:
3671:
3666:
3660:
3651:
3641:
3638:when we take
3625:
3621:
3617:
3612:
3605:
3599:
3594:
3587:
3581:
3573:
3569:
3565:
3561:
3556:
3554:
3550:
3546:
3543:
3515:
3511:
3507:
3495:
3491:
3487:
3483:
3478:
3476:
3472:
3468:
3464:
3435:
3432:
3429:
3425:
3420:
3416:
3412:
3409:
3404:
3400:
3395:
3391:
3388:
3385:
3382:
3379:
3376:
3369:
3352:
3349:
3346:
3342:
3337:
3333:
3329:
3326:
3321:
3318:
3312:
3308:
3305:
3302:
3299:
3296:
3293:
3286:
3264:
3263:
3245:
3242:
3239:
3235:
3230:
3226:
3222:
3219:
3214:
3210:
3205:
3201:
3198:
3195:
3192:
3189:
3186:
3179:
3162:
3159:
3156:
3152:
3147:
3143:
3139:
3136:
3131:
3128:
3122:
3118:
3115:
3112:
3109:
3106:
3103:
3096:
3074:
3073:
3055:
3052:
3049:
3045:
3040:
3036:
3032:
3029:
3024:
3020:
3015:
3011:
3008:
3005:
2999:
2996:
2993:
2988:
2985:
2982:
2976:
2973:
2970:
2967:
2960:
2943:
2940:
2937:
2933:
2928:
2924:
2920:
2917:
2912:
2909:
2903:
2899:
2896:
2893:
2887:
2884:
2881:
2876:
2873:
2870:
2864:
2861:
2858:
2855:
2848:
2826:
2825:
2807:
2804:
2801:
2797:
2792:
2788:
2784:
2781:
2776:
2772:
2767:
2763:
2760:
2757:
2751:
2748:
2745:
2740:
2737:
2734:
2728:
2725:
2722:
2719:
2712:
2695:
2692:
2689:
2685:
2680:
2676:
2672:
2669:
2664:
2661:
2655:
2651:
2648:
2645:
2639:
2636:
2633:
2628:
2625:
2622:
2616:
2613:
2610:
2607:
2600:
2578:
2577:
2558:
2555:
2552:
2548:
2543:
2539:
2535:
2532:
2527:
2523:
2518:
2514:
2511:
2508:
2505:
2502:
2499:
2492:
2474:
2471:
2468:
2464:
2459:
2455:
2451:
2448:
2443:
2440:
2434:
2430:
2427:
2424:
2421:
2418:
2415:
2408:
2386:
2385:
2367:
2364:
2361:
2357:
2352:
2348:
2344:
2341:
2336:
2332:
2327:
2323:
2320:
2317:
2314:
2311:
2308:
2301:
2284:
2281:
2278:
2274:
2269:
2265:
2261:
2258:
2253:
2250:
2244:
2240:
2237:
2234:
2231:
2228:
2225:
2218:
2196:
2195:
2192:
2188:
2186:
2182:
2181:
2178:
2168:
2159:
2153:
2142:
2132:
2122:
2116:
2110:
2106:
2102:
2085:
2082:
2077:
2073:
2066:
2063:
2040:
2034:
2031:
2022:
1997:
1995:
1917:
1914:
1911:
1908:
1901:
1898:
1897:
1896:
1824:
1821:
1818:
1815:
1808:
1805:
1804:
1803:
1802:
1723:
1720:
1717:
1714:
1707:
1704:
1703:
1702:
1624:
1621:
1618:
1615:
1608:
1605:
1604:
1603:
1602:
1523:
1520:
1517:
1514:
1507:
1504:
1503:
1502:
1424:
1421:
1418:
1415:
1408:
1405:
1404:
1403:
1402:
1399:
1393:
1385:
1381:
1369:
1347:
1335:
1330:-axis, while
1319:
1311:
1303:
1295:
1284:= 0.7 radians
1283:
1277:
1265:
1257:
1249:
1238:
1230:
1222:
1211:
1203:
1195:
1188:
1179:
1164:
1161:
1152:
1148:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1094:
1091:
1088:
1085:
1082:
1079:
1059:
1056:
1051:
1048:
1044:
1040:
1037:
1030:The equation
1017:
1014:
1011:
1008:
1000:
981:
975:
970:
967:
963:
942:
939:
934:
931:
927:
918:
914:
897:
894:
884:
871:
862:
856:
850:
847:
841:
832:
829:
826:
820:
814:
806:
802:
781:
775:
772:
769:
763:
760:
740:
734:
728:
725:
722:
716:
710:
707:
684:
678:
673:
669:
648:
645:
640:
636:
627:
623:
619:
614:
601:
595:
592:
589:
583:
580:
560:
557:
554:
548:
542:
539:
519:
516:
513:
510:
507:
497:
491:
487:
483:
479:
478:line segments
468:
466:
465:complex plane
462:
458:
454:
450:
446:
443:is the whole
442:
438:
433:
431:
427:
423:
419:
415:
411:
407:
403:
399:
394:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
345:(also called
344:
340:
332:
328:
324:
320:
315:
304:
299:
297:
292:
290:
285:
284:
282:
281:
276:
273:
271:
268:
266:
263:
261:
258:
256:
255:Regiomontanus
253:
251:
248:
246:
243:
241:
238:
236:
233:
231:
228:
227:
226:
225:
221:
220:
215:
212:
210:
207:
204:
200:
197:
195:
192:
191:
190:
189:
186:
183:
182:
177:
174:
173:
170:
167:
165:
162:
160:
157:
155:
152:
151:
150:
149:
145:
144:
139:
136:
134:
131:
129:
126:
124:
121:
120:
119:
118:
114:
113:
108:
105:
102:
98:
94:
90:
86:
83:
82:
79:
76:
74:
71:
69:
66:
65:
64:
63:
59:
55:
54:
51:
48:
47:
44:
40:
33:
19:
26698:Trigonometry
26613:
26520:
26499:
26476:
26466:
26446:
26433:
26410:
26392:
26371:
26364:
26361:
26342:
26331:
26308:
26287:
26272:
26269:Lars Ahlfors
26226:
26196:. Retrieved
26181:
26172:
26163:
26154:
26130:
26125:
26102:
26092:
26087:
26085:See Merlet,
26075:al-Khwārizmī
26070:
26057:
26040:
26032:
26023:
26012:. Retrieved
25998:
25991:
25980:. Retrieved
25953:
25942:
25915:
25909:
25898:. Retrieved
25889:
25879:
25867:
25851:. Springer.
25846:
25838:
25824:
25810:
25799:. Retrieved
25785:
25752:
25748:
25738:
25727:. Retrieved
25723:the original
25712:
25672:
25658:
25647:. Retrieved
25643:the original
25636:
25578:
25567:
25543:
25533:
25524:
25508:. Springer.
25505:
25499:
25490:
25477:
25468:
25462:
25453:
25444:
25435:
25426:
25392:
25385:
25357:
25343:
25321:
25307:
25298:
25289:
25280:
25268:
25259:
25249:
25240:
25190:Trigonometry
25189:
25182:
25171:. Retrieved
25162:
25152:
25132:
25125:
25108:
25102:
25072:(1): 37–42.
25069:
25065:
25055:
25009:
24987:
24976:. Retrieved
24972:
24963:
24951:
24940:. Retrieved
24912:
24902:
24889:
24880:
24837:
24772:
24764:
24760:
24756:
24747:The prefix "
24746:
24744:the circle.
24741:
24737:
24733:
24729:
24725:
24721:
24719:
24699:
24691:
24675:
24668:al-Khwārizmī
24659:
24649:
24638:
24620:
24613:
24609:
24587:
24578:
24574:
24555:
24547:
24527:
24518:
24496:
24495:) = 1 − sin(
24492:
24473:
24465:
24464:) = 1 − cos(
24461:
24442:
24434:
24399:
24394:
24390:
24386:
24382:
24378:
24374:
24356:
24331:
24328:proved that
24323:
24318:
24317:in his book
24314:
24310:
24306:
24300:
24295:
24294:in his book
24287:
24283:
24281:
24274:
24252:
24220:Omar Khayyám
24208:Al-Khwārizmī
24200:law of sines
24193:
24182:
24176:
24170:Gupta period
24164:
24160:
24133:
24115:
23972:
23881:
23877:
23855:
23851:
23848:
23837:
23826:
23810:
23620:
23560:
23465:
23457:
23453:
23449:
23445:
23441:
23439:
23319:
23304:
23289:
23121:
23103:
23100:
23096:circumradius
22866:
22863:Law of sines
22857:Law of sines
22828:
22814:Applications
22805:
22782:
21815:
21320:
21073:
20273:
20254:this is the
20253:
20186:
19899:
19892:
19529:
18977:
18907:
18777:
18708:
18378:
18095:
18067:
17824:
17500:
16356:
16319:
16284:
16247:
16212:
16175:
16140:
16126:
16092:
15942:as follows:
15899:
15813:
15722:
15715:
14881:
14343:
14046:
13832:
13690:
13489:
13467:
13304:
13241:
13129:
12968:
12839:
12613:
12412:
12411:
12404:
12331:
12027:
11894:
11316:
11193:
11191:th with the
11186:
11178:
11064:
11052:
10125:
8781:
8776:Euler number
8725:
8706:
8623:
8619:
8596:
8174:
8162:
8159:
8105:
7911:
7839:
7832:
7825:
7818:
7811:
7808:
7561:
7399:
7374:
7366:
7361:
6057:
6049:
5966:
5962:square roots
5959:
5566:
5538:
5315:
5165:with period
5091:
4881:
4697:
4495:
4396:
4392:
4382:
4267:
4261:
4222:tangent line
3909:
3874:
3810:
3804:
3800:
3796:
3792:
3788:
3784:
3781:
3764:
3757:
3750:
3731:
3724:
3717:
3690:
3669:
3664:
3639:
3623:
3619:
3615:
3610:
3603:
3597:
3592:
3585:
3579:
3557:
3544:
3541:
3513:
3480:However, in
3479:
3467:angular unit
3460:
2177:Relationship
2173:Description
2157:
2151:
2140:
2130:
2120:
2114:
2108:
2024:. Therefore
1998:
1992:
1391:
1383:
1357:
1345:
1333:
1317:
1309:
1301:
1293:
1281:
1263:
1255:
1247:
1236:
1228:
1220:
1209:
1201:
1193:
1146:
885:
615:
495:
474:
437:acute angles
434:
421:
417:
413:
405:
395:
354:
350:
346:
342:
336:
331:proportional
323:acute angles
84:
50:Trigonometry
43:
26370:Maor, Eli,
26050:to either
26029:Thomas Fale
24894:J. Springer
24775:similarly.
24499:) = versin(
24224:Bhāskara II
24178:Aryabhatiya
24152:Roman Egypt
23975:square wave
23798:square wave
20276:derivatives
19900:By setting
18104:. That is:
18086:above proof
13774:unit vector
8611:holomorphic
7363:G. H. Hardy
5954:right angle
5561:unit circle
4403:. That is,
4374:coordinates
3821:unit circle
3803:osine, and
3700:unit circle
3568:unit circle
3549:derivatives
997:denote the
449:unit circle
410:reciprocals
339:mathematics
325:, they are
321:have equal
240:Brahmagupta
209:Derivatives
138:Unit circle
26682:Categories
26631:Hyperbolic
26564:of cos at
26551:of sin at
26345:, 2nd ed.
26210:References
26198:2017-07-28
26062:Al-Battani
26014:2024-02-06
25982:2024-02-06
25900:2024-02-06
25801:2017-03-15
25729:2007-09-08
25649:2010-07-13
25173:2017-12-29
24978:2020-08-29
24942:2017-08-13
24821:Polar sine
24717:"string".
24664:Al-Battani
24623:logarithms
24590:) = exsec(
24468:) = 2 sin(
24437:) = 2 sin(
24422:excosecant
24335:is not an
24282:The terms
24202:, used in
24140:Hipparchus
21849:Definition
21074:Note: For
19260:Difference
18398:, one has
18383:of period
18070:identities
13772:becomes a
13691:Note that
13309:. The set
7169:Undefined
5643:zero angle
4542:hypotenuse
3636:= sin 180°
3273:hypotenuse
3083:hypotenuse
2827:cotangent
2401:hypotenuse
2211:hypotenuse
1380:hypotenuse
917:reciprocal
915:, not the
420:, and the
404:, and the
371:navigation
369:, such as
250:al-Battani
230:Hipparchus
169:Cotangents
123:Identities
26566:MathWorld
26553:MathWorld
26527:EMS Press
26501:MathWorld
26496:"Tangent"
26471:. (2002).
26451:. (2000).
26438:. (1996).
26389:"Preface"
26058:Astronomy
25777:240294796
25618:, p. 210.
25094:126217060
25086:0025-570X
25037:cite book
24910:(2004) .
24888:(1924) .
24773:cotangens
24720:The word
24639:The word
24629:Etymology
24526:haversin(
24491:coversin(
24414:haversine
24410:coversine
24236:Ulugh Beg
24093:−
24066:−
24047:
24036:∞
24021:∑
24015:π
23941:φ
23925:∞
23910:∑
23734:−
23714:
23696:−
23676:
23658:−
23638:
23597:−
23575:
23537:−
23522:−
23507:−
23456:)/2, and
23396:−
23363:
23346:−
23337:
23249:−
23214:
23185:
23170:−
23060:
23039:
23018:
22968:Δ
22947:
22926:
22905:
22801:arcsecond
22763:≠
22748:π
22743:≤
22737:≤
22729:π
22724:−
22686:−
22650:
22623:
22586:π
22581:≠
22571:π
22568:≤
22562:≤
22521:−
22485:
22458:
22426:π
22393:∞
22381:∞
22378:−
22348:
22321:
22284:π
22265:π
22260:−
22239:∞
22227:∞
22224:−
22194:
22167:
22135:π
22132:≤
22126:≤
22099:≤
22093:≤
22087:−
22057:
22030:
21993:π
21988:≤
21982:≤
21974:π
21969:−
21945:≤
21939:≤
21933:−
21903:
21876:
21834:bijective
21826:monotonic
21818:injective
21779:
21766:−
21754:−
21743:π
21737:
21724:−
21712:−
21701:π
21695:
21653:
21630:
21621:
21615:−
21603:−
21592:π
21586:
21574:−
21563:π
21557:
21551:−
21539:−
21528:π
21522:
21480:
21457:
21451:−
21439:−
21428:π
21422:
21416:−
21404:−
21393:π
21387:
21345:
21273:
21264:
21233:π
21213:π
21210:−
21187:
21155:
21146:
21140:−
21117:
21094:π
21043:
21032:
21005:
20992:−
20968:
20928:
20916:
20905:
20878:
20869:
20842:
20802:
20796:−
20790:
20779:
20752:
20743:
20737:−
20713:
20673:
20662:
20635:
20601:
20566:
20539:
20533:−
20509:
20474:
20468:−
20444:
20417:
20372:∫
20260:integrals
20201:θ
20152:−
20128:θ
20125:
20081:−
20065:θ
20062:
20012:θ
20009:
19955:θ
19932:θ
19917:
19865:
19852:−
19841:
19816:
19797:
19773:
19760:−
19745:
19729:−
19717:−
19711:
19689:
19676:−
19670:
19644:
19625:
19601:
19583:
19574:
19552:
19502:
19493:
19476:
19470:−
19464:
19442:−
19433:
19417:
19408:
19396:
19387:
19366:−
19355:
19339:
19330:
19324:−
19318:
19309:
19288:−
19277:
19233:
19224:
19218:−
19207:
19195:
19164:
19148:
19139:
19133:−
19127:
19118:
19086:
19070:
19061:
19049:
19040:
19008:
18954:
18935:
18890:
18865:
18829:
18796:
18753:
18734:
18679:
18665:π
18645:
18632:
18618:π
18598:
18585:
18571:π
18554:
18541:
18527:π
18510:
18497:
18483:π
18463:
18450:
18436:π
18416:
18349:
18330:−
18324:
18311:
18305:−
18289:−
18283:
18270:
18264:−
18248:−
18242:
18229:
18223:−
18207:−
18201:
18188:
18169:−
18163:
18150:
18144:−
18128:−
18122:
18046:∞
18028:
18010:→
17995:∞
17992:−
17977:
17967:−
17959:→
17923:π
17903:π
17874:
17854:
17836:
17808:∞
17805:−
17790:
17776:π
17772:→
17757:∞
17739:
17729:−
17725:π
17721:→
17685:π
17665:π
17642:−
17614:π
17611:±
17550:
17530:
17512:
17483:±
17455:⊂
17447:…
17435:π
17418:π
17405:π
17400:−
17388:π
17379:−
17373:…
17357:π
17346:π
17316:⊂
17308:…
17302:π
17293:π
17281:π
17278:−
17272:π
17266:−
17260:…
17244:π
17216:π
17213:±
17178:
17155:π
17140:ℜ
17134:π
17131:−
17076:
17041:
17021:π
17009:
16990:
16984:−
16967:π
16955:
16920:
16905:
16893:π
16881:
16872:π
16860:
16841:
16829:π
16817:
16791:
16772:π
16760:
16721:
16709:π
16697:
16678:
16666:π
16654:
16619:
16613:−
16604:π
16592:
16573:
16567:−
16558:π
16546:
16523:π
16503:π
16468:
16456:π
16441:
16422:
16410:π
16395:
16372:π
16330:
16295:
16258:
16223:
16186:
16151:
16071:
16062:
16053:−
16047:
16038:
16022:
16009:
16000:
15985:
15976:
15960:
15849:
15837:
15795:
15786:
15774:
15765:
15750:−
15741:
15688:π
15662:θ
15656:
15595:−
15564:π
15556:θ
15548:
15522:θ
15516:
15510:−
15480:−
15445:−
15423:−
15404:π
15396:θ
15388:
15353:∞
15350:−
15344:∈
15320:−
15314:
15300:π
15289:
15266:∞
15263:→
15237:τ
15231:−
15223:τ
15211:→
15208:τ
15179:τ
15167:τ
15148:−
15123:∫
15107:τ
15095:τ
15076:−
15072:∫
15062:
15050:
15016:π
15002:π
14999:−
14993:∈
14987:
14975:
14940:−
14917:
14905:
14893:
14867:π
14841:θ
14835:
14829:−
14820:π
14814:θ
14808:
14782:θ
14776:
14770:−
14761:π
14755:θ
14749:
14709:π
14703:
14677:π
14671:
14648:θ
14645:
14622:θ
14619:
14593:→
14577:−
14544:θ
14541:
14515:→
14499:−
14469:θ
14466:
14455:, and so
14443:∞
14440:→
14409:π
14406:→
14403:θ
14372:π
14358:π
14355:−
14326:
14317:θ
14294:θ
14252:−
14219:θ
14216:
14193:−
14163:θ
14160:
14144:θ
14141:
14115:
14106:θ
14078:π
14072:θ
14058:π
14055:−
14019:τ
14007:τ
13996:∞
13987:∫
13980:π
13950:π
13927:
13909:τ
13897:τ
13877:∫
13861:θ
13813:π
13708:π
13650:π
13561:→
13503:↦
13444:π
13435:
13391:→
13203:
13191:
13160:
13148:
13094:−
13057:
13028:−
13020:−
12991:
12944:
12935:−
12929:
12908:−
12893:
12878:
12463:
12448:
12384:
12369:
12314:θ
12283:θ
12277:
12251:θ
12245:
12203:∈
12179:π
12154:−
12132:−
12119:∞
12104:∏
12094:
12065:
12039:
12005:∈
11981:π
11954:−
11941:∞
11926:∏
11913:
11859:−
11817:∞
11802:∑
11786:π
11783:
11777:π
11740:−
11670:−
11662:∞
11647:∑
11637:π
11634:
11628:π
11571:∞
11566:∞
11563:−
11553:∑
11543:π
11540:
11521:π
11483:−
11455:−
11444:∞
11429:∑
11376:−
11365:∞
11360:∞
11357:−
11347:∑
11337:π
11334:
11328:π
11286:−
11265:∞
11250:∑
11221:π
11218:
11212:π
11133:−
11123:∑
11117:∞
11114:→
11097:π
11094:
11088:π
11018:⋱
11015:−
10964:−
10913:−
10862:−
10788:⋱
10785:−
10752:−
10719:−
10686:−
10654:
10608:⋱
10592:−
10586:⋅
10558:⋅
10531:−
10525:⋅
10497:⋅
10470:−
10464:⋅
10399:
10353:⋱
10337:−
10331:⋅
10303:⋅
10276:−
10270:⋅
10242:⋅
10215:−
10209:⋅
10144:
10099:π
10073:for
10065:⋯
10062:−
10039:−
10016:−
10000:−
9992:−
9969:−
9896:−
9885:∞
9870:∑
9856:
9814:π
9792:for
9784:⋯
9637:−
9626:∞
9611:∑
9555:∞
9540:∑
9526:
9489:π
9463:for
9455:⋯
9382:−
9359:−
9304:−
9296:−
9255:−
9244:∞
9229:∑
9215:
9173:π
9151:for
9143:⋯
9048:−
8993:−
8954:−
8940:−
8929:∞
8914:∑
8831:∞
8816:∑
8800:
8770:, is the
8688:π
8660:π
8529:−
8518:∞
8503:∑
8489:⋯
8461:−
8411:−
8395:
8327:−
8316:∞
8301:∑
8287:⋯
8259:−
8209:−
8193:
8084:
8056:
8038:
8019:
7997:
7953:
7939:
7927:
7897:π
7877:π
7744:
7738:−
7729:
7708:−
7699:
7644:
7638:−
7629:
7602:
7532:
7505:
7490:
7484:−
7475:
7445:
7433:
7300:−
7277:∑
7225:
7111:∘
7080:π
7011:−
6934:∘
6902:π
6788:∘
6757:π
6649:∘
6618:π
6503:∘
6472:π
6440:−
6359:−
6326:∘
6295:π
6205:∘
6145:θ
6139:
6112:θ
6106:
6079:θ
6073:
5984:cube root
5917:∘
5909:
5895:π
5890:
5849:∘
5841:
5827:π
5822:
5766:∘
5758:
5744:π
5739:
5685:∘
5677:
5663:π
5658:
5605:∘
5597:
5585:
5521:π
5512:θ
5506:
5497:θ
5494:
5466:π
5457:θ
5451:
5442:θ
5439:
5413:π
5385:π
5361:π
5338:π
5296:π
5284:θ
5276:
5267:θ
5264:
5234:π
5222:θ
5214:
5205:θ
5202:
5176:π
5149:π
5106:π
5100:±
5087:animation
5036:θ
5033:
5018:θ
5015:
5002:θ
4999:
4984:θ
4981:
4968:θ
4965:
4957:θ
4954:
4942:θ
4939:
4926:θ
4923:
4915:θ
4912:
4900:θ
4897:
4849:θ
4846:
4803:θ
4800:
4758:θ
4755:
4715:θ
4712:
4677:θ
4674:
4658:θ
4655:
4516:π
4513:≤
4510:θ
4507:≤
4463:θ
4460:
4420:θ
4417:
4168:at point
4090:at point
3954:θ
3925:θ
3849:π
3778:Mnemonics
3514:see below
3433:
3410:−
3405:∘
3392:
3380:
3353:θ
3350:
3330:θ
3327:−
3319:π
3309:
3300:θ
3297:
3265:cosecant
3243:
3220:−
3215:∘
3202:
3190:
3163:θ
3160:
3140:θ
3137:−
3129:π
3119:
3110:θ
3107:
3053:
3030:−
3025:∘
3012:
2997:
2986:
2971:
2944:θ
2941:
2921:θ
2918:−
2910:π
2900:
2888:θ
2885:
2877:θ
2874:
2862:θ
2859:
2805:
2782:−
2777:∘
2764:
2749:
2738:
2723:
2696:θ
2693:
2673:θ
2670:−
2662:π
2652:
2640:θ
2637:
2629:θ
2626:
2614:θ
2611:
2556:
2533:−
2528:∘
2515:
2503:
2475:θ
2472:
2452:θ
2449:−
2441:π
2431:
2422:θ
2419:
2365:
2342:−
2337:∘
2324:
2312:
2285:θ
2282:
2262:θ
2259:−
2251:π
2241:
2232:θ
2229:
2170:Function
2086:θ
2083:−
2078:∘
2067:
2041:θ
2035:
1915:θ
1912:
1899:cotangent
1822:θ
1819:
1721:θ
1718:
1622:θ
1619:
1521:θ
1518:
1422:θ
1419:
1162:−
1124:
1118:⋅
1115:θ
1086:θ
1083:
1057:
1049:−
1038:θ
1018::
1012:
976:
968:−
940:
932:−
895:−
830:∘
773:
764:
729:
723:⋅
711:
679:
646:
584:
543:
511:
445:real line
422:cotangent
265:de Moivre
199:Integrals
115:Reference
85:Functions
18:Cotangent
26657:Exsecant
26562:q-analog
26558:q-Cosine
26549:q-analog
26539:GonioLab
26489:76087042
26459:Archived
26326:(1929).
26262:65-12253
26246:64-60036
26192:Archived
26171:(1620).
26144:Archived
26008:Archived
25795:Archived
25769:45211959
25593:Archived
25559:Archived
25541:(1993).
25485:(1841).
25452:(1981).
25416:Archived
25408:Archived
25377:Archived
25355:(2000).
25320:(eds.),
25213:Archived
25205:Archived
25167:Archived
25117:67-20770
25109:Calculus
24936:Archived
24779:See also
24612:) = csc(
24577:) = sec(
24550:) = sin(
24420:and the
24418:exsecant
24298:(1583).
24258:analysis
24244:Rheticus
24242:(1464),
24165:koti-jyā
23880: (
23854: (
23462:incircle
21846:Function
20341:′
18082:calculus
16359:periodic
15039:, since
14608:. Thus
12840:One has
12536:One has
11201:series:
11181:Herglotz
8121:′
7835:′(0) = 0
7821:′(0) = 1
7777:″
6167:degrees
6164:radians
6010:th roots
5320:and any
4033:The ray
3787:tudents
3482:calculus
3279:opposite
3089:adjacent
2841:opposite
2835:adjacent
2593:adjacent
2587:opposite
2579:tangent
2395:adjacent
2205:opposite
1505:cosecant
1392:adjacent
1384:opposite
1072:implies
471:Notation
414:cosecant
367:geometry
245:al-Hasib
185:Calculus
164:Tangents
26652:Versine
26529:, 2001
26423:61-9103
26254:0167642
26107:Pearson
26071:Algebra
25029:1502474
24761:cosinus
24730:touches
24726:tangens
24722:tangent
24606:
24592:
24566:
24552:
24546:versin(
24544:
24532:
24515:
24501:
24484:
24470:
24460:versin(
24453:
24439:
24406:versine
24284:tangent
24264:. (See
24156:versine
24148:Ptolemy
24124:History
23464:, then
21314:is the
18981:Ptolemy
18375:Periods
8165:(0) = 0
7828:(0) = 1
7814:(0) = 0
6056:Angle,
6016:of the
5373:is the
5322:integer
5071:Tangent
3913:-axis (
3899:be the
3867:radians
3566:of the
3560:radians
3471:degrees
3282:
3270:
3092:
3080:
3075:secant
2844:
2832:
2596:
2584:
2404:
2392:
2387:cosine
2214:
2202:
2191:degrees
2185:radians
2158:Bottom:
2021:radians
2018:
2006:
1806:tangent
1368:similar
1268:
1252:
1241:
1225:
1214:
1198:
700:denote
406:tangent
383:geodesy
327:similar
275:Fourier
235:Ptolemy
201: (
159:Cosines
101:inverse
87: (
73:History
68:Outline
26713:Ratios
26607:Groups
26545:q-Sine
26487:
26421:
26401:
26380:
26353:
26315:
26294:
26260:
26252:
26244:
26234:
26113:
25972:
25930:
25855:
25775:
25767:
25683:
25614:
25585:
25551:
25512:
25400:
25369:
25332:
25254:press.
25197:
25140:
25115:
25092:
25084:
25027:
25017:
24928:
24738:secans
24734:secant
24586:excsc(
24573:exsec(
24424:. The
24416:, the
24412:, the
24404:, the
24395:cosec.
24393:, and
24313:, and
24288:secant
24144:Nicaea
23993:square
23090:where
22997:where
22875:, and
22797:arccos
22793:arcsin
22620:arccsc
22455:arcsec
22318:arccot
22164:arctan
22027:arccos
21873:arcsin
21852:Domain
21302:arsinh
21294:where
21261:arsinh
21143:arsinh
18844:gives
18092:Parity
16535:, and
15311:arctan
15286:arctan
15278:gives
15059:arctan
15047:arctan
14984:arctan
14972:arctan
14914:arctan
14902:arctan
14890:arctan
14323:arctan
14112:arctan
13924:arctan
13853:, put
12618:. The
12616:= 1, 2
12415:: Let
8753:, the
8736:, the
7526:
7499:
7454:
7185:Graphs
6022:cyclic
5134:, and
5067:Cosine
4806:
4395:- and
4265:- and
3870:(90°),
3825:circle
3780:like "
3622:= (180
2189:using
2183:using
2144:, and
1705:secant
1606:cosine
1390:, and
1342:, and
1322:, and
1009:arcsin
624:, not
453:radius
441:domain
418:secant
416:, the
402:cosine
400:, the
357:) are
341:, the
133:Tables
26693:Angle
26667:atan2
26645:Other
26498:from
26391:" to
26047:sinus
25773:S2CID
25765:JSTOR
25090:S2CID
24829:Notes
24713:χορδή
24676:j-y-b
24670:into
24651:sinus
24647:Latin
24616:) − 1
24581:) − 1
24402:chord
24383:tang.
24136:chord
23844:waves
23796:of a
20290:is a
18068:Many
13488:(the
13242:Most
13130:When
12773:, as
12413:Proof
11075:poles
9437:15120
8761:, and
8630:poles
6060:, in
4228:, is
3799:ine,
3558:When
3463:angle
2197:sine
1147:could
270:Euler
260:Viète
154:Sines
78:Usage
26485:LCCN
26419:LCCN
26399:ISBN
26378:ISBN
26351:ISBN
26313:ISBN
26292:ISBN
26258:LCCN
26242:LCCN
26232:ISBN
26137:See
26111:ISBN
26099:See
25970:ISBN
25928:ISBN
25853:ISBN
25681:ISBN
25612:ISBN
25583:ISBN
25549:ISBN
25510:ISBN
25398:ISBN
25367:ISBN
25330:ISBN
25195:ISBN
25138:ISBN
25113:LCCN
25082:ISSN
25047:link
25043:link
25025:OCLC
25015:ISBN
24926:ISBN
24742:cuts
24694:jīvā
24666:and
24660:jaib
24656:toga
24642:sine
24530:) =
24433:crd(
24391:sec.
24387:cot.
24379:cos.
24375:sin.
24330:sin
24286:and
24268:and
24163:and
22887:and
22795:and
22700:>
22683:<
22535:>
22518:<
22423:<
22417:<
22390:<
22384:<
22279:<
22273:<
22236:<
22230:<
21230:<
21224:<
21202:for
21091:<
21085:<
20274:The
20262:and
18908:and
17152:<
17137:<
16068:sinh
16044:cosh
16006:sinh
15982:cosh
15877:<
15871:<
15855:<
15843:<
15828:<
15537:and
14797:and
14634:and
14530:and
14075:<
14069:<
13490:base
12611:for
12481:and
12266:and
10096:<
10080:<
9809:<
9486:<
9470:<
9168:<
8597:The
7830:and
7816:and
7659:and
6020:are
5567:The
5559:The
5482:and
5401:and
5252:and
5063:Sine
4834:and
4743:and
4448:and
4387:and
4372:The
4320:and
4220:The
4062:line
3957:<
3928:>
3875:Let
3839:and
3791:ake
3763:sec
3761:and
3756:cot
3749:cos
3747:are
3743:and
3730:csc
3728:and
3723:tan
3716:sin
3714:are
3710:and
3632:sin
3609:cos
3602:sin
3591:cos
3584:sin
3572:turn
3551:and
3536:and
3500:and
3490:real
3484:and
2113:sin
2109:Top:
2056:and
1406:sine
1366:are
1344:Cot(
1332:Cos(
1316:Tan(
1308:Sin(
1300:Csc(
1292:Sec(
1246:tan
1219:cos
1192:sin
1107:not
955:and
753:not
661:and
494:sin(
398:sine
26073:of
26060:of
26031:'s
25962:doi
25920:doi
25757:doi
25074:doi
24755:'s
24749:co-
24702:jyā
24682:جيب
24397:).
24339:of
24315:tan
24311:cos
24307:sin
24272:.)
24191:.)
24161:jyā
24150:of
24142:of
24044:sin
23868:by
23813:= 1
23711:cot
23673:cot
23635:cot
23572:cot
23440:If
23360:tan
23334:tan
23211:cos
23182:cos
23057:sin
23036:sin
23015:sin
22944:sin
22923:sin
22902:sin
22803:".
22789:cos
22785:sin
22647:csc
22482:sec
22345:cot
22191:tan
22054:cos
21900:sin
21770:csc
21728:sec
21692:tan
21650:cot
21627:cot
21618:csc
21583:tan
21554:sec
21519:sec
21477:csc
21454:sin
21419:cos
21384:sin
21342:cos
21270:tan
21253:as
21184:sec
21152:cot
21114:csc
21040:sin
20996:csc
20965:cot
20925:tan
20913:sec
20875:tan
20866:sec
20839:sec
20799:cot
20787:csc
20749:cot
20740:csc
20710:csc
20670:sec
20626:sec
20598:tan
20563:sin
20536:sin
20506:cos
20471:cos
20441:cos
20414:sin
20122:tan
20059:cos
20006:sin
19971:of
19914:tan
19856:tan
19838:tan
19813:tan
19788:tan
19764:tan
19736:sin
19702:cos
19680:sin
19661:cos
19641:cos
19616:tan
19598:tan
19580:cos
19571:sin
19549:sin
19499:tan
19490:tan
19473:tan
19461:tan
19430:tan
19414:sin
19405:sin
19393:cos
19384:cos
19352:cos
19336:sin
19327:cos
19315:cos
19306:sin
19274:sin
19230:tan
19221:tan
19204:tan
19192:tan
19161:tan
19145:sin
19136:sin
19124:cos
19115:cos
19083:cos
19067:sin
19058:cos
19046:cos
19037:sin
19005:sin
18991:Sum
18945:csc
18926:cot
18881:sec
18856:tan
18820:sin
18811:or
18787:cos
18744:cos
18725:sin
18676:sec
18642:sec
18629:csc
18595:csc
18582:cot
18551:cot
18538:tan
18507:tan
18494:cos
18460:cos
18447:sin
18413:sin
18346:sec
18321:sec
18308:csc
18280:csc
18267:cot
18239:cot
18226:tan
18198:tan
18185:cos
18160:cos
18147:sin
18119:sin
18025:cot
18003:lim
17974:cot
17952:lim
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17833:cot
17787:tan
17765:lim
17736:tan
17714:lim
17547:cos
17527:sin
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17175:cos
17073:sin
17038:cos
17006:sin
16987:sin
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16917:sin
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16838:sin
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16675:tan
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16616:sin
16589:sin
16570:cos
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16419:cos
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16327:csc
16292:sec
16255:cot
16220:tan
16183:cos
16148:sin
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16035:cos
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15973:sin
15957:sin
15846:sin
15834:cos
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15738:cos
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14832:sin
14805:sin
14773:cos
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14700:sin
14668:cos
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14616:cos
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14213:sin
14157:cos
14138:tan
13676:360
13432:exp
13188:sin
13145:cos
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12988:sin
12941:sin
12926:cos
12890:sin
12875:cos
12460:sin
12445:cos
12381:sin
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12274:sin
12242:cos
12091:cos
12062:sin
12036:cot
11910:sin
11780:tan
11631:sec
11531:csc
11331:csc
11215:cot
11107:lim
11091:cot
10651:tan
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10141:sin
10047:945
9853:cot
9766:720
9523:sec
9414:360
9212:csc
9125:315
8797:tan
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8757:th
8740:th
8392:cos
8190:sin
8075:tan
8047:cos
8029:sin
8010:cos
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7968:,
7950:cos
7936:sin
7924:tan
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7626:cos
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7529:cos
7502:sin
7487:sin
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7430:sin
7222:cos
6136:tan
6103:cos
6070:sin
5906:sin
5887:sin
5838:sin
5819:sin
5755:sin
5736:sin
5674:sin
5655:sin
5594:sin
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5491:cot
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5436:tan
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5261:cos
5211:sin
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4996:cos
4978:sec
4962:sin
4951:cos
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4920:cos
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4752:cot
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4232:to
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3389:sec
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3306:sec
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3199:csc
3187:sec
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3116:csc
3104:sec
3050:tan
3009:tan
2994:sin
2983:cos
2968:cot
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2882:sin
2871:cos
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2802:cot
2761:cot
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2735:sin
2720:tan
2690:cot
2649:cot
2634:cos
2623:sin
2608:tan
2553:sec
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2500:cos
2469:sec
2428:sin
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2321:cos
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2226:sin
2064:cos
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2003:or
2001:90°
1909:cot
1816:tan
1715:sec
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1515:csc
1416:sin
1121:sin
1080:sin
1045:sin
964:sin
928:sin
770:sin
761:sin
726:sin
708:sin
670:sin
637:sin
581:sin
540:sin
508:sin
482:arc
353:or
337:In
97:tan
93:cos
89:sin
26684::
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26519:,
26483:,
26465:,
26445:,
26432:,
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26307:,
26271:,
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26250:MR
26248:.
26240:.
26220:;
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25968:.
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25926:.
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25035:{{
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24971:.
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24920:/
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23835:.
23780:A
23466:rs
23452:+
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23126::
23098:.
22891::
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22849:,
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22810:.
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21318:.
21029:ln
20902:ln
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20659:ln
20294:.
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19897:.
19534:.
18987:.
15880:1.
15720:.
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3642:=
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3604:x°
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3401:90
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616:A
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23667:=
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23557:.
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23399:b
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23387:=
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23372:+
23369:A
23353:2
23349:B
23343:A
23296:c
23292:C
23276:.
23270:b
23267:a
23264:2
23257:2
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23244:2
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23236:+
23231:2
23227:a
23220:=
23217:C
23191:,
23188:C
23179:b
23176:a
23173:2
23165:2
23161:b
23157:+
23152:2
23148:a
23144:=
23139:2
23135:c
23092:R
23078:,
23075:R
23072:2
23069:=
23063:C
23053:c
23048:=
23042:B
23032:b
23027:=
23021:A
23011:a
22999:Δ
22985:,
22979:c
22976:b
22973:a
22965:2
22959:=
22954:c
22950:C
22938:=
22933:b
22929:B
22917:=
22912:a
22908:A
22889:C
22885:B
22881:A
22877:c
22873:b
22869:a
22851:c
22847:b
22843:a
22839:C
22835:B
22831:A
22766:0
22760:y
22756:,
22751:2
22740:y
22732:2
22703:1
22697:x
22689:1
22680:x
22659:x
22656:=
22653:y
22626:x
22617:=
22614:y
22589:2
22578:y
22574:,
22565:y
22559:0
22538:1
22532:x
22524:1
22515:x
22494:x
22491:=
22488:y
22461:x
22452:=
22449:y
22420:y
22414:0
22387:x
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22351:y
22324:x
22315:=
22312:y
22287:2
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22268:2
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22170:x
22161:=
22158:y
22129:y
22123:0
22102:1
22096:x
22090:1
22066:x
22063:=
22060:y
22033:x
22024:=
22021:y
21996:2
21985:y
21977:2
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21936:1
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21906:y
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21870:=
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21782:x
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21763:=
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21683:d
21679:d
21674:=
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21656:x
21647:d
21637:,
21633:x
21624:x
21612:=
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21600:2
21596:/
21589:(
21580:)
21577:x
21571:2
21567:/
21560:(
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21542:x
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21464:,
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21425:(
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21407:x
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21390:(
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19768:2
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19634:,
19628:x
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19467:x
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19445:y
19439:x
19436:(
19423:,
19420:y
19411:x
19402:+
19399:y
19390:x
19381:=
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19363:x
19359:(
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19342:y
19333:x
19321:y
19312:x
19303:=
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19215:1
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19090:(
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19073:y
19064:x
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18737:x
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18568:k
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18560:x
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18430:2
18425:+
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18419:(
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18387:π
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18131:x
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18034:x
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17983:x
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17877:(
17867:/
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17860:z
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17839:(
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17760:,
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17539:)
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17050:)
17047:x
17044:(
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17025:/
17018:+
17015:x
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17002:,
16999:)
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16961:x
16958:(
16932:.
16929:)
16926:z
16923:(
16914:)
16911:z
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16890:+
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16875:)
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16850:)
16847:z
16844:(
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16826:+
16823:z
16820:(
16812:2
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16800:)
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16660:z
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16631:.
16628:)
16625:z
16622:(
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16601:+
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16585:,
16582:)
16579:z
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16477:)
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16434:,
16431:)
16428:z
16425:(
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15868:0
15858:x
15852:x
15840:x
15831:x
15825:0
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15789:x
15780:+
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15768:x
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15747:x
15744:(
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15668:.
15665:)
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15525:)
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15439:+
15436:1
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15417:=
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14817:+
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14320:=
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13730:d
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13587:(
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13509:(
13506:e
13500:t
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13450:t
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13423:t
13420:(
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13397:.
13394:U
13387:Z
13382:/
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13373::
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13349:Z
13344:/
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13317:U
13288:b
13284:e
13278:a
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13270:=
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13262:+
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13255:e
13227:.
13223:)
13218:x
13215:i
13211:e
13207:(
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13184:,
13180:)
13175:x
13172:i
13168:e
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13132:x
13111:.
13106:2
13100:x
13097:i
13090:e
13086:+
13081:x
13078:i
13074:e
13067:=
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13044:i
13041:2
13034:x
13031:i
13024:e
13015:x
13012:i
13008:e
13001:=
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12950:.
12947:x
12938:i
12932:x
12923:=
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12911:i
12904:e
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12887:i
12884:+
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12822:=
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12800:=
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12754:x
12751:(
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12737:/
12733:)
12730:x
12727:(
12722:1
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12688:)
12685:x
12682:(
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12668:/
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12661:x
12658:(
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12638:d
12634:/
12630:d
12614:j
12599:)
12596:x
12593:(
12588:j
12584:f
12580:i
12577:=
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12571:d
12567:/
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12560:x
12557:(
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12548:f
12544:d
12524:.
12519:x
12516:i
12512:e
12508:=
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12502:x
12499:(
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12490:f
12469:,
12466:x
12457:i
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12442:=
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12436:x
12433:(
12428:1
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12407:x
12390:.
12387:x
12378:i
12375:+
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12363:=
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12351:e
12311:i
12307:e
12286:)
12280:(
12254:)
12248:(
12211:.
12207:C
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12196:,
12192:)
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12161:/
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12151:n
12148:(
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12129:1
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12068:z
12042:z
12013:.
12009:C
12002:z
11998:,
11994:)
11985:2
11975:2
11971:n
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11960:z
11951:1
11947:(
11936:1
11933:=
11930:n
11922:z
11919:=
11916:z
11875:.
11867:2
11863:x
11854:2
11850:)
11843:2
11840:1
11834:+
11831:n
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11824:1
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11809:=
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11795:2
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11789:x
11756:,
11748:2
11744:x
11735:2
11731:)
11724:2
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11715:+
11712:n
11709:(
11704:)
11701:1
11698:+
11695:n
11692:2
11689:(
11681:n
11677:)
11673:1
11667:(
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11654:=
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11640:x
11607:,
11599:2
11595:)
11591:n
11588:+
11585:x
11582:(
11578:1
11560:=
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11549:=
11546:x
11535:2
11525:2
11499:,
11491:2
11487:n
11478:2
11474:x
11466:n
11462:)
11458:1
11452:(
11439:1
11436:=
11433:n
11425:x
11422:2
11419:+
11414:x
11411:1
11406:=
11400:n
11397:+
11394:x
11387:n
11383:)
11379:1
11373:(
11354:=
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11343:=
11340:x
11302:.
11294:2
11290:n
11281:2
11277:x
11272:1
11260:1
11257:=
11254:n
11246:x
11243:2
11240:+
11235:x
11232:1
11227:=
11224:x
11194:n
11189:)
11187:n
11164:.
11158:n
11155:+
11152:x
11148:1
11141:N
11136:N
11130:=
11127:n
11111:N
11103:=
11100:x
11007:x
10995:7
10975:1
10956:x
10944:5
10924:1
10905:x
10893:3
10873:1
10854:x
10842:1
10822:1
10811:=
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10768:2
10764:x
10749:5
10735:2
10731:x
10716:3
10702:2
10698:x
10683:1
10671:x
10660:=
10657:x
10605:+
10600:2
10596:x
10589:6
10583:5
10569:2
10565:x
10561:4
10555:3
10544:+
10539:2
10535:x
10528:4
10522:3
10508:2
10504:x
10500:2
10494:1
10483:+
10478:2
10474:x
10467:2
10461:1
10447:2
10443:x
10431:+
10428:1
10416:1
10405:=
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10350:+
10345:2
10341:x
10334:7
10328:6
10314:2
10310:x
10306:5
10300:4
10289:+
10284:2
10280:x
10273:5
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10249:x
10245:3
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10228:+
10223:2
10219:x
10212:3
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10192:2
10188:x
10176:+
10173:1
10161:x
10150:=
10147:x
10102:.
10092:|
10088:x
10084:|
10077:0
10068:,
10057:5
10053:x
10044:2
10034:3
10030:x
10021:1
10013:x
10008:3
10005:1
9995:1
9988:x
9984:=
9972:1
9966:n
9963:2
9959:x
9952:!
9949:)
9946:n
9943:2
9940:(
9933:n
9930:2
9926:B
9920:n
9917:2
9913:2
9907:n
9903:)
9899:1
9893:(
9880:0
9877:=
9874:n
9866:=
9859:x
9822:.
9817:2
9805:|
9801:x
9797:|
9787:,
9781:+
9776:6
9772:x
9758:+
9753:4
9749:x
9740:5
9735:+
9730:2
9726:x
9720:2
9717:1
9712:+
9709:1
9706:=
9694:n
9691:2
9687:x
9680:!
9677:)
9674:n
9671:2
9668:(
9661:n
9658:2
9654:E
9648:n
9644:)
9640:1
9634:(
9621:0
9618:=
9615:n
9607:=
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9599:2
9595:x
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9585:)
9582:n
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9576:(
9570:n
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9563:U
9550:0
9547:=
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9536:=
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9482:|
9478:x
9474:|
9467:0
9458:,
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9429:+
9424:3
9420:x
9411:7
9406:+
9403:x
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9385:1
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9374:=
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9356:n
9353:2
9349:x
9342:!
9339:)
9336:n
9333:2
9330:(
9323:n
9320:2
9316:B
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9281:(
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9236:=
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9225:=
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9112:5
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9099:2
9094:+
9089:3
9085:x
9079:3
9076:1
9071:+
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9065:=
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9045:n
9042:2
9038:x
9031:!
9028:)
9025:n
9022:2
9019:(
9012:n
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8988:n
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8976:(
8970:n
8967:2
8963:2
8957:1
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8947:)
8943:1
8937:(
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8532:1
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8225:3
8219:3
8215:x
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8091:,
8087:x
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8065:=
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8033:2
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8022:x
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8000:x
7988:x
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7946:/
7942:x
7933:=
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7826:y
7819:y
7812:y
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7787:=
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7635:=
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7620:x
7617:d
7613:d
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7583:d
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7544:=
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7538:0
7535:(
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7520:0
7517:=
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7511:0
7508:(
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7493:x
7481:=
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7466:x
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7332:(
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5860:3
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5786:=
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5626:=
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5545:k
5541:θ
5524:)
5518:k
5515:+
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5500:=
5469:)
5463:k
5460:+
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5399:B
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5325:k
5318:θ
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5287:+
5280:(
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5225:+
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5120:A
5103:2
5042:.
5026:1
5021:=
5008:,
4992:1
4987:=
4974:,
4945:=
4932:,
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4861:E
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4852:=
4818:D
4813:y
4809:=
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4770:C
4765:x
4761:=
4727:B
4722:y
4718:=
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4618:)
4615:y
4612:,
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4566:y
4562:+
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4524:2
4520:/
4504:0
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4470:y
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4340:x
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4116:y
4112:,
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4075:=
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4048:,
4043:L
4021:.
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3992:x
3988:(
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3960:0
3934:,
3931:0
3910:x
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3837:0
3829:O
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3672:.
3670:O
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3648:π
3644:π
3640:x
3634:π
3628:π
3626:/
3624:x
3620:x
3616:x
3598:x
3593:x
3586:x
3580:x
3576:π
3436:x
3426:1
3421:=
3417:)
3413:x
3396:(
3386:=
3383:x
3343:1
3338:=
3334:)
3322:2
3313:(
3303:=
3276:/
3246:x
3236:1
3231:=
3227:)
3223:x
3206:(
3196:=
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3153:1
3148:=
3144:)
3132:2
3123:(
3113:=
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3056:x
3046:1
3041:=
3037:)
3033:x
3016:(
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3000:x
2989:x
2977:=
2974:x
2934:1
2929:=
2925:)
2913:2
2904:(
2894:=
2865:=
2838:/
2808:x
2798:1
2793:=
2789:)
2785:x
2768:(
2758:=
2752:x
2741:x
2729:=
2726:x
2686:1
2681:=
2677:)
2665:2
2656:(
2646:=
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2590:/
2559:x
2549:1
2544:=
2540:)
2536:x
2519:(
2509:=
2506:x
2465:1
2460:=
2456:)
2444:2
2435:(
2425:=
2398:/
2368:x
2358:1
2353:=
2349:)
2345:x
2328:(
2318:=
2315:x
2275:1
2270:=
2266:)
2254:2
2245:(
2235:=
2208:/
2152:θ
2148:π
2146:2
2141:θ
2137:π
2131:θ
2127:π
2121:θ
2115:θ
2089:)
2070:(
2044:)
2038:(
2015:2
2012:/
2009:π
1971:e
1968:t
1965:i
1962:s
1959:o
1956:p
1953:p
1950:o
1945:t
1942:n
1939:e
1936:c
1933:a
1930:j
1927:d
1924:a
1918:=
1878:t
1875:n
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1869:c
1866:a
1863:j
1860:d
1857:a
1852:e
1849:t
1846:i
1843:s
1840:o
1837:p
1834:p
1831:o
1825:=
1783:t
1780:n
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1774:c
1771:a
1768:j
1765:d
1762:a
1757:e
1754:s
1751:u
1748:n
1745:e
1742:t
1739:o
1736:p
1733:y
1730:h
1724:=
1684:e
1681:s
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1660:y
1657:h
1652:t
1649:n
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1643:c
1640:a
1637:j
1634:d
1631:a
1625:=
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1562:o
1557:e
1554:s
1551:u
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1545:e
1542:t
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1536:p
1533:y
1530:h
1524:=
1484:e
1481:s
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1475:n
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1460:y
1457:h
1452:e
1449:t
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1443:s
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1437:p
1434:p
1431:o
1425:=
1396:θ
1388:θ
1376:θ
1372:θ
1364:θ
1360:θ
1352:x
1348:)
1346:θ
1340:1
1336:)
1334:θ
1328:x
1324:1
1320:)
1318:θ
1312:)
1310:θ
1304:)
1302:θ
1296:)
1294:θ
1288:1
1282:θ
1271:.
1264:b
1260:/
1256:a
1248:A
1237:c
1233:/
1229:b
1221:A
1210:c
1206:/
1202:a
1194:A
1165:1
1130:=
1127:x
1095:,
1092:x
1089:=
1060:x
1052:1
1041:=
1015:x
985:)
982:x
979:(
971:1
943:x
935:1
898:1
872:.
869:)
866:)
863:x
860:(
857:f
854:(
851:f
848:=
845:)
842:x
839:(
836:)
833:f
827:f
824:(
821:=
818:)
815:x
812:(
807:2
803:f
782:.
779:)
776:x
767:(
741:,
738:)
735:x
732:(
720:)
717:x
714:(
688:)
685:x
682:(
674:2
649:x
641:2
602:.
599:)
596:y
593:+
590:x
587:(
561:,
558:y
555:+
552:)
549:x
546:(
520:y
517:+
514:x
498:)
496:x
333:.
302:e
295:t
288:v
205:)
103:)
41:.
34:.
20:)
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