12163:
981:
9762:
11961:
11952:
683:
11970:
10233:
11909:
11900:
1723:
11918:
10700:
63:
3774:
3634:
5685:
394:
12369:
8863:
976:{\displaystyle {\begin{aligned}\tan(\theta )&={\frac {\sin(\theta )}{\cos(\theta )}}={\frac {\text{opposite}}{\text{adjacent}}},\\\cot(\theta )&={\frac {1}{\tan(\theta )}}={\frac {\text{adjacent}}{\text{opposite}}},\\\csc(\theta )&={\frac {1}{\sin(\theta )}}={\frac {\text{hypotenuse}}{\text{opposite}}},\\\sec(\theta )&={\frac {1}{\cos(\theta )}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}.\end{aligned}}}
10254:
14706:
10218:
7105:
6761:
11098:
7265:
7098:
6517:
10695:{\displaystyle {\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}}
7959:
8176:
170:
2549:
6606:
3629:
8848:
9124:
5626:
7653:
12794:
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution. If half-turns are used as the unit for angle, then the value provided
14456:
It was Robert of
Chester's translation from Arabic that resulted in our word "sine". The Hindus had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language, there is also the word jaib meaning "bay" or "inlet". When Robert of Chester
12736:
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled
12431:, the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.
6245:
679:
of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite
6859:
6270:
9487:
10722:
2980:
9280:
662:
11280:
7735:
12795:
by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to
10085:
7964:
9752:
2371:
5402:
7255:
The graph shows both sine and sine squared functions, with the sine in blue and the sine squared in red. Both graphs have the same shape but with different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods.
12434:
Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g.
7253:
3491:
8654:
80:
8963:
583:
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:
5473:
11443:
3928:
of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero. The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign
7495:
11797:
12293:, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
6173:
6756:{\displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}}
2047:
9375:
11553:
1958:
3328:
12254:. With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant.
9135:
3910:
8629:
8446:
587:
11093:{\displaystyle {\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\end{aligned}}}
11115:
9975:
5215:
9606:
12442:
A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or
3015:
Using the unit circle definition has the advantage of drawing a graph of sine and cosine functions. This can be done by rotating counterclockwise a point along the circumference of a circle, depending on the input
5288:
1171:
7093:{\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin(\theta )\cos(\theta ),\\\cos(2\theta )&=\cos ^{2}(\theta )-\sin ^{2}(\theta )\\&=2\cos ^{2}(\theta )-1\\&=1-2\sin ^{2}(\theta )\end{aligned}}}
6512:{\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}}
14166:
12447:
between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.
11655:
7124:
12662:
The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing
6601:
85:
14457:
came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay" or "inlet".
6852:
3195:
Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the
2866:
12111:
over the breast'). Gerard was probably not the first scholar to use this translation; Robert of
Chester appears to have preceded him and there is evidence of even earlier usage. The English form
10727:
10259:
9980:
9611:
9380:
9140:
8968:
8659:
7969:
7740:
6864:
6611:
6275:
3496:
2376:
688:
9970:
2158:
7954:{\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}\end{aligned}}}
4522:
4371:
4570:
4225:
11329:
4416:
5440:
4265:
4079:
2969:
2907:
1210:
12595:
radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases. In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these functions are called
8171:{\displaystyle {\begin{aligned}\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}}}
165:{\displaystyle {\begin{aligned}&\sin(\theta )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\&\cos(\theta )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\\end{aligned}}}
5005:
4827:
2544:{\displaystyle {\begin{aligned}\sin(\theta )&={\frac {|\mathbb {a} \times \mathbb {b} |}{|a||b|}},\\\cos(\theta )&={\frac {\mathbb {a} \cdot \mathbb {b} }{|a||b|}}.\end{aligned}}}
12458:
The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to
4953:
4775:
4673:
4119:
11873:
11324:
4723:
3912:
Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself. These derivatives can be applied to the
1963:
3073:
1618:
1521:
1488:
1408:
14738:
13146:
11450:
1883:
3225:
2909:
because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the
12820:
6008:
6098:
5967:
3624:{\displaystyle {\begin{aligned}\sin(\theta )&=\cos \left({\frac {\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}}
2230:
12789:
12762:
12731:
9597:
6125:
3486:
3190:
2192:
9826:
9794:
8843:{\displaystyle {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\cos(nx)\,dx,\\B_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\sin(nx)\,dx.\end{aligned}}}
5891:
3792:
3151:
2800:
2768:
2643:
1052:
1020:
318:
286:
13096:
12643:
10207:
9856:
9370:
8511:
8302:
5929:
5850:
3122:
3040:
2366:
2344:
2302:
2280:
12733:
in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.
11674:
9899:
3404:
1104:
358:, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the
13044:
13014:
6064:
5770:
5740:
3455:
5812:
3757:
3715:
10125:
9119:{\displaystyle {\begin{aligned}\sin(\theta )&={\frac {e^{i\theta }-e^{-i\theta }}{2i}},\\\cos(\theta )&={\frac {e^{i\theta }+e^{-i\theta }}{2}},\end{aligned}}}
7420:
7385:
10168:
8950:
7452:
7350:
7318:
5670:
5646:
2667:
2322:
1878:
1838:
1072:
469:
449:
425:
254:
14163:
9524:
9316:
7488:
5075:
5040:
4900:
4865:
1858:
13228:
12684:
8506:
8479:
8248:
8221:
2597:
12704:
12593:
8297:
6165:
5621:{\displaystyle L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi \approx 7.6404\ldots }
5098:
1212:. The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator.
12328:(1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "
7679:
1758:
495:
14387:, which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word
10145:
10105:
9564:
9544:
9336:
8649:
8268:
7726:
7706:
6265:
6145:
6032:
5464:
5283:
5263:
5235:
4618:
4591:
4464:
4437:
4313:
4286:
4167:
4140:
3969:) is decreasing (going downward)—in certain intervals. This information can be represented as a Cartesian coordinates system divided into four quadrants.
3967:
3947:
3350:
3214:
3094:
2928:
2734:
2711:
2688:
2067:
1818:
1798:
1778:
577:
550:
523:
7648:{\displaystyle \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}}
5241:. These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the
14341:
12259:
6240:{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right),}
14731:
4906:
of the differential equation with the given initial conditions. It can be interpreted as a phase space trajectory of the system of differential equations
14369:. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word
6780:, the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the
9482:{\displaystyle {\begin{aligned}\sin \theta &=\operatorname {Im} (e^{i\theta }),\\\cos \theta &=\operatorname {Re} (e^{i\theta }).\end{aligned}}}
13916:
14724:
6522:
17:
14033:
12262:(853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.
220:: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the
9275:{\displaystyle {\begin{aligned}e^{i\theta }&=\cos(\theta )+i\sin(\theta ),\\e^{-i\theta }&=\cos(\theta )-i\sin(\theta ).\end{aligned}}}
6787:
3789:
The sine and cosine functions are infinitely differentiable. The derivative of sine is cosine, and the derivative of cosine is negative sine:
14045:
2805:
657:{\displaystyle \sin(\alpha )={\frac {\text{opposite}}{\text{hypotenuse}}},\qquad \cos(\alpha )={\frac {\text{adjacent}}{\text{hypotenuse}}}.}
3920:
of a function can be defined as the inequality of function's first derivative greater or less than equal to zero. It can also be applied to
13832:
12090:
5086:
1109:
11275:{\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}}
3457:. The sine function is odd, whereas the cosine function is even. Both sine and cosine functions are similar, with their difference being
9904:
2083:
12837:
10716:
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:
3768:
11580:
10080:{\displaystyle {\begin{aligned}\operatorname {Re} (z)&=r\cos(\theta ),\\\operatorname {Im} (z)&=r\sin(\theta ),\end{aligned}}}
5443:
1734:
is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. Given that a triangle
9747:{\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}}}
12560:
12499:
12491:
5397:{\displaystyle \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} \left(t,{\frac {1}{\sqrt {2}}}\right),}
3975:
14647:
14627:
14604:
14522:
14497:
14430:
14309:
14272:
14151:
13968:
13466:
14807:
12206:
12037:
12036:'string'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see
359:
14244:
13399:
675:
is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The
7121:
The cosine double angle formula implies that sin and cos are, themselves, shifted and scaled sine waves. Specifically,
14576:
12908:
7288:
of sine is cosine and that the derivative of cosine is the negative of sine. This means the successive derivatives of
7268:
This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.
7248:{\displaystyle \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}}
6781:
4481:
4330:
4528:
4184:
14552:
14331:
14057:
14014:
13589:
12408:
8902:
4377:
5407:
4231:
4038:
2564:
14838:
13960:
12882:
12386:
8880:
6771:
2935:
2873:
2253:
1176:
12832:
14771:
12948:
7655:
where the superscript represents repeated differentiation. This implies the following Taylor series expansion at
4634:
4085:
2080:
is useful for computing the length of an unknown side if two other sides and an angle are known. The law states,
14657:
14413:
Merlet, Jean-Pierre (2004), "A Note on the
History of the Trigonometric Functions", in Ceccarelli, Marco (ed.),
13347:
12480:
are typically either a built-in function or found within the language's standard math library. For example, the
14710:
12390:
8884:
13711:
11820:
12867:
12525:
11285:
4958:
4780:
4678:
1173:. The following table shows the special value of each input for both sine and cosine with the domain between
14562:
14139:
13485:
4909:
4731:
2932:
coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when
1594:
1497:
1464:
1384:
14781:
13267:
13248:
12469:
Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.
12002:
3045:
14115:
14103:
13926:
13656:
13537:
13101:
9342:
in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of
14468:
13727:
13295:
13200:
13179:
8189:
3782:
2646:
672:
11438:{\displaystyle {\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.}
14682:
14489:
14474:
14406:
12918:
12877:
403:, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.
12798:
5972:
14030:
13695:"Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?"
5934:
3672:
2197:
12737:
fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo
11792:{\displaystyle \zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).}
9569:
7544:
6073:
3160:
2163:
12938:
12857:
12847:
12767:
12740:
12709:
12553:
12521:
12485:
9799:
9767:
8185:
6103:
5855:
3464:
3127:
2773:
2741:
2602:
1025:
993:
291:
259:
14042:
13357:
13062:
12610:
9831:
9345:
5896:
5817:
3101:
3019:
2349:
2327:
2285:
2263:
14759:
12953:
12892:
12842:
12571:
Some software libraries provide implementations of sine and cosine using the input angle in half-
12379:
12179:
12162:
12157:
10228:) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.
10173:
9872:
8873:
5238:
3359:
2260:. The sine and cosine functions can be defined in terms of the cross product and dot product. If
1085:
35:
14393:, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf.
13987:
13560:
13019:
12989:
8454:
can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let
6037:
5745:
5715:
3413:
2870:
This definition is consistent with the right-angled triangle definition of sine and cosine when
2042:{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R,}
13952:
12958:
12286:
12199:
12028:
5779:
4903:
3921:
3724:
3682:
3458:
2257:
676:
205:
14088:
13721:
13495:
13412:
13277:
13261:
10110:
7390:
7355:
5776:, their inverses are not exact inverse functions, but partial inverse functions. For example,
3153:, the point returned to its origin. This results that both sine and cosine functions have the
14568:
14196:
13737:
13669:
13602:
13547:
13305:
13242:
13238:
13210:
13194:
11811:
11665:
11548:{\displaystyle \sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).}
10153:
8935:
7425:
7323:
7291:
5655:
5631:
4726:
3913:
3222:, meaning any angle added by the circumference's circle is the angle itself. Mathematically,
2652:
2307:
1953:{\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}.}
1863:
1823:
1075:
1057:
454:
434:
410:
336:
239:
31:
14674:
9494:
9291:
7458:
5045:
5010:
4870:
4835:
1843:
14776:
14690:
14532:
14481:
14451:
14236:
12897:
12666:
12444:
12422:
11803:
8957:
8484:
8457:
8451:
8226:
8199:
4631:
Both sine and cosine functions can be defined by using differential equations. The pair of
3323:{\displaystyle \sin(\theta +2\pi )=\sin(\theta ),\qquad \cos(\theta +2\pi )=\cos(\theta ).}
2570:
332:
12689:
12578:
8273:
6150:
671:
The other trigonometric functions of the angle can be defined similarly; for example, the
8:
14716:
13921:
12862:
12243:
12129:
11661:
11567:
10706:
9600:
7658:
6777:
5673:
5467:
3154:
2233:
1737:
579:. It forms a side of (and is adjacent to) both the angle of interest and the right angle.
474:
14675:
3675:
of the sine function; in other words the only intersection of the sine function and the
14544:
14375:. But since Arabic is written without vowels, later writers interpreted the consonants
14320:
14210:
14201:
14003:
13894:
12852:
12481:
12329:
12305:
10130:
10090:
9761:
9599:, both sine and cosine functions can be expressed in terms of real sines, cosines, and
9549:
9529:
9321:
9129:
8634:
8253:
8181:
7711:
7691:
6250:
6130:
6017:
5449:
5268:
5248:
5220:
4830:
4603:
4576:
4449:
4422:
4298:
4271:
4152:
4125:
4021:
4016:
4006:
4001:
3952:
3932:
3335:
3199:
3079:
2913:
2719:
2696:
2673:
2052:
1803:
1783:
1763:
562:
535:
508:
43:
14127:
3905:{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x),\qquad {\frac {d}{dx}}\cos(x)=-\sin(x).}
14643:
14623:
14600:
14572:
14548:
14518:
14493:
14426:
14327:
14305:
14268:
14240:
14187:
14010:
13964:
13837:
13824:
12297:
12255:
12251:
12098:
9867:
8624:{\displaystyle {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }A_{n}\cos(nx)+B_{n}\sin(nx).}
8441:{\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx).}
3917:
3676:
3219:
347:
13579:
13577:
14613:
14592:
14418:
14297:
14260:
13886:
13811:
12943:
12220:
12183:
11109:
8929:
6011:
6010:, and so on. When only one value is desired, the function may be restricted to its
5709:
3925:
373:
1054:
appear to depend on the choice of a right triangle containing an angle of measure
14686:
14637:
14617:
14586:
14528:
14400:
14254:
14230:
14197:
Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical Tables
14170:
14061:
14049:
14037:
13574:
12235:
12082:
12045:
12041:
10710:
10221:
6067:
4026:
4011:
2563:
The sine and cosine functions may also be defined in a more general way by using
1255:
680:
sides, a reciprocal of a tangent function. These functions can be formulated as:
339:, allowing their extension to arbitrary positive and negative values and even to
12107:(which also means 'bay' or 'fold', and more specifically 'the hanging fold of a
497:
is the angle of interest. The three sides of the triangle are named as follows:
14833:
14670:
14351:
The
English word "sine" comes from a series of mistranslations of the Sanskrit
13323:
13047:
12887:
12513:
12321:
12167:
12094:
11960:
11951:
11571:
9863:
8921:
8193:
7729:
5742:. The inverse function of cosine is arccosine, denoted as "arccos", "acos", or
5649:
4902:. One could interpret the unit circle in the above definitions as defining the
3991:
2077:
1717:
1240:
1079:
428:
340:
217:
183:
14596:
14301:
14264:
12649:
is expressed in half-turns, and consequently the final input to the function,
11969:
10232:
2649:. Let a line through the origin intersect the unit circle, making an angle of
14827:
14443:
14281:
13948:
12928:
12923:
12913:
12278:
12137:
10244:
Applying the series definition of the sine and cosine to a complex argument,
9285:
8953:
7682:
7273:
5210:{\displaystyle \int \sin(x)\,dx=-\cos(x)+C\qquad \int \cos(x)\,dx=\sin(x)+C,}
3718:
3407:
2245:
47:
13877:
Brendan, T. (February 1965). "How
Ptolemy constructed trigonometry tables".
11908:
11899:
14506:
14439:
14191:
12872:
12572:
12247:
12224:
11917:
7277:
3353:
2983:
Animation demonstrating how the sine function (in red) is graphed from the
2070:
1731:
1713:
377:
179:
14422:
14054:
13694:
12239:), via translation from Sanskrit to Arabic and then from Arabic to Latin.
11103:
14514:
13890:
13158:
12507:
12313:
12229:
9339:
8925:
7686:
2993:
2249:
1960:
This is equivalent to the equality of the first three expressions below:
355:
328:
324:
193:
62:
13898:
12540:
module. Complex sine and cosine functions are also available within the
8952:, the definition of both sine and cosine functions can be extended in a
8180:
Both sine and cosine functions with multiple angles may appear as their
3759:. The decimal expansion of the Dottie number is approximately 0.739085.
1722:
13819:
12902:
12524:, such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly,
12516:
value, specifying the angle in radians. Each function returns the same
12393: in this section. Unsourced material may be challenged and removed.
12290:
12187:
12086:
8887: in this section. Unsourced material may be challenged and removed.
8188:. The trigonometric polynomial's ample applications may be acquired in
7961:
Taking the derivative of each term gives the Taylor series for cosine:
7285:
5242:
552:. The hypotenuse is always the longest side of a right-angled triangle.
233:
225:
13363:
13335:
12520:
as it accepts. Many other trigonometric functions are also defined in
12178:
While the early study of trigonometry can be traced to antiquity, the
12102:
12085:), which means 'bosom', 'pocket', or 'fold'. When the Arabic texts of
9862:
Sine and cosine are used to connect the real and imaginary parts of a
8508:
be any coefficients, then the trigonometric series can be defined as:
3949:) denotes a graph is increasing (going upward) and the negative sign (
3773:
12933:
12517:
12073:
11879:
5773:
3633:
1166:{\textstyle \sin 45^{\circ }=\cos 45^{\circ }={\frac {\sqrt {2}}{2}}}
1078:, and so the ratios are the same for each of them. For example, each
12368:
12182:
as they are in use today were developed in the medieval period. The
8862:
5712:
of sine is arcsine or inverse sine, denoted as "arcsin", "asin", or
5684:
2974:
14802:
13957:
Mathematics Across
Cultures: The History of Non-western Mathematics
13525:
13513:
13473:
13442:
13430:
13418:
13404:
13402:
12428:
12012:
8250:
be any coefficients, then the trigonometric polynomial of a degree
5092:
3075:, the point is rotated counterclockwise and stopped exactly on the
505:
is the side opposite to the angle of interest; in this case, it is
393:
221:
13454:
12191:
5893:, and so on. It follows that the arcsine function is multivalued:
14797:
14656:
Zimmermann, Paul (2006), "Can we trust floating-point numbers?",
13311:
13283:
13216:
12549:
12195:
12171:
11878:
The complex sine function is also related to the level curves of
8631:
In the case of a
Fourier series with a given integrable function
3717:. The only real fixed point of the cosine function is called the
2979:
1250:
12427:
There is no standard algorithm for calculating sine and cosine.
11650:{\displaystyle \Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},}
1082:
of the 45-45-90 right triangle is 1 unit, and its hypotenuse is
14705:
14561:
Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007),
13760:
13594:
13592:
12452:
8184:, resulting in a polynomial. Such a polynomial is known as the
3996:
3218:
coordinate. In other words, both sine and cosine functions are
1245:
10217:
14812:
14415:
International
Symposium on History of Machines and Mechanisms
12258:(c. 780–850) produced tables of sines, cosines and tangents.
12242:
All six trigonometric functions in current use were known in
12031:
10170:, Euler's formula in terms of polar coordinates is stated as
7104:
6856:
Sine and cosine satisfy the following double-angle formulas:
6170:
The inverse function of both sine and cosine are defined as:
351:
323:
The definitions of sine and cosine have been extended to any
229:
213:
209:
14357:(chord-half). Āryabhaṭa frequently abbreviated this term to
13851:, Chapter 3, for an earlier etymology crediting Gerard. See
13750:
13748:
13746:
7108:
Sine function in blue and sine squared function in red. The
6070:. The standard range of principal values for arcsin is from
5095:
with a certain bounded interval. Their antiderivatives are:
3721:. The Dottie number is the unique real root of the equation
327:
value in terms of the lengths of certain line segments in a
13382:
12281:; these were further promulgated by Euler (see below). The
12108:
8915:
7641:
6784:, the sum of a squared sine and a squared cosine equals 1:
6596:{\displaystyle \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x}
39:
14746:
1074:. However, this is not the case as all such triangles are
532:
is the side opposite the right angle; in this case, it is
13743:
10127:
represents the magnitude and angle of the complex number
9128:
Alternatively, both functions can be defined in terms of
7272:
Both sine and cosine functions can be defined by using a
346:
The sine and cosine functions are commonly used to model
331:. More modern definitions express the sine and cosine as
13465:
sfnp error: no target: CITEREFVarbergRigdonPurcell2007 (
8192:, and its extension of a periodic function known as the
7280:
involving the higher-order derivatives. As mentioned in
6847:{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1.}
5080:
14296:, Springer Undergraduate Mathematics Series, Springer,
12054:, which is meaningless in that language and written as
11104:
Partial fraction and product expansions of complex sine
7264:
2971:, even under the new definition using the unit circle.
2861:{\displaystyle \sin(\theta )=y,\qquad \cos(\theta )=x.}
2738:
coordinates of this point of intersection are equal to
30:"Sine" and "Cosine" redirect here. For other uses, see
14066:
12801:
12770:
12743:
12712:
12455:
algorithm is commonly used in scientific calculators.
12213:
12207:
11288:
10176:
6106:
6076:
5091:
Their area under a curve can be obtained by using the
3467:
3048:
2938:
2876:
1179:
1112:
366:
360:
14211:"An Eloquent Formula for the Perimeter of an Ellipse"
13644:
13608:
13104:
13065:
13022:
12992:
12692:
12669:
12613:
12581:
11823:
11677:
11583:
11453:
11332:
11118:
10725:
10257:
10156:
10133:
10113:
10093:
9978:
9907:
9875:
9834:
9802:
9770:
9609:
9572:
9552:
9532:
9497:
9378:
9348:
9324:
9294:
9138:
8966:
8938:
8657:
8637:
8514:
8487:
8460:
8305:
8276:
8256:
8229:
8202:
7967:
7738:
7714:
7694:
7661:
7498:
7461:
7428:
7393:
7358:
7326:
7294:
7127:
6862:
6790:
6609:
6525:
6519:
By definition, both functions satisfy the equations:
6273:
6253:
6176:
6153:
6133:
6040:
6020:
5975:
5937:
5899:
5858:
5820:
5782:
5748:
5718:
5658:
5634:
5476:
5452:
5410:
5291:
5271:
5251:
5223:
5101:
5048:
5013:
4961:
4912:
4873:
4838:
4783:
4734:
4681:
4637:
4606:
4579:
4531:
4484:
4452:
4425:
4380:
4333:
4301:
4274:
4234:
4187:
4155:
4128:
4088:
4041:
3955:
3935:
3795:
3727:
3685:
3494:
3416:
3362:
3338:
3228:
3202:
3163:
3130:
3104:
3082:
3022:
2916:
2808:
2776:
2744:
2722:
2699:
2676:
2655:
2605:
2573:
2374:
2352:
2330:
2310:
2288:
2266:
2200:
2166:
2086:
2055:
1966:
1886:
1866:
1846:
1826:
1806:
1786:
1766:
1740:
1597:
1500:
1467:
1387:
1088:
1060:
1028:
996:
686:
590:
565:
538:
511:
477:
457:
437:
413:
294:
262:
242:
83:
14560:
14376:
14370:
14252:
13803:
13784:
13661:
13659:
13632:
13583:
13531:
13519:
13479:
13460:
13448:
13436:
13424:
13408:
13369:
13341:
13329:
13317:
13289:
13222:
12144:(1620), which also includes a similar definition of
12076:
12067:
12055:
12049:
13947:
13946:Jacques Sesiano, "Islamic mathematics", p. 157, in
13772:
13620:
13253:
13251:
13186:
13184:
13182:
12205:The sine and cosine functions can be traced to the
7685:to show that the following identities hold for all
7281:
388:
14319:
14002:
13858:
13675:
13140:
13090:
13038:
13008:
12814:
12783:
12756:
12725:
12698:
12678:
12637:
12587:
11867:
11791:
11649:
11547:
11447:Using product expansion technique, one can derive
11437:
11318:
11274:
11108:Using the partial fraction expansion technique in
11092:
10694:
10201:
10162:
10139:
10119:
10099:
10079:
9965:{\displaystyle z=r(\cos(\theta )+i\sin(\theta )),}
9964:
9893:
9850:
9820:
9788:
9746:
9591:
9558:
9538:
9518:
9481:
9364:
9330:
9310:
9274:
9118:
8944:
8842:
8651:, the coefficients of a trigonometric series are:
8643:
8623:
8500:
8473:
8440:
8291:
8262:
8242:
8215:
8170:
7953:
7720:
7700:
7673:
7647:
7482:
7446:
7414:
7379:
7344:
7312:
7247:
7092:
6846:
6755:
6595:
6511:
6259:
6239:
6159:
6139:
6119:
6092:
6058:
6026:
6002:
5961:
5923:
5885:
5844:
5806:
5764:
5734:
5664:
5640:
5620:
5470:. In the case of a full period, its arc length is
5458:
5434:
5396:
5277:
5257:
5229:
5209:
5069:
5034:
4999:
4947:
4894:
4859:
4821:
4769:
4717:
4667:
4612:
4585:
4564:
4516:
4458:
4431:
4410:
4365:
4307:
4280:
4259:
4219:
4161:
4134:
4113:
4073:
3961:
3941:
3904:
3751:
3709:
3623:
3480:
3449:
3398:
3344:
3322:
3208:
3184:
3145:
3116:
3088:
3067:
3034:
2963:
2922:
2901:
2860:
2794:
2762:
2728:
2705:
2682:
2661:
2637:
2591:
2543:
2360:
2338:
2316:
2296:
2274:
2224:
2186:
2153:{\displaystyle a^{2}+b^{2}-2ab\cos(\gamma )=c^{2}}
2152:
2061:
2041:
1952:
1872:
1852:
1832:
1812:
1792:
1772:
1752:
1612:
1515:
1482:
1402:
1204:
1165:
1098:
1066:
1046:
1014:
975:
656:
571:
544:
517:
489:
463:
443:
419:
312:
280:
248:
164:
13910:
13908:
13840:'s 1145 translation of the tables of al-Khwārizmī
13501:
13157:The anglicized form is first recorded in 1593 in
12066:). Since Arabic is written without short vowels,
8920:Both sine and cosine can be extended further via
7728:is the angle in radians. More generally, for all
7454:, continuing to repeat those four functions. The
6066:will evaluate only to a single value, called its
5307:
4517:{\displaystyle 270^{\circ }<x<360^{\circ }}
4366:{\displaystyle 180^{\circ }<x<270^{\circ }}
3762:
2975:Graph of a function and its elementary properties
232:of the length of the adjacent leg to that of the
14825:
14253:Bourchtein, Ludmila; Bourchtein, Andrei (2022),
14186:
14000:
13598:
4565:{\displaystyle {\frac {3\pi }{2}}<x<2\pi }
4220:{\displaystyle 90^{\circ }<x<180^{\circ }}
2567:, a circle of radius one centered at the origin
666:
407:To define the sine and cosine of an acute angle
14317:
12575:, a half-turn being an angle of 180 degrees or
8852:
5444:incomplete elliptic integral of the second kind
4411:{\displaystyle \pi <x<{\frac {3\pi }{2}}}
256:, the sine and cosine functions are denoted as
14364:
14358:
14352:
14322:The New College Latin & English Dictionary
13905:
13046:denotes the inverse of a function, instead of
12566:
12022:
12021:'bow-string' or more specifically its synonym
12016:
5435:{\displaystyle \operatorname {E} (\varphi ,k)}
4260:{\displaystyle {\frac {\pi }{2}}<x<\pi }
4074:{\displaystyle 0^{\circ }<x<90^{\circ }}
2964:{\textstyle 0<\theta <{\frac {\pi }{2}}}
2902:{\textstyle 0<\theta <{\frac {\pi }{2}}}
1205:{\textstyle 0<\alpha <{\frac {\pi }{2}}}
14732:
14438:
13982:
13980:
13754:
12265:The first published use of the abbreviations
14382:
12332:", as well as the near-modern abbreviations
12277:is by the 16th-century French mathematician
12061:
6127:, and the standard range for arccos is from
5087:List of integrals of trigonometric functions
3777:The quadrants of the unit circle and of sin(
14388:
13925:. Vol. 254. p. 74. Archived from
12358:
12123:
4668:{\displaystyle (\cos \theta ,\sin \theta )}
4114:{\displaystyle 0<x<{\frac {\pi }{2}}}
3124:, the point is at the circle's halfway. If
559:is the remaining side; in this case, it is
14739:
14725:
14655:
14072:
13977:
12122:derives from an abbreviation of the Latin
6420:
6416:
6303:
6299:
3769:Differentiation of trigonometric functions
2368:, then sine and cosine can be defined as:
383:
61:
14083:
14081:
13914:
12409:Learn how and when to remove this message
8903:Learn how and when to remove this message
8826:
8737:
7492:th derivative, evaluated at the point 0:
7259:
5340:
5173:
5120:
2558:
2499:
2491:
2418:
2410:
2354:
2332:
2290:
2268:
985:
14349:(3rd ed.), Boston: Addison-Wesley,
13990:. Encyclopedia Britannica. 17 June 2024.
13802:Various sources credit the first use of
12260:Muhammad ibn Jābir al-Harrānī al-Battānī
12161:
12003:History of trigonometry § Etymology
11868:{\displaystyle \Delta u(x_{1},x_{2})=0.}
11557:
11319:{\textstyle {\frac {\pi }{\sin(\pi z)}}}
11112:, one can find that the infinite series
10231:
10216:
9760:
8916:Complex exponential function definitions
7263:
7103:
5704:functions graphed on the Cartesian plane
5683:
3772:
3632:
2978:
2553:
1721:
392:
14669:
14642:(4th ed.), John Wiley & Sons,
14622:(3rd ed.), John Wiley & Sons,
14466:
14256:Theory of Infinite Sequences and Series
13940:
13876:
13717:
13650:
13638:
12838:Bhaskara I's sine approximation formula
12316:computed the derivative of sine in his
5000:{\displaystyle x'(\theta )=-y(\theta )}
4822:{\displaystyle x'(\theta )=-y(\theta )}
4718:{\displaystyle (x(\theta ),y(\theta ))}
2992:coordinate (red dot) of a point on the
14:
14826:
14747:Trigonometric and hyperbolic functions
14480:
14412:
14280:
14078:
14043:Historical Notes for Calculus Teachers
14005:Elements of the History of Mathematics
13864:
13844:
13614:
12764:involves inaccuracies in representing
12561:double-precision floating-point format
12170:with axes for looking up the sine and
9972:and the real and imaginary parts are:
4948:{\displaystyle y'(\theta )=x(\theta )}
4770:{\displaystyle y'(\theta )=x(\theta )}
3068:{\textstyle \theta ={\frac {\pi }{2}}}
3042:. In a sine function, if the input is
1726:Law of sines and cosines' illustration
1613:{\displaystyle {\frac {\sqrt {3}}{2}}}
1516:{\displaystyle {\frac {\sqrt {2}}{2}}}
1483:{\displaystyle {\frac {\sqrt {2}}{2}}}
1403:{\displaystyle {\frac {\sqrt {3}}{2}}}
14720:
14635:
14612:
14584:
14538:
14505:
14291:
14228:
14208:
13790:
13681:
13665:
13626:
13543:
13507:
13491:
13353:
13301:
13273:
13257:
13234:
13206:
13190:
13163:Horologiographia, the Art of Dialling
11944:Arcsine function in the complex plane
7282:§ Continuity and differentiation
5081:Integral and the usage in mensuration
5007:starting from the initial conditions
2232:, the resulting equation becomes the
14398:
14339:
14098:
14096:
14031:Why the sine has a simple derivative
13852:
13848:
13778:
13766:
13733:
13584:Varberg, Purcell & Rigdon (2007)
13532:Varberg, Purcell & Rigdon (2007)
13520:Varberg, Purcell & Rigdon (2007)
13480:Varberg, Purcell & Rigdon (2007)
13461:Varberg, Rigdon & Purcell (2007)
13449:Varberg, Purcell & Rigdon (2007)
13437:Varberg, Purcell & Rigdon (2007)
13425:Varberg, Purcell & Rigdon (2007)
13370:Varberg, Purcell & Rigdon (2007)
13342:Varberg, Purcell & Rigdon (2007)
13330:Varberg, Purcell & Rigdon (2007)
13318:Varberg, Purcell & Rigdon (2007)
13290:Varberg, Purcell & Rigdon (2007)
13223:Varberg, Purcell & Rigdon (2007)
13141:{\displaystyle \sin(x)\cdot \sin(x)}
12391:adding citations to reliable sources
12362:
10212:
9828:are the real and imaginary parts of
9756:
8924:, a set of numbers composed of both
8885:adding citations to reliable sources
8856:
5679:
2239:
451:; in the accompanying figure, angle
38:. "Sine" is not to be confused with
12423:Lookup table § Computing sines
12326:Introductio in analysin infinitorum
12062:
6765:
27:Fundamental trigonometric functions
24:
13409:Bourchtein & Bourchtein (2022)
12909:Proofs of trigonometric identities
12905:—a generalization to vertex angles
11968:
11959:
11950:
11916:
11907:
11898:
11892:Sine function in the complex plane
11824:
11721:
11596:
11584:
11497:
11396:
11391:
11217:
11138:
11133:
10549:
10300:
8554:
8104:
7875:
6782:Pythagorean trigonometric identity
6014:. With this restriction, for each
5688:The usual principal values of the
5635:
5544:
5508:
5411:
5357:
1820:, and angles opposite those sides
431:that contains an angle of measure
212:. The sine and cosine of an acute
25:
14850:
14698:
14318:Traupman, Ph.D., John C. (1966),
14093:
12815:{\textstyle {\frac {\pi }{2048}}}
12655:can be interpreted in radians by
12011:is derived, indirectly, from the
11885:
7681:. One can then use the theory of
4725:to the two-dimensional system of
335:, or as the solutions of certain
14704:
14486:Approximation Theory and Methods
13098:means the squared sine function
12883:List of trigonometric identities
12367:
8861:
6772:List of trigonometric identities
6003:{\displaystyle \arcsin(0)=2\pi }
2599:, formulated as the equation of
389:Right-angled triangle definition
216:are defined in the context of a
14659:Grand Challenges of Informatics
14156:
14144:
14132:
14120:
14108:
14023:
13994:
13961:Springer Science+Business Media
13870:
13796:
13687:
13553:
13375:
13151:
13053:
12980:
12378:needs additional citations for
12246:by the 9th century, as was the
12101:, he used the Latin equivalent
11326:. Similarly, one can show that
11282:both converge and are equal to
8872:needs additional citations for
7186:
6733:
6725:
6655:
6647:
6559:
6093:{\textstyle -{\frac {\pi }{2}}}
5962:{\displaystyle \arcsin(0)=\pi }
5466:. It cannot be expressed using
5154:
3847:
3667:converges to the Dottie number.
3274:
2833:
2225:{\displaystyle \cos(\gamma )=0}
622:
18:Cosine (trigonometric function)
14636:——— (2017),
14588:Calculus for Computer Graphics
14179:
14164:ALLEGRO Angle Sensor Datasheet
13599:Abramowitz & Stegun (1970)
13135:
13129:
13117:
13111:
13085:
13079:
12784:{\textstyle {\frac {\pi }{2}}}
12757:{\textstyle {\frac {\pi }{2}}}
12726:{\textstyle {\frac {\pi }{2}}}
12629:
12620:
12484:defines sine functions within
12296:In a paper published in 1682,
11856:
11830:
11783:
11771:
11736:
11724:
11706:
11696:
11687:
11681:
11660:which in turn is found in the
11638:
11629:
11611:
11599:
11593:
11587:
11469:
11460:
11420:
11407:
11368:
11359:
11310:
11301:
11235:
11225:
11156:
11146:
11083:
11077:
11068:
11062:
11047:
11041:
11032:
11026:
11007:
10998:
10989:
10983:
10971:
10962:
10953:
10947:
10931:
10916:
10903:
10897:
10888:
10882:
10867:
10861:
10852:
10846:
10827:
10818:
10809:
10803:
10791:
10782:
10773:
10767:
10751:
10736:
10685:
10676:
10587:
10578:
10567:
10557:
10523:
10517:
10344:
10329:
10318:
10308:
10274:
10268:
10236:Vector field rendering of sin(
10067:
10061:
10042:
10036:
10020:
10014:
9995:
9989:
9956:
9953:
9947:
9932:
9926:
9917:
9888:
9876:
9815:
9809:
9783:
9777:
9592:{\displaystyle i={\sqrt {-1}}}
9469:
9453:
9421:
9405:
9262:
9256:
9241:
9235:
9196:
9190:
9175:
9169:
9058:
9052:
8983:
8977:
8823:
8814:
8805:
8799:
8734:
8725:
8716:
8710:
8615:
8606:
8584:
8575:
8432:
8423:
8380:
8371:
8315:
8309:
8286:
8280:
8142:
8133:
8122:
8112:
7984:
7978:
7919:
7904:
7893:
7883:
7755:
7749:
7533:
7527:
7519:
7504:
7477:
7462:
7441:
7435:
7409:
7403:
7374:
7368:
7339:
7333:
7307:
7301:
7236:
7227:
7206:
7200:
7177:
7168:
7147:
7141:
7083:
7077:
7036:
7030:
7001:
6995:
6976:
6970:
6947:
6938:
6922:
6916:
6907:
6901:
6882:
6873:
6835:
6829:
6810:
6804:
6716:
6713:
6707:
6698:
6638:
6635:
6629:
6620:
6584:
6581:
6575:
6566:
6550:
6547:
6541:
6532:
6490:
6484:
6441:
6435:
6417:
6407:
6401:
6376:
6370:
6324:
6318:
6300:
6290:
6284:
6120:{\textstyle {\frac {\pi }{2}}}
6053:
6047:
6034:in the domain, the expression
5988:
5982:
5950:
5944:
5912:
5906:
5874:
5865:
5833:
5827:
5795:
5789:
5562:
5547:
5526:
5511:
5429:
5417:
5335:
5329:
5195:
5189:
5170:
5164:
5145:
5139:
5117:
5111:
5058:
5052:
5023:
5017:
4994:
4988:
4976:
4970:
4942:
4936:
4927:
4921:
4883:
4877:
4848:
4842:
4816:
4810:
4798:
4792:
4764:
4758:
4749:
4743:
4712:
4709:
4703:
4694:
4688:
4682:
4662:
4638:
3896:
3890:
3875:
3869:
3841:
3835:
3823:
3817:
3763:Continuity and differentiation
3740:
3734:
3698:
3692:
3572:
3566:
3511:
3505:
3481:{\textstyle {\frac {\pi }{2}}}
3444:
3438:
3429:
3420:
3393:
3387:
3375:
3366:
3314:
3308:
3296:
3281:
3268:
3262:
3250:
3235:
3185:{\displaystyle -1\leq y\leq 1}
3002:. The cosine (in blue) is the
2846:
2840:
2821:
2815:
2789:
2783:
2757:
2751:
2669:with the positive half of the
2586:
2574:
2527:
2519:
2514:
2506:
2477:
2471:
2451:
2443:
2438:
2430:
2423:
2405:
2391:
2385:
2213:
2207:
2187:{\displaystyle \gamma =\pi /2}
2134:
2128:
1041:
1035:
1009:
1003:
943:
937:
915:
909:
877:
871:
849:
843:
811:
805:
783:
777:
745:
739:
728:
722:
703:
697:
635:
629:
603:
597:
307:
301:
275:
269:
138:
132:
101:
95:
13:
1:
14808:Jyā, koti-jyā and utkrama-jyā
14218:American Mathematical Society
12968:
12868:Lemniscate elliptic functions
12512:. The parameter of each is a
12115:was introduced in the 1590s.
12038:jyā, koti-jyā and utkrama-jyā
11991:
9821:{\displaystyle \sin(\theta )}
9789:{\displaystyle \cos(\theta )}
5886:{\displaystyle \sin(2\pi )=0}
5772:. As sine and cosine are not
3146:{\displaystyle \theta =2\pi }
2795:{\displaystyle \sin(\theta )}
2763:{\displaystyle \cos(\theta )}
2638:{\displaystyle x^{2}+y^{2}=1}
1047:{\displaystyle \cos(\alpha )}
1015:{\displaystyle \sin(\alpha )}
667:Other trigonometric functions
313:{\displaystyle \cos(\theta )}
281:{\displaystyle \sin(\theta )}
13172:
13091:{\displaystyle \sin ^{2}(x)}
12973:
12638:{\displaystyle \sin(\pi x),}
12283:Opus palatinum de triangulis
11996:
10705:where sinh and cosh are the
10202:{\textstyle z=re^{i\theta }}
9851:{\displaystyle e^{i\theta }}
9365:{\displaystyle e^{i\theta }}
8853:Complex numbers relationship
5924:{\displaystyle \arcsin(0)=0}
5845:{\displaystyle \sin(\pi )=0}
3117:{\displaystyle \theta =\pi }
3035:{\displaystyle \theta >0}
2361:{\displaystyle \mathbb {b} }
2339:{\displaystyle \mathbb {a} }
2297:{\displaystyle \mathbb {b} }
2275:{\displaystyle \mathbb {a} }
7:
14681:(2nd, reprinted ed.),
14377:
14371:
13814:'s 1116 translation of the
13755:Merzbach & Boyer (2011)
12825:
12567:Turns based implementations
12552:'s math functions call the
12214:
12208:
12186:function was discovered by
12077:
12068:
12056:
12050:
9894:{\displaystyle (r,\theta )}
3783:Cartesian coordinate system
3399:{\displaystyle f(-x)=-f(x)}
2996:(in green), at an angle of
2647:Cartesian coordinate system
1099:{\displaystyle {\sqrt {2}}}
367:
361:
10:
14855:
14683:Cambridge University Press
14513:(3rd ed.), New York:
14490:Cambridge University Press
14475:Princeton University Press
14407:Princeton University Press
14116:OpenCL Documentation sinpi
14089:MATLAB Documentation sinpi
13039:{\displaystyle \cos ^{-1}}
13009:{\displaystyle \sin ^{-1}}
12919:Sine and cosine transforms
12878:List of periodic functions
12472:In programming languages,
12420:
12155:
12151:
12032:
12000:
10707:hyperbolic sine and cosine
6769:
6059:{\displaystyle \arcsin(x)}
5765:{\displaystyle \cos ^{-1}}
5735:{\displaystyle \sin ^{-1}}
5245:of the sine curve between
5084:
3766:
3637:The fixed point iteration
3450:{\displaystyle f(-x)=f(x)}
1711:
29:
14790:
14752:
14597:10.1007/978-3-031-28117-4
14511:Real and complex analysis
14302:10.1007/978-1-4471-0027-0
14265:10.1007/978-3-030-79431-6
14128:Julia Documentation sinpi
14001:Nicolás Bourbaki (1994).
12986:The superscript of −1 in
5807:{\displaystyle \sin(0)=0}
3985:
3982:
3979:
3974:
3924:, according to which the
3916:, according to which the
3752:{\displaystyle \cos(x)=x}
3710:{\displaystyle \sin(0)=0}
2802:, respectively; that is,
1230:
1223:
1217:
228:), and the cosine is the
175:
74:
69:
60:
55:
14448:A History of Mathematics
14343:A History of Mathematics
14340:Katz, Victor J. (2008),
14232:Algebra and Trigonometry
14140:CUDA Documentation sinpi
13915:Gingerich, Owen (1986).
13586:, p. 491–492.
12858:Generalized trigonometry
12848:Dixon elliptic functions
12359:Software implementations
10120:{\displaystyle \varphi }
8186:trigonometric polynomial
7415:{\displaystyle -\cos(x)}
7380:{\displaystyle -\sin(x)}
14839:Trigonometric functions
14292:Howie, John M. (2003),
14229:Axler, Sheldon (2012),
14152:ARM Documentation sinpi
13879:The Mathematics Teacher
13561:"Sine-squared function"
12954:Trigonometric functions
12843:Discrete sine transform
12180:trigonometric functions
12158:History of trigonometry
12097:in the 12th century by
12072:was interpreted as the
10163:{\displaystyle \theta }
8945:{\displaystyle \theta }
7447:{\displaystyle \sin(x)}
7345:{\displaystyle \cos(x)}
7313:{\displaystyle \sin(x)}
6247:where for some integer
5665:{\displaystyle \varpi }
5641:{\displaystyle \Gamma }
5239:constant of integration
2662:{\displaystyle \theta }
2317:{\displaystyle \theta }
1873:{\displaystyle \gamma }
1833:{\displaystyle \alpha }
1707:
1067:{\displaystyle \alpha }
464:{\displaystyle \alpha }
444:{\displaystyle \alpha }
420:{\displaystyle \alpha }
384:Elementary descriptions
249:{\displaystyle \theta }
206:trigonometric functions
36:Cosine (disambiguation)
14541:History of Mathematics
14539:Smith, D. E. (1958) ,
14402:Trigonometric Delights
14389:
14383:
14365:
14359:
14353:
14209:Adlaj, Semjon (2012),
13827:'s translation of the
13804:
13769:, p. 35–36.
13699:math.stackexchange.com
13332:, p. 41–42.
13142:
13092:
13040:
13010:
12959:Trigonometric integral
12833:Āryabhaṭa's sine table
12816:
12785:
12758:
12727:
12700:
12680:
12639:
12589:
12287:Georg Joachim Rheticus
12175:
12124:
12023:
12017:
11973:
11964:
11955:
11921:
11912:
11903:
11869:
11793:
11651:
11549:
11501:
11439:
11400:
11320:
11276:
11221:
11142:
11094:
10696:
10553:
10304:
10241:
10229:
10203:
10164:
10141:
10121:
10101:
10081:
9966:
9895:
9859:
9852:
9822:
9790:
9748:
9593:
9560:
9540:
9520:
9519:{\displaystyle z=x+iy}
9483:
9366:
9332:
9312:
9311:{\displaystyle e^{ix}}
9276:
9120:
8946:
8844:
8645:
8625:
8558:
8502:
8475:
8442:
8406:
8354:
8299:—is defined as:
8293:
8264:
8244:
8217:
8172:
8108:
7955:
7879:
7722:
7702:
7675:
7649:
7484:
7483:{\displaystyle (4n+k)}
7448:
7416:
7381:
7346:
7314:
7269:
7260:Series and polynomials
7249:
7118:
7094:
6848:
6757:
6597:
6513:
6261:
6241:
6161:
6141:
6121:
6094:
6060:
6028:
6004:
5963:
5925:
5887:
5846:
5808:
5766:
5736:
5705:
5666:
5642:
5622:
5460:
5436:
5398:
5279:
5259:
5231:
5211:
5071:
5070:{\displaystyle x(0)=1}
5036:
5035:{\displaystyle y(0)=0}
5001:
4949:
4904:phase space trajectory
4896:
4895:{\displaystyle x(0)=1}
4861:
4860:{\displaystyle y(0)=0}
4823:
4771:
4727:differential equations
4719:
4669:
4614:
4587:
4566:
4518:
4460:
4433:
4412:
4367:
4309:
4282:
4261:
4221:
4163:
4136:
4115:
4075:
3963:
3943:
3922:second derivative test
3906:
3786:
3753:
3711:
3671:Zero is the only real
3668:
3625:
3482:
3451:
3400:
3346:
3324:
3210:
3186:
3147:
3118:
3090:
3069:
3036:
3012:
2965:
2924:
2903:
2862:
2796:
2764:
2730:
2707:
2684:
2663:
2639:
2593:
2559:Unit circle definition
2545:
2362:
2340:
2318:
2298:
2276:
2258:Euclidean vector space
2252:are operations on two
2226:
2188:
2154:
2063:
2043:
1954:
1874:
1854:
1853:{\displaystyle \beta }
1834:
1814:
1794:
1774:
1754:
1727:
1614:
1517:
1484:
1404:
1206:
1167:
1100:
1068:
1048:
1016:
990:As stated, the values
986:Special angle measures
977:
658:
573:
546:
519:
491:
465:
445:
421:
404:
337:differential equations
314:
282:
250:
166:
14772:Inverse trigonometric
14569:Pearson Prentice Hall
14482:Powell, Michael J. D.
14452:John Wiley & Sons
14423:10.1007/1-4020-2204-2
14237:John Wiley & Sons
14104:R Documentation sinpi
13143:
13093:
13041:
13011:
12817:
12786:
12759:
12728:
12701:
12681:
12679:{\displaystyle 2\pi }
12640:
12590:
12165:
12093:were translated into
11972:
11963:
11954:
11920:
11911:
11902:
11870:
11794:
11666:Riemann zeta-function
11652:
11558:Usage of complex sine
11550:
11481:
11440:
11377:
11321:
11277:
11201:
11119:
11095:
10697:
10533:
10284:
10235:
10220:
10204:
10165:
10142:
10122:
10102:
10082:
9967:
9896:
9853:
9823:
9791:
9764:
9749:
9594:
9561:
9541:
9521:
9484:
9367:
9333:
9313:
9277:
9121:
8947:
8845:
8646:
8626:
8538:
8503:
8501:{\displaystyle B_{n}}
8476:
8474:{\displaystyle A_{n}}
8443:
8386:
8334:
8294:
8265:
8245:
8243:{\displaystyle b_{n}}
8218:
8216:{\displaystyle a_{n}}
8173:
8088:
7956:
7859:
7723:
7703:
7676:
7650:
7485:
7449:
7417:
7382:
7347:
7315:
7267:
7250:
7107:
7095:
6849:
6758:
6598:
6514:
6262:
6242:
6162:
6142:
6122:
6095:
6061:
6029:
6005:
5964:
5926:
5888:
5847:
5809:
5767:
5737:
5687:
5667:
5643:
5623:
5461:
5437:
5399:
5280:
5260:
5232:
5212:
5072:
5037:
5002:
4950:
4897:
4862:
4824:
4772:
4720:
4670:
4615:
4588:
4567:
4519:
4461:
4434:
4413:
4368:
4310:
4283:
4262:
4222:
4164:
4137:
4116:
4076:
3964:
3944:
3914:first derivative test
3907:
3776:
3754:
3712:
3636:
3626:
3483:
3452:
3401:
3347:
3325:
3211:
3187:
3148:
3119:
3091:
3070:
3037:
2982:
2966:
2925:
2904:
2863:
2797:
2765:
2731:
2708:
2685:
2664:
2640:
2594:
2592:{\displaystyle (0,0)}
2554:Analytic descriptions
2546:
2363:
2341:
2324:is the angle between
2319:
2299:
2277:
2227:
2189:
2155:
2064:
2044:
1955:
1875:
1855:
1835:
1815:
1795:
1775:
1755:
1725:
1615:
1518:
1485:
1405:
1207:
1168:
1101:
1069:
1049:
1017:
978:
659:
574:
547:
520:
492:
466:
446:
422:
396:
315:
283:
251:
176:Fields of application
167:
32:Sine (disambiguation)
14713:at Wikimedia Commons
14677:Trigonometric Series
14585:Vince, John (2023),
14470:Mathematics in India
13953:D'Ambrosio, Ubiratan
13891:10.5951/MT.58.2.0141
13102:
13063:
13020:
12990:
12939:Sine–Gordon equation
12898:Optical sine theorem
12893:Madhava's sine table
12799:
12768:
12741:
12710:
12699:{\displaystyle \pi }
12690:
12667:
12611:
12588:{\displaystyle \pi }
12579:
12536:within the built-in
12445:linearly interpolate
12387:improve this article
12166:Quadrant from 1840s
11982:Imaginary component
11930:Imaginary component
11821:
11810:is a 2D solution of
11804:holomorphic function
11675:
11581:
11451:
11330:
11286:
11116:
10723:
10255:
10174:
10154:
10150:For any real number
10131:
10111:
10091:
9976:
9905:
9873:
9832:
9800:
9768:
9607:
9601:hyperbolic functions
9570:
9550:
9530:
9495:
9376:
9346:
9322:
9292:
9284:When plotted on the
9136:
8964:
8958:exponential function
8936:
8881:improve this article
8655:
8635:
8512:
8485:
8458:
8452:trigonometric series
8303:
8292:{\displaystyle T(x)}
8274:
8254:
8227:
8200:
7965:
7736:
7712:
7692:
7659:
7496:
7459:
7426:
7391:
7356:
7324:
7292:
7125:
6860:
6788:
6607:
6523:
6271:
6251:
6174:
6160:{\displaystyle \pi }
6151:
6131:
6104:
6074:
6038:
6018:
5973:
5935:
5897:
5856:
5818:
5780:
5746:
5716:
5656:
5632:
5474:
5468:elementary functions
5450:
5408:
5289:
5269:
5249:
5221:
5099:
5046:
5011:
4959:
4910:
4871:
4836:
4781:
4732:
4679:
4635:
4604:
4577:
4529:
4482:
4450:
4423:
4378:
4331:
4299:
4272:
4232:
4185:
4153:
4126:
4086:
4039:
3953:
3933:
3793:
3725:
3683:
3492:
3465:
3414:
3406:, and is said to be
3360:
3336:
3226:
3200:
3161:
3128:
3102:
3080:
3046:
3020:
2936:
2914:
2874:
2806:
2774:
2742:
2720:
2697:
2674:
2653:
2603:
2571:
2372:
2350:
2328:
2308:
2286:
2264:
2198:
2164:
2084:
2053:
1964:
1884:
1864:
1844:
1824:
1804:
1784:
1764:
1738:
1595:
1498:
1465:
1385:
1177:
1110:
1086:
1058:
1026:
994:
684:
588:
563:
536:
509:
475:
471:in a right triangle
455:
435:
411:
292:
260:
240:
81:
14326:, Toronto: Bantam,
14055:V. Frederick Rickey
13922:Scientific American
13917:"Islamic Astronomy"
12863:Hyperbolic function
12822:would be incurred.
12559:library, and use a
12318:Harmonia Mensurarum
12244:Islamic mathematics
12130:complementary angle
12027:(both adopted from
11946:
11894:
11662:functional equation
11568:functional equation
9318:for real values of
8795:
8706:
7674:{\displaystyle x=0}
7117:axis is in radians.
6778:Pythagorean theorem
5674:lemniscate constant
5306:
3657:with initial value
2234:Pythagorean theorem
1753:{\displaystyle ABC}
490:{\displaystyle ABC}
70:General information
14782:Inverse hyperbolic
14545:Dover Publications
14399:Maor, Eli (1998),
14286:Canon triangulorum
14202:Dover Publications
14188:Abramowitz, Milton
14169:2019-04-17 at the
14060:2011-07-20 at the
14048:2011-07-20 at the
14036:2011-07-20 at the
13138:
13088:
13036:
13006:
12812:
12781:
12754:
12723:
12696:
12676:
12635:
12607:would evaluate to
12585:
12482:C standard library
12306:algebraic function
12219:functions used in
12194:(180–125 BCE) and
12176:
12142:Canon triangulorum
11974:
11965:
11956:
11942:
11922:
11913:
11904:
11890:
11865:
11812:Laplace's equation
11789:
11647:
11566:) is found in the
11545:
11435:
11316:
11272:
11090:
11088:
10692:
10690:
10242:
10230:
10199:
10160:
10137:
10117:
10097:
10077:
10075:
9962:
9891:
9860:
9848:
9818:
9786:
9744:
9742:
9589:
9556:
9536:
9516:
9479:
9477:
9362:
9328:
9308:
9272:
9270:
9116:
9114:
8942:
8932:. For real number
8840:
8838:
8778:
8689:
8641:
8621:
8498:
8471:
8438:
8289:
8270:—denoted as
8260:
8240:
8213:
8182:linear combination
8168:
8166:
7951:
7949:
7718:
7698:
7671:
7645:
7640:
7480:
7444:
7412:
7377:
7342:
7310:
7270:
7245:
7119:
7090:
7088:
6844:
6753:
6751:
6593:
6509:
6507:
6257:
6237:
6157:
6137:
6117:
6090:
6056:
6024:
6000:
5959:
5921:
5883:
5842:
5804:
5762:
5732:
5706:
5662:
5638:
5618:
5456:
5432:
5394:
5292:
5275:
5255:
5227:
5207:
5067:
5032:
4997:
4945:
4892:
4857:
4831:initial conditions
4819:
4767:
4715:
4665:
4610:
4583:
4562:
4514:
4456:
4429:
4408:
4363:
4325:3rd quadrant, III
4305:
4278:
4257:
4217:
4159:
4132:
4111:
4071:
3959:
3939:
3902:
3787:
3749:
3707:
3669:
3621:
3619:
3478:
3447:
3396:
3342:
3320:
3206:
3182:
3143:
3114:
3086:
3065:
3032:
3013:
2961:
2920:
2899:
2858:
2792:
2760:
2726:
2703:
2680:
2659:
2635:
2589:
2541:
2539:
2358:
2336:
2314:
2294:
2272:
2222:
2184:
2160:In the case where
2150:
2069:is the triangle's
2059:
2039:
1950:
1880:. The law states,
1870:
1850:
1830:
1810:
1790:
1770:
1750:
1728:
1610:
1513:
1480:
1400:
1202:
1163:
1096:
1064:
1044:
1012:
973:
971:
654:
569:
542:
515:
487:
461:
441:
417:
405:
372:functions used in
350:phenomena such as
310:
278:
246:
162:
160:
75:General definition
44:Sign (mathematics)
14821:
14820:
14709:Media related to
14649:978-1-119-32113-2
14629:978-1-119-32113-2
14606:978-3-031-28117-4
14524:978-0-07-054234-1
14499:978-0-521-29514-7
14432:978-1-4020-2203-6
14311:978-1-4471-0027-0
14274:978-3-030-79431-6
14073:Zimmermann (2006)
13970:978-1-4020-0260-1
13838:Robert of Chester
13825:Gerard of Cremona
13372:, p. 42, 47.
13344:, p. 41, 43.
12810:
12779:
12752:
12721:
12419:
12418:
12411:
12252:solving triangles
12125:complementi sinus
12099:Gerard of Cremona
11989:
11988:
11937:
11936:
11758:
11642:
11535:
11430:
11372:
11314:
11270:
11190:
11177:
10658:
10594:
10467:
10426:
10351:
10213:Complex arguments
10140:{\displaystyle z}
10100:{\displaystyle r}
9868:polar coordinates
9757:Polar coordinates
9587:
9559:{\displaystyle y}
9539:{\displaystyle x}
9331:{\displaystyle x}
9107:
9037:
8930:imaginary numbers
8913:
8912:
8905:
8776:
8687:
8644:{\displaystyle f}
8523:
8263:{\displaystyle N}
8190:its interpolation
8149:
8070:
8045:
8020:
7926:
7841:
7816:
7791:
7721:{\displaystyle x}
7701:{\displaystyle x}
7627:
7601:
7578:
7555:
7243:
7184:
6731:
6686:
6667:
6653:
6462:
6345:
6260:{\displaystyle k}
6228:
6227:
6224:
6201:
6200:
6197:
6140:{\displaystyle 0}
6115:
6088:
6027:{\displaystyle x}
5680:Inverse functions
5598:
5580:
5579:
5536:
5504:
5459:{\displaystyle k}
5384:
5383:
5355:
5338:
5278:{\displaystyle t}
5258:{\displaystyle 0}
5230:{\displaystyle C}
4629:
4628:
4613:{\displaystyle +}
4586:{\displaystyle -}
4545:
4476:4th quadrant, IV
4459:{\displaystyle -}
4432:{\displaystyle -}
4406:
4308:{\displaystyle -}
4281:{\displaystyle +}
4243:
4179:2nd quadrant, II
4162:{\displaystyle +}
4135:{\displaystyle +}
4109:
3962:{\displaystyle -}
3942:{\displaystyle +}
3861:
3809:
3677:identity function
3648: = cos(
3601:
3540:
3476:
3345:{\displaystyle f}
3209:{\displaystyle y}
3089:{\displaystyle y}
3063:
2959:
2923:{\displaystyle y}
2897:
2729:{\displaystyle y}
2706:{\displaystyle x}
2683:{\displaystyle x}
2532:
2456:
2304:are vectors, and
2240:Vector definition
2062:{\displaystyle R}
2025:
2004:
1983:
1945:
1924:
1903:
1813:{\displaystyle c}
1793:{\displaystyle b}
1773:{\displaystyle a}
1705:
1704:
1608:
1604:
1511:
1507:
1478:
1474:
1398:
1394:
1200:
1161:
1157:
1094:
964:
962:
957:
947:
894:
893:
890:
881:
828:
827:
824:
815:
762:
761:
758:
749:
649:
648:
645:
617:
616:
613:
572:{\displaystyle b}
545:{\displaystyle h}
518:{\displaystyle a}
190:
189:
156:
154:
149:
119:
117:
112:
16:(Redirected from
14846:
14741:
14734:
14727:
14718:
14717:
14708:
14693:
14680:
14666:
14664:
14652:
14632:
14609:
14581:
14567:(9th ed.),
14557:
14535:
14502:
14477:
14467:Plofker (2009),
14461:
14450:(3rd ed.),
14440:Merzbach, Uta C.
14435:
14409:
14395:
14392:
14386:
14380:
14374:
14368:
14362:
14356:
14348:
14336:
14325:
14314:
14294:Complex Analysis
14288:
14277:
14249:
14246:978-0470-58579-5
14225:
14215:
14205:
14204:, Ninth printing
14192:Stegun, Irene A.
14173:
14160:
14154:
14148:
14142:
14136:
14130:
14124:
14118:
14112:
14106:
14100:
14091:
14085:
14076:
14070:
14064:
14027:
14021:
14020:
14008:
13998:
13992:
13991:
13984:
13975:
13974:
13944:
13938:
13937:
13935:
13934:
13912:
13903:
13902:
13874:
13868:
13862:
13856:
13812:Plato Tiburtinus
13807:
13800:
13794:
13788:
13782:
13776:
13770:
13764:
13758:
13752:
13741:
13731:
13725:
13715:
13709:
13708:
13706:
13705:
13691:
13685:
13679:
13673:
13663:
13654:
13648:
13642:
13636:
13630:
13624:
13618:
13612:
13606:
13596:
13587:
13581:
13572:
13571:
13569:
13567:
13557:
13551:
13541:
13535:
13529:
13523:
13517:
13511:
13505:
13499:
13489:
13483:
13477:
13471:
13470:
13458:
13452:
13446:
13440:
13434:
13428:
13422:
13416:
13406:
13397:
13396:
13394:
13393:
13379:
13373:
13367:
13361:
13351:
13345:
13339:
13333:
13327:
13321:
13315:
13309:
13299:
13293:
13287:
13281:
13271:
13265:
13255:
13246:
13232:
13226:
13220:
13214:
13204:
13198:
13188:
13166:
13155:
13149:
13147:
13145:
13144:
13139:
13097:
13095:
13094:
13089:
13075:
13074:
13057:
13051:
13045:
13043:
13042:
13037:
13035:
13034:
13015:
13013:
13012:
13007:
13005:
13004:
12984:
12944:Sinusoidal model
12821:
12819:
12818:
12813:
12811:
12803:
12790:
12788:
12787:
12782:
12780:
12772:
12763:
12761:
12760:
12755:
12753:
12745:
12732:
12730:
12729:
12724:
12722:
12714:
12705:
12703:
12702:
12697:
12685:
12683:
12682:
12677:
12658:
12654:
12644:
12642:
12641:
12636:
12606:
12602:
12598:
12594:
12592:
12591:
12586:
12558:
12547:
12543:
12539:
12535:
12531:
12511:
12503:
12495:
12479:
12475:
12465:
12461:
12438:
12414:
12407:
12403:
12400:
12394:
12371:
12363:
12300:proved that sin
12221:Indian astronomy
12217:
12211:
12127:
12080:
12071:
12065:
12064:
12059:
12053:
12035:
12034:
12026:
12020:
11947:
11941:
11895:
11889:
11874:
11872:
11871:
11866:
11855:
11854:
11842:
11841:
11798:
11796:
11795:
11790:
11767:
11763:
11759:
11751:
11720:
11719:
11656:
11654:
11653:
11648:
11643:
11641:
11618:
11554:
11552:
11551:
11546:
11541:
11537:
11536:
11534:
11533:
11524:
11523:
11514:
11500:
11495:
11444:
11442:
11441:
11436:
11431:
11429:
11428:
11427:
11402:
11399:
11394:
11373:
11371:
11355:
11354:
11344:
11343:
11334:
11325:
11323:
11322:
11317:
11315:
11313:
11290:
11281:
11279:
11278:
11273:
11271:
11269:
11268:
11267:
11255:
11254:
11244:
11243:
11242:
11223:
11220:
11215:
11191:
11183:
11178:
11176:
11165:
11164:
11163:
11144:
11141:
11136:
11110:complex analysis
11099:
11097:
11096:
11091:
11089:
11013:
10833:
10711:entire functions
10701:
10699:
10698:
10693:
10691:
10663:
10659:
10654:
10653:
10652:
10634:
10633:
10620:
10612:
10608:
10607:
10595:
10593:
10576:
10575:
10574:
10555:
10552:
10547:
10506:
10502:
10472:
10468:
10463:
10462:
10458:
10439:
10431:
10427:
10425:
10417:
10416:
10415:
10397:
10396:
10383:
10375:
10371:
10370:
10352:
10350:
10327:
10326:
10325:
10306:
10303:
10298:
10208:
10206:
10205:
10200:
10198:
10197:
10169:
10167:
10166:
10161:
10146:
10144:
10143:
10138:
10126:
10124:
10123:
10118:
10106:
10104:
10103:
10098:
10086:
10084:
10083:
10078:
10076:
9971:
9969:
9968:
9963:
9900:
9898:
9897:
9892:
9857:
9855:
9854:
9849:
9847:
9846:
9827:
9825:
9824:
9819:
9795:
9793:
9792:
9787:
9753:
9751:
9750:
9745:
9743:
9598:
9596:
9595:
9590:
9588:
9580:
9565:
9563:
9562:
9557:
9545:
9543:
9542:
9537:
9526:for real values
9525:
9523:
9522:
9517:
9488:
9486:
9485:
9480:
9478:
9468:
9467:
9420:
9419:
9371:
9369:
9368:
9363:
9361:
9360:
9337:
9335:
9334:
9329:
9317:
9315:
9314:
9309:
9307:
9306:
9281:
9279:
9278:
9273:
9271:
9221:
9220:
9155:
9154:
9125:
9123:
9122:
9117:
9115:
9108:
9103:
9102:
9101:
9083:
9082:
9069:
9038:
9036:
9028:
9027:
9026:
9008:
9007:
8994:
8951:
8949:
8948:
8943:
8908:
8901:
8897:
8894:
8888:
8865:
8857:
8849:
8847:
8846:
8841:
8839:
8794:
8786:
8777:
8769:
8760:
8759:
8705:
8697:
8688:
8680:
8671:
8670:
8650:
8648:
8647:
8642:
8630:
8628:
8627:
8622:
8599:
8598:
8568:
8567:
8557:
8552:
8534:
8533:
8524:
8516:
8507:
8505:
8504:
8499:
8497:
8496:
8480:
8478:
8477:
8472:
8470:
8469:
8447:
8445:
8444:
8439:
8416:
8415:
8405:
8400:
8364:
8363:
8353:
8348:
8330:
8329:
8298:
8296:
8295:
8290:
8269:
8267:
8266:
8261:
8249:
8247:
8246:
8241:
8239:
8238:
8222:
8220:
8219:
8214:
8212:
8211:
8177:
8175:
8174:
8169:
8167:
8163:
8162:
8150:
8148:
8131:
8130:
8129:
8110:
8107:
8102:
8081:
8071:
8069:
8061:
8060:
8051:
8046:
8044:
8036:
8035:
8026:
8021:
8019:
8011:
8010:
8001:
7960:
7958:
7957:
7952:
7950:
7946:
7945:
7927:
7925:
7902:
7901:
7900:
7881:
7878:
7873:
7852:
7842:
7840:
7832:
7831:
7822:
7817:
7815:
7807:
7806:
7797:
7792:
7790:
7782:
7781:
7772:
7727:
7725:
7724:
7719:
7707:
7705:
7704:
7699:
7680:
7678:
7677:
7672:
7654:
7652:
7651:
7646:
7644:
7643:
7628:
7625:
7602:
7599:
7579:
7576:
7556:
7553:
7523:
7522:
7491:
7489:
7487:
7486:
7481:
7453:
7451:
7450:
7445:
7421:
7419:
7418:
7413:
7386:
7384:
7383:
7378:
7351:
7349:
7348:
7343:
7319:
7317:
7316:
7311:
7254:
7252:
7251:
7246:
7244:
7239:
7213:
7196:
7195:
7185:
7180:
7154:
7137:
7136:
7116:
7114:
7099:
7097:
7096:
7091:
7089:
7073:
7072:
7048:
7026:
7025:
7007:
6991:
6990:
6966:
6965:
6853:
6851:
6850:
6845:
6825:
6824:
6800:
6799:
6766:Other identities
6762:
6760:
6759:
6754:
6752:
6732:
6729:
6687:
6679:
6668:
6660:
6654:
6651:
6602:
6600:
6599:
6594:
6518:
6516:
6515:
6510:
6508:
6467:
6463:
6460:
6350:
6346:
6343:
6266:
6264:
6263:
6258:
6246:
6244:
6243:
6238:
6233:
6229:
6225:
6222:
6221:
6206:
6202:
6198:
6195:
6194:
6166:
6164:
6163:
6158:
6146:
6144:
6143:
6138:
6126:
6124:
6123:
6118:
6116:
6108:
6099:
6097:
6096:
6091:
6089:
6081:
6065:
6063:
6062:
6057:
6033:
6031:
6030:
6025:
6012:principal branch
6009:
6007:
6006:
6001:
5968:
5966:
5965:
5960:
5930:
5928:
5927:
5922:
5892:
5890:
5889:
5884:
5851:
5849:
5848:
5843:
5813:
5811:
5810:
5805:
5771:
5769:
5768:
5763:
5761:
5760:
5741:
5739:
5738:
5733:
5731:
5730:
5710:inverse function
5703:
5695:
5671:
5669:
5668:
5663:
5647:
5645:
5644:
5639:
5627:
5625:
5624:
5619:
5599:
5594:
5586:
5581:
5572:
5571:
5570:
5569:
5557:
5542:
5537:
5535:
5534:
5533:
5521:
5506:
5505:
5503:
5502:
5490:
5484:
5465:
5463:
5462:
5457:
5441:
5439:
5438:
5433:
5403:
5401:
5400:
5395:
5390:
5386:
5385:
5379:
5375:
5356:
5351:
5339:
5325:
5324:
5309:
5305:
5300:
5284:
5282:
5281:
5276:
5264:
5262:
5261:
5256:
5236:
5234:
5233:
5228:
5216:
5214:
5213:
5208:
5076:
5074:
5073:
5068:
5041:
5039:
5038:
5033:
5006:
5004:
5003:
4998:
4969:
4954:
4952:
4951:
4946:
4920:
4901:
4899:
4898:
4893:
4866:
4864:
4863:
4858:
4828:
4826:
4825:
4820:
4791:
4776:
4774:
4773:
4768:
4742:
4724:
4722:
4721:
4716:
4675:is the solution
4674:
4672:
4671:
4666:
4619:
4617:
4616:
4611:
4592:
4590:
4589:
4584:
4571:
4569:
4568:
4563:
4546:
4541:
4533:
4523:
4521:
4520:
4515:
4513:
4512:
4494:
4493:
4465:
4463:
4462:
4457:
4438:
4436:
4435:
4430:
4417:
4415:
4414:
4409:
4407:
4402:
4394:
4372:
4370:
4369:
4364:
4362:
4361:
4343:
4342:
4314:
4312:
4311:
4306:
4287:
4285:
4284:
4279:
4266:
4264:
4263:
4258:
4244:
4236:
4226:
4224:
4223:
4218:
4216:
4215:
4197:
4196:
4168:
4166:
4165:
4160:
4141:
4139:
4138:
4133:
4120:
4118:
4117:
4112:
4110:
4102:
4080:
4078:
4077:
4072:
4070:
4069:
4051:
4050:
4033:1st quadrant, I
3972:
3971:
3968:
3966:
3965:
3960:
3948:
3946:
3945:
3940:
3911:
3909:
3908:
3903:
3862:
3860:
3849:
3810:
3808:
3797:
3758:
3756:
3755:
3750:
3716:
3714:
3713:
3708:
3666:
3656:
3630:
3628:
3627:
3622:
3620:
3613:
3609:
3602:
3594:
3552:
3548:
3541:
3533:
3487:
3485:
3484:
3479:
3477:
3469:
3456:
3454:
3453:
3448:
3405:
3403:
3402:
3397:
3351:
3349:
3348:
3343:
3329:
3327:
3326:
3321:
3217:
3215:
3213:
3212:
3207:
3191:
3189:
3188:
3183:
3152:
3150:
3149:
3144:
3123:
3121:
3120:
3115:
3097:
3095:
3093:
3092:
3087:
3074:
3072:
3071:
3066:
3064:
3056:
3041:
3039:
3038:
3033:
3010:
3008:
3001:
2991:
2989:
2970:
2968:
2967:
2962:
2960:
2952:
2931:
2929:
2927:
2926:
2921:
2908:
2906:
2905:
2900:
2898:
2890:
2867:
2865:
2864:
2859:
2801:
2799:
2798:
2793:
2769:
2767:
2766:
2761:
2737:
2735:
2733:
2732:
2727:
2714:
2712:
2710:
2709:
2704:
2691:
2689:
2687:
2686:
2681:
2668:
2666:
2665:
2660:
2644:
2642:
2641:
2636:
2628:
2627:
2615:
2614:
2598:
2596:
2595:
2590:
2550:
2548:
2547:
2542:
2540:
2533:
2531:
2530:
2522:
2517:
2509:
2503:
2502:
2494:
2488:
2457:
2455:
2454:
2446:
2441:
2433:
2427:
2426:
2421:
2413:
2408:
2402:
2367:
2365:
2364:
2359:
2357:
2345:
2343:
2342:
2337:
2335:
2323:
2321:
2320:
2315:
2303:
2301:
2300:
2295:
2293:
2281:
2279:
2278:
2273:
2271:
2231:
2229:
2228:
2223:
2193:
2191:
2190:
2185:
2180:
2159:
2157:
2156:
2151:
2149:
2148:
2109:
2108:
2096:
2095:
2068:
2066:
2065:
2060:
2048:
2046:
2045:
2040:
2026:
2024:
2010:
2005:
2003:
1989:
1984:
1982:
1968:
1959:
1957:
1956:
1951:
1946:
1941:
1930:
1925:
1920:
1909:
1904:
1899:
1888:
1879:
1877:
1876:
1871:
1859:
1857:
1856:
1851:
1839:
1837:
1836:
1831:
1819:
1817:
1816:
1811:
1799:
1797:
1796:
1791:
1779:
1777:
1776:
1771:
1759:
1757:
1756:
1751:
1689:
1687:
1686:
1683:
1680:
1672:
1667:
1664:
1662:
1661:
1658:
1655:
1639:
1637:
1636:
1633:
1630:
1619:
1617:
1616:
1611:
1609:
1600:
1599:
1589:
1587:
1586:
1583:
1580:
1572:
1571:
1569:
1568:
1565:
1562:
1558:
1550:
1547:
1545:
1544:
1541:
1538:
1522:
1520:
1519:
1514:
1512:
1503:
1502:
1489:
1487:
1486:
1481:
1479:
1470:
1469:
1459:
1457:
1456:
1453:
1450:
1442:
1437:
1434:
1432:
1431:
1428:
1425:
1409:
1407:
1406:
1401:
1399:
1390:
1389:
1376:
1374:
1373:
1370:
1367:
1359:
1357:
1356:
1353:
1350:
1342:
1341:
1339:
1338:
1335:
1332:
1328:
1320:
1317:
1315:
1314:
1311:
1308:
1215:
1214:
1211:
1209:
1208:
1203:
1201:
1193:
1172:
1170:
1169:
1164:
1162:
1153:
1152:
1147:
1146:
1128:
1127:
1105:
1103:
1102:
1097:
1095:
1090:
1073:
1071:
1070:
1065:
1053:
1051:
1050:
1045:
1021:
1019:
1018:
1013:
982:
980:
979:
974:
972:
965:
963:
960:
958:
955:
953:
948:
946:
926:
895:
891:
888:
887:
882:
880:
860:
829:
825:
822:
821:
816:
814:
794:
763:
759:
756:
755:
750:
748:
731:
714:
663:
661:
660:
655:
650:
646:
643:
642:
618:
614:
611:
610:
578:
576:
575:
570:
551:
549:
548:
543:
524:
522:
521:
516:
496:
494:
493:
488:
470:
468:
467:
462:
450:
448:
447:
442:
426:
424:
423:
418:
402:
374:Indian astronomy
370:
364:
319:
317:
316:
311:
287:
285:
284:
279:
255:
253:
252:
247:
171:
169:
168:
163:
161:
157:
155:
152:
150:
147:
145:
124:
120:
118:
115:
113:
110:
108:
87:
65:
53:
52:
21:
14854:
14853:
14849:
14848:
14847:
14845:
14844:
14843:
14824:
14823:
14822:
14817:
14786:
14765:Sine and cosine
14748:
14745:
14701:
14696:
14671:Zygmund, Antoni
14665:, p. 14/31
14662:
14650:
14630:
14607:
14579:
14555:
14543:, vol. I,
14525:
14500:
14433:
14363:or its synonym
14346:
14334:
14312:
14275:
14247:
14213:
14182:
14177:
14176:
14171:Wayback Machine
14161:
14157:
14149:
14145:
14137:
14133:
14125:
14121:
14113:
14109:
14101:
14094:
14086:
14079:
14071:
14067:
14062:Wayback Machine
14050:Wayback Machine
14038:Wayback Machine
14028:
14024:
14017:
13999:
13995:
13986:
13985:
13978:
13971:
13955:, eds. (2000).
13945:
13941:
13932:
13930:
13913:
13906:
13875:
13871:
13863:
13859:
13801:
13797:
13789:
13785:
13777:
13773:
13765:
13761:
13753:
13744:
13732:
13728:
13716:
13712:
13703:
13701:
13693:
13692:
13688:
13680:
13676:
13664:
13657:
13649:
13645:
13637:
13633:
13625:
13621:
13613:
13609:
13597:
13590:
13582:
13575:
13565:
13563:
13559:
13558:
13554:
13542:
13538:
13530:
13526:
13518:
13514:
13506:
13502:
13490:
13486:
13478:
13474:
13464:
13459:
13455:
13447:
13443:
13435:
13431:
13423:
13419:
13407:
13400:
13391:
13389:
13381:
13380:
13376:
13368:
13364:
13352:
13348:
13340:
13336:
13328:
13324:
13316:
13312:
13300:
13296:
13288:
13284:
13272:
13268:
13256:
13249:
13233:
13229:
13221:
13217:
13205:
13201:
13189:
13180:
13175:
13170:
13169:
13156:
13152:
13103:
13100:
13099:
13070:
13066:
13064:
13061:
13060:
13058:
13054:
13027:
13023:
13021:
13018:
13017:
12997:
12993:
12991:
12988:
12987:
12985:
12981:
12976:
12971:
12965:
12963:
12853:Euler's formula
12828:
12802:
12800:
12797:
12796:
12771:
12769:
12766:
12765:
12744:
12742:
12739:
12738:
12713:
12711:
12708:
12707:
12691:
12688:
12687:
12668:
12665:
12664:
12656:
12650:
12612:
12609:
12608:
12604:
12603:. For example,
12600:
12596:
12580:
12577:
12576:
12569:
12556:
12545:
12541:
12537:
12533:
12529:
12505:
12497:
12489:
12477:
12473:
12463:
12459:
12436:
12425:
12415:
12404:
12398:
12395:
12384:
12372:
12361:
12330:Euler's formula
12289:, a student of
12236:Surya Siddhanta
12160:
12154:
12005:
11999:
11994:
11979:Real component
11939:
11927:Real component
11888:
11850:
11846:
11837:
11833:
11822:
11819:
11818:
11750:
11749:
11745:
11709:
11705:
11676:
11673:
11672:
11622:
11617:
11582:
11579:
11578:
11560:
11529:
11525:
11519:
11515:
11513:
11506:
11502:
11496:
11485:
11452:
11449:
11448:
11423:
11419:
11406:
11401:
11395:
11381:
11350:
11346:
11345:
11339:
11335:
11333:
11331:
11328:
11327:
11294:
11289:
11287:
11284:
11283:
11263:
11259:
11250:
11246:
11245:
11238:
11234:
11224:
11222:
11216:
11205:
11182:
11166:
11159:
11155:
11145:
11143:
11137:
11123:
11117:
11114:
11113:
11106:
11087:
11086:
11011:
11010:
10934:
10907:
10906:
10831:
10830:
10754:
10726:
10724:
10721:
10720:
10689:
10688:
10661:
10660:
10642:
10638:
10626:
10622:
10621:
10619:
10610:
10609:
10600:
10596:
10577:
10570:
10566:
10556:
10554:
10548:
10537:
10526:
10508:
10507:
10495:
10491:
10470:
10469:
10451:
10447:
10440:
10438:
10429:
10428:
10418:
10405:
10401:
10389:
10385:
10384:
10382:
10373:
10372:
10357:
10353:
10328:
10321:
10317:
10307:
10305:
10299:
10288:
10277:
10258:
10256:
10253:
10252:
10222:Domain coloring
10215:
10190:
10186:
10175:
10172:
10171:
10155:
10152:
10151:
10132:
10129:
10128:
10112:
10109:
10108:
10092:
10089:
10088:
10074:
10073:
10045:
10027:
10026:
9998:
9979:
9977:
9974:
9973:
9906:
9903:
9902:
9874:
9871:
9870:
9839:
9835:
9833:
9830:
9829:
9801:
9798:
9797:
9769:
9766:
9765:
9759:
9741:
9740:
9688:
9676:
9675:
9623:
9610:
9608:
9605:
9604:
9579:
9571:
9568:
9567:
9551:
9548:
9547:
9531:
9528:
9527:
9496:
9493:
9492:
9476:
9475:
9460:
9456:
9440:
9428:
9427:
9412:
9408:
9392:
9379:
9377:
9374:
9373:
9353:
9349:
9347:
9344:
9343:
9338:traces out the
9323:
9320:
9319:
9299:
9295:
9293:
9290:
9289:
9288:, the function
9269:
9268:
9222:
9210:
9206:
9203:
9202:
9156:
9147:
9143:
9139:
9137:
9134:
9133:
9130:Euler's formula
9113:
9112:
9091:
9087:
9075:
9071:
9070:
9068:
9061:
9043:
9042:
9029:
9016:
9012:
9000:
8996:
8995:
8993:
8986:
8967:
8965:
8962:
8961:
8956:in terms of an
8937:
8934:
8933:
8918:
8909:
8898:
8892:
8889:
8878:
8866:
8855:
8837:
8836:
8787:
8782:
8768:
8761:
8755:
8751:
8748:
8747:
8698:
8693:
8679:
8672:
8666:
8662:
8658:
8656:
8653:
8652:
8636:
8633:
8632:
8594:
8590:
8563:
8559:
8553:
8542:
8529:
8525:
8515:
8513:
8510:
8509:
8492:
8488:
8486:
8483:
8482:
8465:
8461:
8459:
8456:
8455:
8411:
8407:
8401:
8390:
8359:
8355:
8349:
8338:
8325:
8321:
8304:
8301:
8300:
8275:
8272:
8271:
8255:
8252:
8251:
8234:
8230:
8228:
8225:
8224:
8207:
8203:
8201:
8198:
8197:
8165:
8164:
8155:
8151:
8132:
8125:
8121:
8111:
8109:
8103:
8092:
8079:
8078:
8062:
8056:
8052:
8050:
8037:
8031:
8027:
8025:
8012:
8006:
8002:
8000:
7987:
7968:
7966:
7963:
7962:
7948:
7947:
7932:
7928:
7903:
7896:
7892:
7882:
7880:
7874:
7863:
7850:
7849:
7833:
7827:
7823:
7821:
7808:
7802:
7798:
7796:
7783:
7777:
7773:
7771:
7758:
7739:
7737:
7734:
7733:
7730:complex numbers
7713:
7710:
7709:
7693:
7690:
7689:
7660:
7657:
7656:
7639:
7638:
7624:
7622:
7613:
7612:
7598:
7596:
7590:
7589:
7575:
7573:
7567:
7566:
7552:
7550:
7540:
7539:
7503:
7499:
7497:
7494:
7493:
7460:
7457:
7456:
7455:
7427:
7424:
7423:
7392:
7389:
7388:
7357:
7354:
7353:
7325:
7322:
7321:
7293:
7290:
7289:
7262:
7214:
7212:
7191:
7187:
7155:
7153:
7132:
7128:
7126:
7123:
7122:
7110:
7109:
7087:
7086:
7068:
7064:
7046:
7045:
7021:
7017:
7005:
7004:
6986:
6982:
6961:
6957:
6950:
6929:
6928:
6885:
6863:
6861:
6858:
6857:
6820:
6816:
6795:
6791:
6789:
6786:
6785:
6774:
6768:
6750:
6749:
6728:
6726:
6689:
6688:
6678:
6659:
6650:
6648:
6610:
6608:
6605:
6604:
6524:
6521:
6520:
6506:
6505:
6465:
6464:
6459:
6421:
6392:
6391:
6348:
6347:
6342:
6304:
6274:
6272:
6269:
6268:
6252:
6249:
6248:
6220:
6216:
6193:
6189:
6175:
6172:
6171:
6152:
6149:
6148:
6132:
6129:
6128:
6107:
6105:
6102:
6101:
6080:
6075:
6072:
6071:
6068:principal value
6039:
6036:
6035:
6019:
6016:
6015:
5974:
5971:
5970:
5936:
5933:
5932:
5898:
5895:
5894:
5857:
5854:
5853:
5819:
5816:
5815:
5781:
5778:
5777:
5753:
5749:
5747:
5744:
5743:
5723:
5719:
5717:
5714:
5713:
5697:
5689:
5682:
5657:
5654:
5653:
5633:
5630:
5629:
5587:
5585:
5565:
5561:
5553:
5543:
5541:
5529:
5525:
5517:
5507:
5498:
5494:
5489:
5485:
5483:
5475:
5472:
5471:
5451:
5448:
5447:
5409:
5406:
5405:
5374:
5367:
5363:
5350:
5320:
5316:
5308:
5301:
5296:
5290:
5287:
5286:
5270:
5267:
5266:
5250:
5247:
5246:
5222:
5219:
5218:
5100:
5097:
5096:
5089:
5083:
5047:
5044:
5043:
5012:
5009:
5008:
4962:
4960:
4957:
4956:
4913:
4911:
4908:
4907:
4872:
4869:
4868:
4837:
4834:
4833:
4784:
4782:
4779:
4778:
4735:
4733:
4730:
4729:
4680:
4677:
4676:
4636:
4633:
4632:
4605:
4602:
4601:
4578:
4575:
4574:
4534:
4532:
4530:
4527:
4526:
4508:
4504:
4489:
4485:
4483:
4480:
4479:
4451:
4448:
4447:
4424:
4421:
4420:
4395:
4393:
4379:
4376:
4375:
4357:
4353:
4338:
4334:
4332:
4329:
4328:
4300:
4297:
4296:
4273:
4270:
4269:
4235:
4233:
4230:
4229:
4211:
4207:
4192:
4188:
4186:
4183:
4182:
4154:
4151:
4150:
4127:
4124:
4123:
4101:
4087:
4084:
4083:
4065:
4061:
4046:
4042:
4040:
4037:
4036:
3954:
3951:
3950:
3934:
3931:
3930:
3853:
3848:
3801:
3796:
3794:
3791:
3790:
3771:
3765:
3726:
3723:
3722:
3684:
3681:
3680:
3665: = −1
3664:
3658:
3653:
3647:
3638:
3618:
3617:
3593:
3592:
3588:
3575:
3557:
3556:
3532:
3531:
3527:
3514:
3495:
3493:
3490:
3489:
3468:
3466:
3463:
3462:
3415:
3412:
3411:
3361:
3358:
3357:
3337:
3334:
3333:
3227:
3224:
3223:
3201:
3198:
3197:
3196:
3162:
3159:
3158:
3129:
3126:
3125:
3103:
3100:
3099:
3081:
3078:
3077:
3076:
3055:
3047:
3044:
3043:
3021:
3018:
3017:
3004:
3003:
2997:
2985:
2984:
2977:
2951:
2937:
2934:
2933:
2915:
2912:
2911:
2910:
2889:
2875:
2872:
2871:
2807:
2804:
2803:
2775:
2772:
2771:
2743:
2740:
2739:
2721:
2718:
2717:
2716:
2698:
2695:
2694:
2693:
2675:
2672:
2671:
2670:
2654:
2651:
2650:
2623:
2619:
2610:
2606:
2604:
2601:
2600:
2572:
2569:
2568:
2561:
2556:
2538:
2537:
2526:
2518:
2513:
2505:
2504:
2498:
2490:
2489:
2487:
2480:
2462:
2461:
2450:
2442:
2437:
2429:
2428:
2422:
2417:
2409:
2404:
2403:
2401:
2394:
2375:
2373:
2370:
2369:
2353:
2351:
2348:
2347:
2331:
2329:
2326:
2325:
2309:
2306:
2305:
2289:
2287:
2284:
2283:
2267:
2265:
2262:
2261:
2242:
2199:
2196:
2195:
2176:
2165:
2162:
2161:
2144:
2140:
2104:
2100:
2091:
2087:
2085:
2082:
2081:
2054:
2051:
2050:
2014:
2009:
1993:
1988:
1972:
1967:
1965:
1962:
1961:
1931:
1929:
1910:
1908:
1889:
1887:
1885:
1882:
1881:
1865:
1862:
1861:
1845:
1842:
1841:
1825:
1822:
1821:
1805:
1802:
1801:
1785:
1782:
1781:
1765:
1762:
1761:
1739:
1736:
1735:
1720:
1712:Main articles:
1710:
1684:
1681:
1678:
1677:
1675:
1670:
1665:
1659:
1656:
1653:
1652:
1650:
1634:
1631:
1628:
1627:
1625:
1598:
1596:
1593:
1592:
1584:
1581:
1578:
1577:
1575:
1566:
1563:
1560:
1559:
1556:
1554:
1553:
1548:
1542:
1539:
1536:
1535:
1533:
1501:
1499:
1496:
1495:
1468:
1466:
1463:
1462:
1454:
1451:
1448:
1447:
1445:
1440:
1435:
1429:
1426:
1423:
1422:
1420:
1388:
1386:
1383:
1382:
1371:
1368:
1365:
1364:
1362:
1354:
1351:
1348:
1347:
1345:
1336:
1333:
1330:
1329:
1326:
1324:
1323:
1318:
1312:
1309:
1306:
1305:
1303:
1192:
1178:
1175:
1174:
1151:
1142:
1138:
1123:
1119:
1111:
1108:
1107:
1089:
1087:
1084:
1083:
1059:
1056:
1055:
1027:
1024:
1023:
995:
992:
991:
988:
970:
969:
959:
954:
952:
930:
925:
918:
900:
899:
886:
864:
859:
852:
834:
833:
820:
798:
793:
786:
768:
767:
754:
732:
715:
713:
706:
687:
685:
682:
681:
669:
641:
609:
589:
586:
585:
564:
561:
560:
537:
534:
533:
510:
507:
506:
476:
473:
472:
456:
453:
452:
436:
433:
432:
427:, start with a
412:
409:
408:
398:
391:
386:
341:complex numbers
333:infinite series
293:
290:
289:
261:
258:
257:
241:
238:
237:
236:. For an angle
159:
158:
151:
146:
144:
122:
121:
114:
109:
107:
84:
82:
79:
78:
56:Sine and cosine
51:
28:
23:
22:
15:
12:
11:
5:
14852:
14842:
14841:
14836:
14819:
14818:
14816:
14815:
14810:
14805:
14800:
14794:
14792:
14788:
14787:
14785:
14784:
14779:
14774:
14769:
14768:
14767:
14756:
14754:
14750:
14749:
14744:
14743:
14736:
14729:
14721:
14715:
14714:
14700:
14699:External links
14697:
14695:
14694:
14667:
14653:
14648:
14633:
14628:
14614:Young, Cynthia
14610:
14605:
14582:
14578:978-0131469686
14577:
14558:
14553:
14536:
14523:
14503:
14498:
14478:
14464:
14444:Boyer, Carl B.
14436:
14431:
14410:
14396:
14337:
14332:
14315:
14310:
14289:
14282:Gunter, Edmund
14278:
14273:
14250:
14245:
14226:
14206:
14183:
14181:
14178:
14175:
14174:
14155:
14143:
14131:
14119:
14107:
14092:
14077:
14065:
14022:
14015:
13993:
13988:"trigonometry"
13976:
13969:
13949:Selin, Helaine
13939:
13904:
13885:(2): 141–149.
13869:
13857:
13855:, p. 210.
13842:
13841:
13835:
13822:
13795:
13793:, p. 202.
13783:
13781:, p. 253.
13771:
13759:
13742:
13726:
13718:Plofker (2009)
13710:
13686:
13674:
13655:
13651:Zygmund (1968)
13643:
13639:Zygmund (1968)
13631:
13619:
13617:, p. 150.
13607:
13588:
13573:
13552:
13536:
13534:, p. 365.
13524:
13522:, p. 366.
13512:
13500:
13484:
13482:, p. 199.
13472:
13453:
13451:, p. 157.
13441:
13439:, p. 155.
13429:
13427:, p. 115.
13417:
13398:
13383:"OEIS A003957"
13374:
13362:
13346:
13334:
13322:
13310:
13294:
13282:
13266:
13247:
13227:
13215:
13199:
13177:
13176:
13174:
13171:
13168:
13167:
13150:
13137:
13134:
13131:
13128:
13125:
13122:
13119:
13116:
13113:
13110:
13107:
13087:
13084:
13081:
13078:
13073:
13069:
13052:
13048:exponentiation
13033:
13030:
13026:
13003:
13000:
12996:
12978:
12977:
12975:
12972:
12970:
12967:
12962:
12961:
12956:
12951:
12946:
12941:
12936:
12931:
12926:
12921:
12916:
12911:
12906:
12900:
12895:
12890:
12888:Madhava series
12885:
12880:
12875:
12870:
12865:
12860:
12855:
12850:
12845:
12840:
12835:
12829:
12827:
12824:
12809:
12806:
12778:
12775:
12751:
12748:
12720:
12717:
12695:
12675:
12672:
12634:
12631:
12628:
12625:
12622:
12619:
12616:
12584:
12568:
12565:
12514:floating point
12417:
12416:
12375:
12373:
12366:
12360:
12357:
12322:Leonhard Euler
12168:Ottoman Turkey
12156:Main article:
12153:
12150:
12095:Medieval Latin
12042:transliterated
12001:Main article:
11998:
11995:
11993:
11990:
11987:
11986:
11983:
11980:
11976:
11975:
11966:
11957:
11935:
11934:
11931:
11928:
11924:
11923:
11914:
11905:
11887:
11886:Complex graphs
11884:
11876:
11875:
11864:
11861:
11858:
11853:
11849:
11845:
11840:
11836:
11832:
11829:
11826:
11800:
11799:
11788:
11785:
11782:
11779:
11776:
11773:
11770:
11766:
11762:
11757:
11754:
11748:
11744:
11741:
11738:
11735:
11732:
11729:
11726:
11723:
11718:
11715:
11712:
11708:
11704:
11701:
11698:
11695:
11692:
11689:
11686:
11683:
11680:
11658:
11657:
11646:
11640:
11637:
11634:
11631:
11628:
11625:
11621:
11616:
11613:
11610:
11607:
11604:
11601:
11598:
11595:
11592:
11589:
11586:
11572:Gamma function
11559:
11556:
11544:
11540:
11532:
11528:
11522:
11518:
11512:
11509:
11505:
11499:
11494:
11491:
11488:
11484:
11480:
11477:
11474:
11471:
11468:
11465:
11462:
11459:
11456:
11434:
11426:
11422:
11418:
11415:
11412:
11409:
11405:
11398:
11393:
11390:
11387:
11384:
11380:
11376:
11370:
11367:
11364:
11361:
11358:
11353:
11349:
11342:
11338:
11312:
11309:
11306:
11303:
11300:
11297:
11293:
11266:
11262:
11258:
11253:
11249:
11241:
11237:
11233:
11230:
11227:
11219:
11214:
11211:
11208:
11204:
11200:
11197:
11194:
11189:
11186:
11181:
11175:
11172:
11169:
11162:
11158:
11154:
11151:
11148:
11140:
11135:
11132:
11129:
11126:
11122:
11105:
11102:
11101:
11100:
11085:
11082:
11079:
11076:
11073:
11070:
11067:
11064:
11061:
11058:
11055:
11052:
11049:
11046:
11043:
11040:
11037:
11034:
11031:
11028:
11025:
11022:
11019:
11016:
11014:
11012:
11009:
11006:
11003:
11000:
10997:
10994:
10991:
10988:
10985:
10982:
10979:
10976:
10973:
10970:
10967:
10964:
10961:
10958:
10955:
10952:
10949:
10946:
10943:
10940:
10937:
10935:
10933:
10930:
10927:
10924:
10921:
10918:
10915:
10912:
10909:
10908:
10905:
10902:
10899:
10896:
10893:
10890:
10887:
10884:
10881:
10878:
10875:
10872:
10869:
10866:
10863:
10860:
10857:
10854:
10851:
10848:
10845:
10842:
10839:
10836:
10834:
10832:
10829:
10826:
10823:
10820:
10817:
10814:
10811:
10808:
10805:
10802:
10799:
10796:
10793:
10790:
10787:
10784:
10781:
10778:
10775:
10772:
10769:
10766:
10763:
10760:
10757:
10755:
10753:
10750:
10747:
10744:
10741:
10738:
10735:
10732:
10729:
10728:
10703:
10702:
10687:
10684:
10681:
10678:
10675:
10672:
10669:
10666:
10664:
10662:
10657:
10651:
10648:
10645:
10641:
10637:
10632:
10629:
10625:
10618:
10615:
10613:
10611:
10606:
10603:
10599:
10592:
10589:
10586:
10583:
10580:
10573:
10569:
10565:
10562:
10559:
10551:
10546:
10543:
10540:
10536:
10532:
10529:
10527:
10525:
10522:
10519:
10516:
10513:
10510:
10509:
10505:
10501:
10498:
10494:
10490:
10487:
10484:
10481:
10478:
10475:
10473:
10471:
10466:
10461:
10457:
10454:
10450:
10446:
10443:
10437:
10434:
10432:
10430:
10424:
10421:
10414:
10411:
10408:
10404:
10400:
10395:
10392:
10388:
10381:
10378:
10376:
10374:
10369:
10366:
10363:
10360:
10356:
10349:
10346:
10343:
10340:
10337:
10334:
10331:
10324:
10320:
10316:
10313:
10310:
10302:
10297:
10294:
10291:
10287:
10283:
10280:
10278:
10276:
10273:
10270:
10267:
10264:
10261:
10260:
10214:
10211:
10196:
10193:
10189:
10185:
10182:
10179:
10159:
10136:
10116:
10096:
10072:
10069:
10066:
10063:
10060:
10057:
10054:
10051:
10048:
10046:
10044:
10041:
10038:
10035:
10032:
10029:
10028:
10025:
10022:
10019:
10016:
10013:
10010:
10007:
10004:
10001:
9999:
9997:
9994:
9991:
9988:
9985:
9982:
9981:
9961:
9958:
9955:
9952:
9949:
9946:
9943:
9940:
9937:
9934:
9931:
9928:
9925:
9922:
9919:
9916:
9913:
9910:
9890:
9887:
9884:
9881:
9878:
9864:complex number
9845:
9842:
9838:
9817:
9814:
9811:
9808:
9805:
9785:
9782:
9779:
9776:
9773:
9758:
9755:
9739:
9736:
9733:
9730:
9727:
9724:
9721:
9718:
9715:
9712:
9709:
9706:
9703:
9700:
9697:
9694:
9691:
9689:
9687:
9684:
9681:
9678:
9677:
9674:
9671:
9668:
9665:
9662:
9659:
9656:
9653:
9650:
9647:
9644:
9641:
9638:
9635:
9632:
9629:
9626:
9624:
9622:
9619:
9616:
9613:
9612:
9586:
9583:
9578:
9575:
9555:
9535:
9515:
9512:
9509:
9506:
9503:
9500:
9474:
9471:
9466:
9463:
9459:
9455:
9452:
9449:
9446:
9443:
9441:
9439:
9436:
9433:
9430:
9429:
9426:
9423:
9418:
9415:
9411:
9407:
9404:
9401:
9398:
9395:
9393:
9391:
9388:
9385:
9382:
9381:
9359:
9356:
9352:
9327:
9305:
9302:
9298:
9267:
9264:
9261:
9258:
9255:
9252:
9249:
9246:
9243:
9240:
9237:
9234:
9231:
9228:
9225:
9223:
9219:
9216:
9213:
9209:
9205:
9204:
9201:
9198:
9195:
9192:
9189:
9186:
9183:
9180:
9177:
9174:
9171:
9168:
9165:
9162:
9159:
9157:
9153:
9150:
9146:
9142:
9141:
9111:
9106:
9100:
9097:
9094:
9090:
9086:
9081:
9078:
9074:
9067:
9064:
9062:
9060:
9057:
9054:
9051:
9048:
9045:
9044:
9041:
9035:
9032:
9025:
9022:
9019:
9015:
9011:
9006:
9003:
8999:
8992:
8989:
8987:
8985:
8982:
8979:
8976:
8973:
8970:
8969:
8941:
8922:complex number
8917:
8914:
8911:
8910:
8869:
8867:
8860:
8854:
8851:
8835:
8832:
8829:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8798:
8793:
8790:
8785:
8781:
8775:
8772:
8767:
8764:
8762:
8758:
8754:
8750:
8749:
8746:
8743:
8740:
8736:
8733:
8730:
8727:
8724:
8721:
8718:
8715:
8712:
8709:
8704:
8701:
8696:
8692:
8686:
8683:
8678:
8675:
8673:
8669:
8665:
8661:
8660:
8640:
8620:
8617:
8614:
8611:
8608:
8605:
8602:
8597:
8593:
8589:
8586:
8583:
8580:
8577:
8574:
8571:
8566:
8562:
8556:
8551:
8548:
8545:
8541:
8537:
8532:
8528:
8522:
8519:
8495:
8491:
8468:
8464:
8437:
8434:
8431:
8428:
8425:
8422:
8419:
8414:
8410:
8404:
8399:
8396:
8393:
8389:
8385:
8382:
8379:
8376:
8373:
8370:
8367:
8362:
8358:
8352:
8347:
8344:
8341:
8337:
8333:
8328:
8324:
8320:
8317:
8314:
8311:
8308:
8288:
8285:
8282:
8279:
8259:
8237:
8233:
8210:
8206:
8194:Fourier series
8161:
8158:
8154:
8147:
8144:
8141:
8138:
8135:
8128:
8124:
8120:
8117:
8114:
8106:
8101:
8098:
8095:
8091:
8087:
8084:
8082:
8080:
8077:
8074:
8068:
8065:
8059:
8055:
8049:
8043:
8040:
8034:
8030:
8024:
8018:
8015:
8009:
8005:
7999:
7996:
7993:
7990:
7988:
7986:
7983:
7980:
7977:
7974:
7971:
7970:
7944:
7941:
7938:
7935:
7931:
7924:
7921:
7918:
7915:
7912:
7909:
7906:
7899:
7895:
7891:
7888:
7885:
7877:
7872:
7869:
7866:
7862:
7858:
7855:
7853:
7851:
7848:
7845:
7839:
7836:
7830:
7826:
7820:
7814:
7811:
7805:
7801:
7795:
7789:
7786:
7780:
7776:
7770:
7767:
7764:
7761:
7759:
7757:
7754:
7751:
7748:
7745:
7742:
7741:
7717:
7697:
7670:
7667:
7664:
7642:
7637:
7634:
7631:
7623:
7621:
7618:
7615:
7614:
7611:
7608:
7605:
7597:
7595:
7592:
7591:
7588:
7585:
7582:
7574:
7572:
7569:
7568:
7565:
7562:
7559:
7551:
7549:
7546:
7545:
7543:
7538:
7535:
7532:
7529:
7526:
7521:
7518:
7515:
7512:
7509:
7506:
7502:
7479:
7476:
7473:
7470:
7467:
7464:
7443:
7440:
7437:
7434:
7431:
7411:
7408:
7405:
7402:
7399:
7396:
7376:
7373:
7370:
7367:
7364:
7361:
7341:
7338:
7335:
7332:
7329:
7309:
7306:
7303:
7300:
7297:
7261:
7258:
7242:
7238:
7235:
7232:
7229:
7226:
7223:
7220:
7217:
7211:
7208:
7205:
7202:
7199:
7194:
7190:
7183:
7179:
7176:
7173:
7170:
7167:
7164:
7161:
7158:
7152:
7149:
7146:
7143:
7140:
7135:
7131:
7085:
7082:
7079:
7076:
7071:
7067:
7063:
7060:
7057:
7054:
7051:
7049:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7024:
7020:
7016:
7013:
7010:
7008:
7006:
7003:
7000:
6997:
6994:
6989:
6985:
6981:
6978:
6975:
6972:
6969:
6964:
6960:
6956:
6953:
6951:
6949:
6946:
6943:
6940:
6937:
6934:
6931:
6930:
6927:
6924:
6921:
6918:
6915:
6912:
6909:
6906:
6903:
6900:
6897:
6894:
6891:
6888:
6886:
6884:
6881:
6878:
6875:
6872:
6869:
6866:
6865:
6843:
6840:
6837:
6834:
6831:
6828:
6823:
6819:
6815:
6812:
6809:
6806:
6803:
6798:
6794:
6770:Main article:
6767:
6764:
6748:
6745:
6742:
6739:
6736:
6727:
6724:
6721:
6718:
6715:
6712:
6709:
6706:
6703:
6700:
6697:
6694:
6691:
6690:
6685:
6682:
6677:
6674:
6671:
6666:
6663:
6658:
6649:
6646:
6643:
6640:
6637:
6634:
6631:
6628:
6625:
6622:
6619:
6616:
6613:
6612:
6592:
6589:
6586:
6583:
6580:
6577:
6574:
6571:
6568:
6565:
6562:
6558:
6555:
6552:
6549:
6546:
6543:
6540:
6537:
6534:
6531:
6528:
6504:
6501:
6498:
6495:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6468:
6466:
6461: or
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6437:
6434:
6431:
6428:
6425:
6422:
6419:
6415:
6412:
6409:
6406:
6403:
6400:
6397:
6394:
6393:
6390:
6387:
6384:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6360:
6357:
6354:
6351:
6349:
6344: or
6341:
6338:
6335:
6332:
6329:
6326:
6323:
6320:
6317:
6314:
6311:
6308:
6305:
6302:
6298:
6295:
6292:
6289:
6286:
6283:
6280:
6277:
6276:
6256:
6236:
6232:
6219:
6215:
6212:
6209:
6205:
6192:
6188:
6185:
6182:
6179:
6156:
6136:
6114:
6111:
6087:
6084:
6079:
6055:
6052:
6049:
6046:
6043:
6023:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5958:
5955:
5952:
5949:
5946:
5943:
5940:
5920:
5917:
5914:
5911:
5908:
5905:
5902:
5882:
5879:
5876:
5873:
5870:
5867:
5864:
5861:
5841:
5838:
5835:
5832:
5829:
5826:
5823:
5803:
5800:
5797:
5794:
5791:
5788:
5785:
5759:
5756:
5752:
5729:
5726:
5722:
5681:
5678:
5661:
5650:gamma function
5637:
5617:
5614:
5611:
5608:
5605:
5602:
5597:
5593:
5590:
5584:
5578:
5575:
5568:
5564:
5560:
5556:
5552:
5549:
5546:
5540:
5532:
5528:
5524:
5520:
5516:
5513:
5510:
5501:
5497:
5493:
5488:
5482:
5479:
5455:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5393:
5389:
5382:
5378:
5373:
5370:
5366:
5362:
5359:
5354:
5349:
5346:
5343:
5337:
5334:
5331:
5328:
5323:
5319:
5315:
5312:
5304:
5299:
5295:
5274:
5254:
5226:
5206:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5172:
5169:
5166:
5163:
5160:
5157:
5153:
5150:
5147:
5144:
5141:
5138:
5135:
5132:
5129:
5126:
5123:
5119:
5116:
5113:
5110:
5107:
5104:
5085:Main article:
5082:
5079:
5066:
5063:
5060:
5057:
5054:
5051:
5031:
5028:
5025:
5022:
5019:
5016:
4996:
4993:
4990:
4987:
4984:
4981:
4978:
4975:
4972:
4968:
4965:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4919:
4916:
4891:
4888:
4885:
4882:
4879:
4876:
4856:
4853:
4850:
4847:
4844:
4841:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4790:
4787:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4741:
4738:
4714:
4711:
4708:
4705:
4702:
4699:
4696:
4693:
4690:
4687:
4684:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4627:
4626:
4623:
4620:
4609:
4599:
4596:
4593:
4582:
4572:
4561:
4558:
4555:
4552:
4549:
4544:
4540:
4537:
4524:
4511:
4507:
4503:
4500:
4497:
4492:
4488:
4477:
4473:
4472:
4469:
4466:
4455:
4445:
4442:
4439:
4428:
4418:
4405:
4401:
4398:
4392:
4389:
4386:
4383:
4373:
4360:
4356:
4352:
4349:
4346:
4341:
4337:
4326:
4322:
4321:
4318:
4315:
4304:
4294:
4291:
4288:
4277:
4267:
4256:
4253:
4250:
4247:
4242:
4239:
4227:
4214:
4210:
4206:
4203:
4200:
4195:
4191:
4180:
4176:
4175:
4172:
4169:
4158:
4148:
4145:
4142:
4131:
4121:
4108:
4105:
4100:
4097:
4094:
4091:
4081:
4068:
4064:
4060:
4057:
4054:
4049:
4045:
4034:
4030:
4029:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3988:
3987:
3984:
3981:
3978:
3958:
3938:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3859:
3856:
3852:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3807:
3804:
3800:
3767:Main article:
3764:
3761:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3662:
3651:
3642:
3616:
3612:
3608:
3605:
3600:
3597:
3591:
3587:
3584:
3581:
3578:
3576:
3574:
3571:
3568:
3565:
3562:
3559:
3558:
3555:
3551:
3547:
3544:
3539:
3536:
3530:
3526:
3523:
3520:
3517:
3515:
3513:
3510:
3507:
3504:
3501:
3498:
3497:
3488:. This means,
3475:
3472:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3352:is said to be
3341:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3205:
3181:
3178:
3175:
3172:
3169:
3166:
3142:
3139:
3136:
3133:
3113:
3110:
3107:
3085:
3062:
3059:
3054:
3051:
3031:
3028:
3025:
2976:
2973:
2958:
2955:
2950:
2947:
2944:
2941:
2919:
2896:
2893:
2888:
2885:
2882:
2879:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2791:
2788:
2785:
2782:
2779:
2759:
2756:
2753:
2750:
2747:
2725:
2702:
2679:
2658:
2634:
2631:
2626:
2622:
2618:
2613:
2609:
2588:
2585:
2582:
2579:
2576:
2560:
2557:
2555:
2552:
2536:
2529:
2525:
2521:
2516:
2512:
2508:
2501:
2497:
2493:
2486:
2483:
2481:
2479:
2476:
2473:
2470:
2467:
2464:
2463:
2460:
2453:
2449:
2445:
2440:
2436:
2432:
2425:
2420:
2416:
2412:
2407:
2400:
2397:
2395:
2393:
2390:
2387:
2384:
2381:
2378:
2377:
2356:
2334:
2313:
2292:
2270:
2241:
2238:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2183:
2179:
2175:
2172:
2169:
2147:
2143:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2107:
2103:
2099:
2094:
2090:
2078:law of cosines
2058:
2038:
2035:
2032:
2029:
2023:
2020:
2017:
2013:
2008:
2002:
1999:
1996:
1992:
1987:
1981:
1978:
1975:
1971:
1949:
1944:
1940:
1937:
1934:
1928:
1923:
1919:
1916:
1913:
1907:
1902:
1898:
1895:
1892:
1869:
1849:
1829:
1809:
1789:
1769:
1749:
1746:
1743:
1718:Law of cosines
1709:
1706:
1703:
1702:
1699:
1696:
1693:
1690:
1673:
1668:
1648:
1644:
1643:
1640:
1623:
1620:
1607:
1603:
1590:
1573:
1551:
1531:
1527:
1526:
1523:
1510:
1506:
1493:
1490:
1477:
1473:
1460:
1443:
1438:
1418:
1414:
1413:
1410:
1397:
1393:
1380:
1377:
1360:
1343:
1321:
1301:
1297:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1271:
1270:
1267:
1264:
1261:
1258:
1253:
1248:
1243:
1237:
1236:
1229:
1222:
1199:
1196:
1191:
1188:
1185:
1182:
1160:
1156:
1150:
1145:
1141:
1137:
1134:
1131:
1126:
1122:
1118:
1115:
1093:
1063:
1043:
1040:
1037:
1034:
1031:
1011:
1008:
1005:
1002:
999:
987:
984:
968:
951:
945:
942:
939:
936:
933:
929:
924:
921:
919:
917:
914:
911:
908:
905:
902:
901:
898:
885:
879:
876:
873:
870:
867:
863:
858:
855:
853:
851:
848:
845:
842:
839:
836:
835:
832:
819:
813:
810:
807:
804:
801:
797:
792:
789:
787:
785:
782:
779:
776:
773:
770:
769:
766:
753:
747:
744:
741:
738:
735:
730:
727:
724:
721:
718:
712:
709:
707:
705:
702:
699:
696:
693:
690:
689:
668:
665:
653:
640:
637:
634:
631:
628:
625:
621:
608:
605:
602:
599:
596:
593:
581:
580:
568:
553:
541:
526:
514:
486:
483:
480:
460:
440:
429:right triangle
416:
397:For the angle
390:
387:
385:
382:
309:
306:
303:
300:
297:
277:
274:
271:
268:
265:
245:
218:right triangle
188:
187:
184:Fourier series
177:
173:
172:
143:
140:
137:
134:
131:
128:
125:
123:
106:
103:
100:
97:
94:
91:
88:
86:
76:
72:
71:
67:
66:
58:
57:
26:
9:
6:
4:
3:
2:
14851:
14840:
14837:
14835:
14832:
14831:
14829:
14814:
14811:
14809:
14806:
14804:
14801:
14799:
14796:
14795:
14793:
14789:
14783:
14780:
14778:
14775:
14773:
14770:
14766:
14763:
14762:
14761:
14760:Trigonometric
14758:
14757:
14755:
14751:
14742:
14737:
14735:
14730:
14728:
14723:
14722:
14719:
14712:
14711:Sine function
14707:
14703:
14702:
14692:
14688:
14684:
14679:
14678:
14672:
14668:
14661:
14660:
14654:
14651:
14645:
14641:
14640:
14634:
14631:
14625:
14621:
14620:
14615:
14611:
14608:
14602:
14598:
14594:
14590:
14589:
14583:
14580:
14574:
14570:
14566:
14565:
14559:
14556:
14554:0-486-20429-4
14550:
14546:
14542:
14537:
14534:
14530:
14526:
14520:
14516:
14512:
14508:
14507:Rudin, Walter
14504:
14501:
14495:
14491:
14487:
14483:
14479:
14476:
14472:
14471:
14465:
14463:
14460:
14459:
14453:
14449:
14445:
14441:
14437:
14434:
14428:
14424:
14420:
14416:
14411:
14408:
14404:
14403:
14397:
14394:
14391:
14385:
14379:
14373:
14367:
14361:
14355:
14345:
14344:
14338:
14335:
14333:0-553-27619-0
14329:
14324:
14323:
14316:
14313:
14307:
14303:
14299:
14295:
14290:
14287:
14283:
14279:
14276:
14270:
14266:
14262:
14258:
14257:
14251:
14248:
14242:
14238:
14234:
14233:
14227:
14223:
14219:
14212:
14207:
14203:
14199:
14198:
14193:
14189:
14185:
14184:
14172:
14168:
14165:
14159:
14153:
14147:
14141:
14135:
14129:
14123:
14117:
14111:
14105:
14099:
14097:
14090:
14084:
14082:
14074:
14069:
14063:
14059:
14056:
14052:
14051:
14047:
14044:
14039:
14035:
14032:
14026:
14018:
14016:9783540647676
14012:
14007:
14006:
13997:
13989:
13983:
13981:
13972:
13966:
13962:
13958:
13954:
13950:
13943:
13929:on 2013-10-19
13928:
13924:
13923:
13918:
13911:
13909:
13900:
13896:
13892:
13888:
13884:
13880:
13873:
13866:
13865:Gunter (1620)
13861:
13854:
13850:
13846:
13845:Merlet (2004)
13839:
13836:
13834:
13830:
13826:
13823:
13821:
13817:
13813:
13810:
13809:
13806:
13799:
13792:
13787:
13780:
13775:
13768:
13763:
13756:
13751:
13749:
13747:
13739:
13735:
13730:
13723:
13719:
13714:
13700:
13696:
13690:
13683:
13678:
13671:
13667:
13662:
13660:
13653:, p. 11.
13652:
13647:
13640:
13635:
13629:, p. 88.
13628:
13623:
13616:
13615:Powell (1981)
13611:
13604:
13600:
13595:
13593:
13585:
13580:
13578:
13562:
13556:
13549:
13545:
13540:
13533:
13528:
13521:
13516:
13509:
13504:
13497:
13493:
13488:
13481:
13476:
13468:
13463:, p. 42.
13462:
13457:
13450:
13445:
13438:
13433:
13426:
13421:
13414:
13410:
13405:
13403:
13388:
13384:
13378:
13371:
13366:
13359:
13355:
13350:
13343:
13338:
13331:
13326:
13320:, p. 47.
13319:
13314:
13307:
13303:
13298:
13292:, p. 41.
13291:
13286:
13279:
13275:
13270:
13263:
13259:
13254:
13252:
13244:
13240:
13236:
13231:
13225:, p. 42.
13224:
13219:
13212:
13208:
13203:
13196:
13192:
13187:
13185:
13183:
13178:
13164:
13160:
13154:
13132:
13126:
13123:
13120:
13114:
13108:
13105:
13082:
13076:
13071:
13067:
13056:
13049:
13031:
13028:
13024:
13001:
12998:
12994:
12983:
12979:
12966:
12960:
12957:
12955:
12952:
12950:
12947:
12945:
12942:
12940:
12937:
12935:
12932:
12930:
12929:Sine quadrant
12927:
12925:
12924:Sine integral
12922:
12920:
12917:
12915:
12914:Sinc function
12912:
12910:
12907:
12904:
12901:
12899:
12896:
12894:
12891:
12889:
12886:
12884:
12881:
12879:
12876:
12874:
12871:
12869:
12866:
12864:
12861:
12859:
12856:
12854:
12851:
12849:
12846:
12844:
12841:
12839:
12836:
12834:
12831:
12830:
12823:
12807:
12804:
12792:
12776:
12773:
12749:
12746:
12734:
12718:
12715:
12693:
12673:
12670:
12660:
12653:
12648:
12632:
12626:
12623:
12617:
12614:
12582:
12574:
12564:
12562:
12555:
12551:
12544:module, e.g.
12527:
12523:
12519:
12515:
12509:
12501:
12493:
12487:
12483:
12470:
12467:
12456:
12454:
12449:
12446:
12440:
12432:
12430:
12424:
12413:
12410:
12402:
12392:
12388:
12382:
12381:
12376:This section
12374:
12370:
12365:
12364:
12356:
12355:
12351:
12347:
12343:
12339:
12335:
12331:
12327:
12323:
12319:
12315:
12311:
12307:
12303:
12299:
12294:
12292:
12288:
12284:
12280:
12279:Albert Girard
12276:
12272:
12268:
12263:
12261:
12257:
12253:
12249:
12245:
12240:
12238:
12237:
12232:
12231:
12226:
12222:
12218:
12216:
12210:
12203:
12202:(90–165 CE).
12201:
12197:
12193:
12189:
12185:
12181:
12173:
12169:
12164:
12159:
12149:
12147:
12143:
12139:
12138:Edmund Gunter
12135:
12131:
12128:'sine of the
12126:
12121:
12116:
12114:
12110:
12106:
12105:
12100:
12096:
12092:
12088:
12084:
12079:
12075:
12070:
12058:
12052:
12047:
12043:
12039:
12030:
12029:Ancient Greek
12025:
12019:
12014:
12010:
12004:
11984:
11981:
11978:
11977:
11971:
11967:
11962:
11958:
11953:
11949:
11948:
11945:
11940:
11932:
11929:
11926:
11925:
11919:
11915:
11910:
11906:
11901:
11897:
11896:
11893:
11883:
11881:
11862:
11859:
11851:
11847:
11843:
11838:
11834:
11827:
11817:
11816:
11815:
11813:
11809:
11805:
11786:
11780:
11777:
11774:
11768:
11764:
11760:
11755:
11752:
11746:
11742:
11739:
11733:
11730:
11727:
11716:
11713:
11710:
11702:
11699:
11693:
11690:
11684:
11678:
11671:
11670:
11669:
11667:
11663:
11644:
11635:
11632:
11626:
11623:
11619:
11614:
11608:
11605:
11602:
11590:
11577:
11576:
11575:
11573:
11569:
11565:
11555:
11542:
11538:
11530:
11526:
11520:
11516:
11510:
11507:
11503:
11492:
11489:
11486:
11482:
11478:
11475:
11472:
11466:
11463:
11457:
11454:
11445:
11432:
11424:
11416:
11413:
11410:
11403:
11388:
11385:
11382:
11378:
11374:
11365:
11362:
11356:
11351:
11347:
11340:
11336:
11307:
11304:
11298:
11295:
11291:
11264:
11260:
11256:
11251:
11247:
11239:
11231:
11228:
11212:
11209:
11206:
11202:
11198:
11195:
11192:
11187:
11184:
11179:
11173:
11170:
11167:
11160:
11152:
11149:
11130:
11127:
11124:
11120:
11111:
11080:
11074:
11071:
11065:
11059:
11056:
11053:
11050:
11044:
11038:
11035:
11029:
11023:
11020:
11017:
11015:
11004:
11001:
10995:
10992:
10986:
10980:
10977:
10974:
10968:
10965:
10959:
10956:
10950:
10944:
10941:
10938:
10936:
10928:
10925:
10922:
10919:
10913:
10910:
10900:
10894:
10891:
10885:
10879:
10876:
10873:
10870:
10864:
10858:
10855:
10849:
10843:
10840:
10837:
10835:
10824:
10821:
10815:
10812:
10806:
10800:
10797:
10794:
10788:
10785:
10779:
10776:
10770:
10764:
10761:
10758:
10756:
10748:
10745:
10742:
10739:
10733:
10730:
10719:
10718:
10717:
10714:
10712:
10708:
10682:
10679:
10673:
10670:
10667:
10665:
10655:
10649:
10646:
10643:
10639:
10635:
10630:
10627:
10623:
10616:
10614:
10604:
10601:
10597:
10590:
10584:
10581:
10571:
10563:
10560:
10544:
10541:
10538:
10534:
10530:
10528:
10520:
10514:
10511:
10503:
10499:
10496:
10492:
10488:
10485:
10482:
10479:
10476:
10474:
10464:
10459:
10455:
10452:
10448:
10444:
10441:
10435:
10433:
10422:
10419:
10412:
10409:
10406:
10402:
10398:
10393:
10390:
10386:
10379:
10377:
10367:
10364:
10361:
10358:
10354:
10347:
10341:
10338:
10335:
10332:
10322:
10314:
10311:
10295:
10292:
10289:
10285:
10281:
10279:
10271:
10265:
10262:
10251:
10250:
10249:
10247:
10239:
10234:
10227:
10223:
10219:
10210:
10194:
10191:
10187:
10183:
10180:
10177:
10157:
10148:
10134:
10114:
10094:
10070:
10064:
10058:
10055:
10052:
10049:
10047:
10039:
10033:
10030:
10023:
10017:
10011:
10008:
10005:
10002:
10000:
9992:
9986:
9983:
9959:
9950:
9944:
9941:
9938:
9935:
9929:
9923:
9920:
9914:
9911:
9908:
9885:
9882:
9879:
9869:
9865:
9843:
9840:
9836:
9812:
9806:
9803:
9780:
9774:
9771:
9763:
9754:
9737:
9734:
9731:
9728:
9725:
9722:
9719:
9716:
9713:
9710:
9707:
9704:
9701:
9698:
9695:
9692:
9690:
9685:
9682:
9679:
9672:
9669:
9666:
9663:
9660:
9657:
9654:
9651:
9648:
9645:
9642:
9639:
9636:
9633:
9630:
9627:
9625:
9620:
9617:
9614:
9602:
9584:
9581:
9576:
9573:
9553:
9533:
9513:
9510:
9507:
9504:
9501:
9498:
9489:
9472:
9464:
9461:
9457:
9450:
9447:
9444:
9442:
9437:
9434:
9431:
9424:
9416:
9413:
9409:
9402:
9399:
9396:
9394:
9389:
9386:
9383:
9357:
9354:
9350:
9341:
9325:
9303:
9300:
9296:
9287:
9286:complex plane
9282:
9265:
9259:
9253:
9250:
9247:
9244:
9238:
9232:
9229:
9226:
9224:
9217:
9214:
9211:
9207:
9199:
9193:
9187:
9184:
9181:
9178:
9172:
9166:
9163:
9160:
9158:
9151:
9148:
9144:
9131:
9126:
9109:
9104:
9098:
9095:
9092:
9088:
9084:
9079:
9076:
9072:
9065:
9063:
9055:
9049:
9046:
9039:
9033:
9030:
9023:
9020:
9017:
9013:
9009:
9004:
9001:
8997:
8990:
8988:
8980:
8974:
8971:
8959:
8955:
8954:complex plane
8939:
8931:
8927:
8923:
8907:
8904:
8896:
8886:
8882:
8876:
8875:
8870:This section
8868:
8864:
8859:
8858:
8850:
8833:
8830:
8827:
8820:
8817:
8811:
8808:
8802:
8796:
8791:
8788:
8783:
8779:
8773:
8770:
8765:
8763:
8756:
8752:
8744:
8741:
8738:
8731:
8728:
8722:
8719:
8713:
8707:
8702:
8699:
8694:
8690:
8684:
8681:
8676:
8674:
8667:
8663:
8638:
8618:
8612:
8609:
8603:
8600:
8595:
8591:
8587:
8581:
8578:
8572:
8569:
8564:
8560:
8549:
8546:
8543:
8539:
8535:
8530:
8526:
8520:
8517:
8493:
8489:
8466:
8462:
8453:
8448:
8435:
8429:
8426:
8420:
8417:
8412:
8408:
8402:
8397:
8394:
8391:
8387:
8383:
8377:
8374:
8368:
8365:
8360:
8356:
8350:
8345:
8342:
8339:
8335:
8331:
8326:
8322:
8318:
8312:
8306:
8283:
8277:
8257:
8235:
8231:
8208:
8204:
8195:
8191:
8187:
8183:
8178:
8159:
8156:
8152:
8145:
8139:
8136:
8126:
8118:
8115:
8099:
8096:
8093:
8089:
8085:
8083:
8075:
8072:
8066:
8063:
8057:
8053:
8047:
8041:
8038:
8032:
8028:
8022:
8016:
8013:
8007:
8003:
7997:
7994:
7991:
7989:
7981:
7975:
7972:
7942:
7939:
7936:
7933:
7929:
7922:
7916:
7913:
7910:
7907:
7897:
7889:
7886:
7870:
7867:
7864:
7860:
7856:
7854:
7846:
7843:
7837:
7834:
7828:
7824:
7818:
7812:
7809:
7803:
7799:
7793:
7787:
7784:
7778:
7774:
7768:
7765:
7762:
7760:
7752:
7746:
7743:
7731:
7715:
7708:—where
7695:
7688:
7684:
7683:Taylor series
7668:
7665:
7662:
7635:
7632:
7629:
7619:
7616:
7609:
7606:
7603:
7593:
7586:
7583:
7580:
7570:
7563:
7560:
7557:
7547:
7541:
7536:
7530:
7524:
7516:
7513:
7510:
7507:
7500:
7474:
7471:
7468:
7465:
7438:
7432:
7429:
7406:
7400:
7397:
7394:
7371:
7365:
7362:
7359:
7336:
7330:
7327:
7304:
7298:
7295:
7287:
7283:
7279:
7275:
7274:Taylor series
7266:
7257:
7240:
7233:
7230:
7224:
7221:
7218:
7215:
7209:
7203:
7197:
7192:
7188:
7181:
7174:
7171:
7165:
7162:
7159:
7156:
7150:
7144:
7138:
7133:
7129:
7113:
7106:
7102:
7100:
7080:
7074:
7069:
7065:
7061:
7058:
7055:
7052:
7050:
7042:
7039:
7033:
7027:
7022:
7018:
7014:
7011:
7009:
6998:
6992:
6987:
6983:
6979:
6973:
6967:
6962:
6958:
6954:
6952:
6944:
6941:
6935:
6932:
6925:
6919:
6913:
6910:
6904:
6898:
6895:
6892:
6889:
6887:
6879:
6876:
6870:
6867:
6854:
6841:
6838:
6832:
6826:
6821:
6817:
6813:
6807:
6801:
6796:
6792:
6783:
6779:
6776:According to
6773:
6763:
6746:
6743:
6740:
6737:
6734:
6722:
6719:
6710:
6704:
6701:
6695:
6692:
6683:
6680:
6675:
6672:
6669:
6664:
6661:
6656:
6644:
6641:
6632:
6626:
6623:
6617:
6614:
6590:
6587:
6578:
6572:
6569:
6563:
6560:
6556:
6553:
6544:
6538:
6535:
6529:
6526:
6502:
6499:
6496:
6493:
6487:
6481:
6478:
6475:
6472:
6469:
6456:
6453:
6450:
6447:
6444:
6438:
6432:
6429:
6426:
6423:
6413:
6410:
6404:
6398:
6395:
6388:
6385:
6382:
6379:
6373:
6367:
6364:
6361:
6358:
6355:
6352:
6339:
6336:
6333:
6330:
6327:
6321:
6315:
6312:
6309:
6306:
6296:
6293:
6287:
6281:
6278:
6254:
6234:
6230:
6217:
6213:
6210:
6207:
6203:
6190:
6186:
6183:
6180:
6177:
6168:
6154:
6134:
6112:
6109:
6085:
6082:
6077:
6069:
6050:
6044:
6041:
6021:
6013:
5997:
5994:
5991:
5985:
5979:
5976:
5956:
5953:
5947:
5941:
5938:
5918:
5915:
5909:
5903:
5900:
5880:
5877:
5871:
5868:
5862:
5859:
5839:
5836:
5830:
5824:
5821:
5801:
5798:
5792:
5786:
5783:
5775:
5757:
5754:
5750:
5727:
5724:
5720:
5711:
5701:
5693:
5686:
5677:
5675:
5659:
5651:
5615:
5612:
5609:
5606:
5603:
5600:
5595:
5591:
5588:
5582:
5576:
5573:
5566:
5558:
5554:
5550:
5538:
5530:
5522:
5518:
5514:
5499:
5495:
5491:
5486:
5480:
5477:
5469:
5453:
5446:with modulus
5445:
5426:
5423:
5420:
5414:
5391:
5387:
5380:
5376:
5371:
5368:
5364:
5360:
5352:
5347:
5344:
5341:
5332:
5326:
5321:
5317:
5313:
5310:
5302:
5297:
5293:
5272:
5252:
5244:
5240:
5224:
5204:
5201:
5198:
5192:
5186:
5183:
5180:
5177:
5174:
5167:
5161:
5158:
5155:
5151:
5148:
5142:
5136:
5133:
5130:
5127:
5124:
5121:
5114:
5108:
5105:
5102:
5094:
5088:
5078:
5064:
5061:
5055:
5049:
5029:
5026:
5020:
5014:
4991:
4985:
4982:
4979:
4973:
4966:
4963:
4939:
4933:
4930:
4924:
4917:
4914:
4905:
4889:
4886:
4880:
4874:
4854:
4851:
4845:
4839:
4832:
4813:
4807:
4804:
4801:
4795:
4788:
4785:
4761:
4755:
4752:
4746:
4739:
4736:
4728:
4706:
4700:
4697:
4691:
4685:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4624:
4621:
4607:
4600:
4597:
4594:
4580:
4573:
4559:
4556:
4553:
4550:
4547:
4542:
4538:
4535:
4525:
4509:
4505:
4501:
4498:
4495:
4490:
4486:
4478:
4475:
4474:
4470:
4467:
4453:
4446:
4443:
4440:
4426:
4419:
4403:
4399:
4396:
4390:
4387:
4384:
4381:
4374:
4358:
4354:
4350:
4347:
4344:
4339:
4335:
4327:
4324:
4323:
4319:
4316:
4302:
4295:
4292:
4289:
4275:
4268:
4254:
4251:
4248:
4245:
4240:
4237:
4228:
4212:
4208:
4204:
4201:
4198:
4193:
4189:
4181:
4178:
4177:
4173:
4170:
4156:
4149:
4146:
4143:
4129:
4122:
4106:
4103:
4098:
4095:
4092:
4089:
4082:
4066:
4062:
4058:
4055:
4052:
4047:
4043:
4035:
4032:
4031:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3989:
3977:
3973:
3970:
3956:
3936:
3927:
3923:
3919:
3915:
3899:
3893:
3887:
3884:
3881:
3878:
3872:
3866:
3863:
3857:
3854:
3850:
3844:
3838:
3832:
3829:
3826:
3820:
3814:
3811:
3805:
3802:
3798:
3784:
3781:), using the
3780:
3775:
3770:
3760:
3746:
3743:
3737:
3731:
3728:
3720:
3719:Dottie number
3704:
3701:
3695:
3689:
3686:
3678:
3674:
3661:
3654:
3645:
3641:
3635:
3631:
3614:
3610:
3606:
3603:
3598:
3595:
3589:
3585:
3582:
3579:
3577:
3569:
3563:
3560:
3553:
3549:
3545:
3542:
3537:
3534:
3528:
3524:
3521:
3518:
3516:
3508:
3502:
3499:
3473:
3470:
3460:
3441:
3435:
3432:
3426:
3423:
3417:
3409:
3390:
3384:
3381:
3378:
3372:
3369:
3363:
3355:
3339:
3330:
3317:
3311:
3305:
3302:
3299:
3293:
3290:
3287:
3284:
3278:
3275:
3271:
3265:
3259:
3256:
3253:
3247:
3244:
3241:
3238:
3232:
3229:
3221:
3203:
3193:
3179:
3176:
3173:
3170:
3167:
3164:
3156:
3140:
3137:
3134:
3131:
3111:
3108:
3105:
3083:
3060:
3057:
3052:
3049:
3029:
3026:
3023:
3007:
3000:
2995:
2988:
2981:
2972:
2956:
2953:
2948:
2945:
2942:
2939:
2917:
2894:
2891:
2886:
2883:
2880:
2877:
2868:
2855:
2852:
2849:
2843:
2837:
2834:
2830:
2827:
2824:
2818:
2812:
2809:
2786:
2780:
2777:
2754:
2748:
2745:
2723:
2700:
2677:
2656:
2648:
2632:
2629:
2624:
2620:
2616:
2611:
2607:
2583:
2580:
2577:
2566:
2551:
2534:
2523:
2510:
2495:
2484:
2482:
2474:
2468:
2465:
2458:
2447:
2434:
2414:
2398:
2396:
2388:
2382:
2379:
2311:
2259:
2255:
2251:
2247:
2246:cross product
2237:
2235:
2219:
2216:
2210:
2204:
2201:
2181:
2177:
2173:
2170:
2167:
2145:
2141:
2137:
2131:
2125:
2122:
2119:
2116:
2113:
2110:
2105:
2101:
2097:
2092:
2088:
2079:
2074:
2072:
2056:
2036:
2033:
2030:
2027:
2021:
2018:
2015:
2011:
2006:
2000:
1997:
1994:
1990:
1985:
1979:
1976:
1973:
1969:
1947:
1942:
1938:
1935:
1932:
1926:
1921:
1917:
1914:
1911:
1905:
1900:
1896:
1893:
1890:
1867:
1847:
1827:
1807:
1787:
1767:
1747:
1744:
1741:
1733:
1724:
1719:
1715:
1700:
1697:
1694:
1691:
1674:
1669:
1649:
1646:
1645:
1641:
1624:
1621:
1605:
1601:
1591:
1574:
1552:
1532:
1529:
1528:
1524:
1508:
1504:
1494:
1491:
1475:
1471:
1461:
1444:
1439:
1419:
1416:
1415:
1411:
1395:
1391:
1381:
1378:
1361:
1344:
1322:
1302:
1299:
1298:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1272:
1268:
1265:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1238:
1234:
1227:
1221:
1216:
1213:
1197:
1194:
1189:
1186:
1183:
1180:
1158:
1154:
1148:
1143:
1139:
1135:
1132:
1129:
1124:
1120:
1116:
1113:
1106:; therefore,
1091:
1081:
1077:
1061:
1038:
1032:
1029:
1006:
1000:
997:
983:
966:
949:
940:
934:
931:
927:
922:
920:
912:
906:
903:
896:
883:
874:
868:
865:
861:
856:
854:
846:
840:
837:
830:
817:
808:
802:
799:
795:
790:
788:
780:
774:
771:
764:
751:
742:
736:
733:
725:
719:
716:
710:
708:
700:
694:
691:
678:
674:
664:
651:
638:
632:
626:
623:
619:
606:
600:
594:
591:
566:
558:
557:adjacent side
554:
539:
531:
527:
512:
504:
503:opposite side
500:
499:
498:
484:
481:
478:
458:
438:
430:
414:
401:
395:
381:
379:
375:
371:
369:
363:
357:
353:
349:
344:
342:
338:
334:
330:
326:
321:
304:
298:
295:
272:
266:
263:
243:
235:
231:
227:
223:
219:
215:
211:
207:
203:
199:
195:
185:
181:
178:
174:
141:
135:
129:
126:
104:
98:
92:
89:
77:
73:
68:
64:
59:
54:
49:
48:sign function
45:
41:
37:
33:
19:
14764:
14676:
14658:
14639:Trigonometry
14638:
14619:Trigonometry
14618:
14591:, Springer,
14587:
14563:
14540:
14510:
14485:
14469:
14462:
14458:
14455:
14447:
14417:, Springer,
14414:
14401:
14350:
14342:
14321:
14293:
14285:
14259:, Springer,
14255:
14231:
14221:
14217:
14200:, New York:
14195:
14158:
14146:
14134:
14122:
14110:
14068:
14041:
14025:
14009:. Springer.
14004:
13996:
13956:
13942:
13931:. Retrieved
13927:the original
13920:
13882:
13878:
13872:
13860:
13833:al-Khwārizmī
13828:
13815:
13798:
13791:Smith (1958)
13786:
13774:
13762:
13729:
13713:
13702:. Retrieved
13698:
13689:
13684:, p. 2.
13682:Rudin (1987)
13677:
13666:Howie (2003)
13646:
13641:, p. 1.
13634:
13627:Rudin (1987)
13622:
13610:
13564:. Retrieved
13555:
13544:Young (2017)
13539:
13527:
13515:
13508:Adlaj (2012)
13503:
13492:Vince (2023)
13487:
13475:
13456:
13444:
13432:
13420:
13390:. Retrieved
13386:
13377:
13365:
13354:Young (2012)
13349:
13337:
13325:
13313:
13302:Young (2017)
13297:
13285:
13274:Axler (2012)
13269:
13258:Axler (2012)
13235:Young (2017)
13230:
13218:
13207:Young (2017)
13202:
13191:Young (2017)
13162:
13153:
13055:
12982:
12964:
12873:Law of sines
12793:
12735:
12661:
12651:
12646:
12570:
12546:cmath.sin(z)
12471:
12468:
12457:
12450:
12441:
12433:
12426:
12405:
12396:
12385:Please help
12380:verification
12377:
12353:
12349:
12345:
12341:
12337:
12333:
12325:
12317:
12309:
12301:
12295:
12282:
12274:
12270:
12266:
12264:
12256:Al-Khwārizmī
12248:law of sines
12241:
12234:
12228:
12225:Gupta period
12204:
12177:
12145:
12141:
12133:
12119:
12117:
12112:
12103:
12091:al-Khwārizmī
12040:). This was
12008:
12006:
11943:
11938:
11891:
11877:
11807:
11801:
11659:
11563:
11561:
11446:
11107:
10715:
10709:. These are
10704:
10245:
10243:
10237:
10225:
10149:
9861:
9490:
9283:
9127:
8960:as follows:
8919:
8899:
8890:
8879:Please help
8874:verification
8871:
8449:
8179:
7687:real numbers
7278:power series
7271:
7120:
7111:
7101:
6855:
6775:
6169:
5707:
5699:
5691:
5237:denotes the
5090:
4630:
3918:monotonicity
3788:
3778:
3670:
3659:
3649:
3643:
3639:
3331:
3194:
3014:
3005:
2998:
2986:
2869:
2562:
2243:
2075:
2071:circumradius
1732:law of sines
1729:
1714:Law of sines
1232:
1225:
1219:
989:
670:
582:
556:
529:
502:
406:
399:
378:Gupta period
345:
322:
201:
197:
191:
180:Trigonometry
14515:McGraw-Hill
14180:Works cited
13853:Katz (2008)
13849:Maor (1998)
13808:to either
13779:Katz (2008)
13767:Maor (1998)
13734:Maor (1998)
13159:Thomas Fale
12949:SOH-CAH-TOA
12534:math.cos(x)
12530:math.sin(x)
12508:long double
12399:August 2024
12314:Roger Cotes
12230:Aryabhatiya
12223:during the
12200:Roman Egypt
9340:unit circle
8893:August 2024
5931:, but also
5814:, but also
4622:Increasing
4595:Increasing
4468:Increasing
4441:Decreasing
4317:Decreasing
4290:Decreasing
4171:Decreasing
4144:Increasing
3673:fixed point
3332:A function
3011:coordinate.
2994:unit circle
2565:unit circle
2250:dot product
2194:from which
1760:with sides
376:during the
356:light waves
329:unit circle
194:mathematics
14828:Categories
14777:Hyperbolic
13933:2010-07-13
13820:Al-Battani
13736:, p.
13720:, p.
13704:2019-08-12
13668:, p.
13601:, p.
13546:, p.
13494:, p.
13411:, p.
13392:2019-05-26
13356:, p.
13304:, p.
13276:, p.
13260:, p.
13237:, p.
13209:, p.
13193:, p.
12969:References
12903:Polar sine
12421:See also:
12304:is not an
12291:Copernicus
12250:, used in
12188:Hipparchus
12087:Al-Battani
11992:Background
11985:Magnitude
11933:Magnitude
7626:when
7600:when
7577:when
7554:when
7286:derivative
6226:hypotenuse
6199:hypotenuse
5243:arc length
2692:axis. The
956:hypotenuse
889:hypotenuse
677:reciprocal
647:hypotenuse
615:hypotenuse
530:hypotenuse
234:hypotenuse
226:hypotenuse
153:hypotenuse
116:hypotenuse
14354:jyā-ardha
14224:(8): 1097
13816:Astronomy
13566:August 9,
13173:Citations
13127:
13121:⋅
13109:
13077:
13029:−
12999:−
12974:Footnotes
12934:Sine wave
12805:π
12774:π
12747:π
12716:π
12694:π
12674:π
12624:π
12618:
12583:π
12518:data type
12174:of angles
12146:cotangens
12118:The word
12074:homograph
12007:The word
11997:Etymology
11880:pendulums
11825:Δ
11778:−
11769:ζ
11753:π
11743:
11731:−
11722:Γ
11714:−
11703:π
11679:ζ
11633:π
11627:
11620:π
11606:−
11597:Γ
11585:Γ
11511:−
11498:∞
11483:∏
11476:π
11464:π
11458:
11414:−
11397:∞
11392:∞
11389:−
11379:∑
11363:π
11357:
11337:π
11305:π
11299:
11292:π
11257:−
11229:−
11218:∞
11203:∑
11193:−
11171:−
11150:−
11139:∞
11134:∞
11131:−
11121:∑
11075:
11060:
11051:−
11039:
11024:
10996:
10981:
10975:−
10960:
10945:
10914:
10895:
10880:
10859:
10844:
10816:
10801:
10780:
10765:
10734:
10674:
10644:−
10561:−
10550:∞
10535:∑
10515:
10489:
10480:−
10445:
10407:−
10399:−
10312:−
10301:∞
10286:∑
10266:
10248:, gives:
10195:θ
10158:θ
10115:φ
10065:θ
10059:
10034:
10018:θ
10012:
9987:
9951:θ
9945:
9930:θ
9924:
9886:θ
9866:with its
9844:θ
9813:θ
9807:
9781:θ
9775:
9732:
9723:
9714:−
9708:
9699:
9683:
9667:
9658:
9643:
9634:
9618:
9582:−
9465:θ
9451:
9438:θ
9435:
9417:θ
9403:
9390:θ
9387:
9358:θ
9260:θ
9254:
9245:−
9239:θ
9233:
9218:θ
9212:−
9194:θ
9188:
9173:θ
9167:
9152:θ
9099:θ
9093:−
9080:θ
9056:θ
9050:
9024:θ
9018:−
9010:−
9005:θ
8981:θ
8975:
8940:θ
8812:
8792:π
8780:∫
8774:π
8723:
8703:π
8691:∫
8685:π
8604:
8573:
8555:∞
8540:∑
8421:
8388:∑
8369:
8336:∑
8116:−
8105:∞
8090:∑
8076:⋯
8048:−
7998:−
7976:
7887:−
7876:∞
7861:∑
7847:⋯
7819:−
7769:−
7747:
7617:−
7525:
7433:
7401:
7395:−
7366:
7360:−
7331:
7299:
7234:θ
7225:
7204:θ
7198:
7175:θ
7166:
7160:−
7145:θ
7139:
7081:θ
7075:
7059:−
7040:−
7034:θ
7028:
6999:θ
6993:
6980:−
6974:θ
6968:
6945:θ
6936:
6920:θ
6914:
6905:θ
6899:
6880:θ
6871:
6833:θ
6827:
6808:θ
6802:
6747:π
6744:≤
6741:θ
6738:≤
6723:θ
6711:θ
6705:
6696:
6681:π
6676:≤
6673:θ
6670:≤
6662:π
6657:−
6645:θ
6633:θ
6627:
6618:
6573:
6564:
6539:
6530:
6500:π
6482:
6476:−
6451:π
6433:
6418:⟺
6399:
6386:π
6368:
6362:−
6359:π
6334:π
6316:
6301:⟺
6282:
6214:
6187:
6178:θ
6155:π
6110:π
6083:π
6078:−
6045:
5998:π
5980:
5957:π
5942:
5904:
5872:π
5863:
5831:π
5825:
5787:
5774:injective
5755:−
5725:−
5660:ϖ
5636:Γ
5616:…
5610:≈
5607:ϖ
5596:ϖ
5592:π
5577:π
5545:Γ
5509:Γ
5496:π
5421:φ
5415:
5361:
5327:
5294:∫
5187:
5162:
5156:∫
5137:
5131:−
5109:
5103:∫
4992:θ
4983:−
4974:θ
4940:θ
4925:θ
4829:with the
4814:θ
4805:−
4796:θ
4762:θ
4747:θ
4707:θ
4692:θ
4660:θ
4657:
4648:θ
4645:
4581:−
4560:π
4539:π
4510:∘
4491:∘
4454:−
4427:−
4400:π
4382:π
4359:∘
4340:∘
4303:−
4255:π
4238:π
4213:∘
4194:∘
4104:π
4067:∘
4048:∘
4027:Convexity
4012:Convexity
3957:−
3926:concavity
3888:
3882:−
3867:
3833:
3815:
3732:
3690:
3607:θ
3604:−
3596:π
3586:
3570:θ
3564:
3546:θ
3543:−
3535:π
3525:
3509:θ
3503:
3471:π
3424:−
3382:−
3370:−
3312:θ
3306:
3294:π
3285:θ
3279:
3266:θ
3260:
3248:π
3239:θ
3233:
3177:≤
3171:≤
3165:−
3141:π
3132:θ
3112:π
3106:θ
3098:axis. If
3058:π
3050:θ
3024:θ
2954:π
2946:θ
2892:π
2884:θ
2844:θ
2838:
2819:θ
2813:
2787:θ
2781:
2755:θ
2749:
2657:θ
2496:⋅
2475:θ
2469:
2415:×
2389:θ
2383:
2312:θ
2211:γ
2205:
2174:π
2168:γ
2132:γ
2126:
2111:−
2022:γ
2019:
2001:β
1998:
1980:α
1977:
1939:γ
1936:
1918:β
1915:
1897:α
1894:
1868:γ
1848:β
1828:α
1555:66
1325:33
1195:π
1187:α
1144:∘
1136:
1125:∘
1117:
1062:α
1039:α
1033:
1007:α
1001:
941:θ
935:
913:θ
907:
875:θ
869:
847:θ
841:
809:θ
803:
781:θ
775:
743:θ
737:
726:θ
720:
701:θ
695:
633:α
627:
601:α
595:
459:α
439:α
415:α
305:θ
299:
273:θ
267:
244:θ
136:θ
130:
99:θ
93:
14803:Exsecant
14673:(1968),
14616:(2012),
14564:Calculus
14509:(1987),
14484:(1981),
14446:(2011),
14284:(1620),
14194:(1970),
14167:Archived
14058:Archived
14046:Archived
14034:Archived
13899:27967990
13387:oeis.org
12826:See also
12605:sinpi(x)
12528:defines
12429:IEEE 754
12320:(1722).
12215:koṭi-jyā
12013:Sanskrit
11664:for the
11570:for the
9566:, where
6223:adjacent
6196:opposite
5093:integral
4967:′
4918:′
4789:′
4740:′
4625:Concave
4293:Concave
4174:Concave
4147:Concave
4022:Monotony
4007:Monotony
3976:Quadrant
3220:periodic
3157:between
1269:Decimal
1263:Decimal
1251:Gradians
961:adjacent
892:opposite
826:opposite
823:adjacent
760:adjacent
757:opposite
644:adjacent
612:opposite
368:koṭi-jyā
348:periodic
222:triangle
148:adjacent
111:opposite
14798:Versine
14691:0236587
14533:0924157
13829:Algebra
12652:πx
12550:CPython
12437:sin(10)
12298:Leibniz
12196:Ptolemy
12172:versine
12152:History
12134:cosinus
10224:of sin(
5698:arccos(
5690:arcsin(
5672:is the
5648:is the
5442:is the
4598:Convex
4471:Convex
4444:Convex
4320:Convex
3997:Radians
3992:Degrees
3986:Cosine
3459:shifted
2645:in the
2254:vectors
1688:
1676:
1663:
1651:
1638:
1626:
1622:0.8660
1588:
1576:
1570:
1546:
1534:
1525:0.7071
1492:0.7071
1458:
1446:
1433:
1421:
1412:0.8660
1375:
1363:
1358:
1346:
1340:
1316:
1304:
1246:Radians
1241:Degrees
1218:Angle,
1076:similar
673:tangent
46:or the
14753:Groups
14689:
14646:
14626:
14603:
14575:
14551:
14531:
14521:
14496:
14429:
14330:
14308:
14271:
14243:
14040:", in
14013:
13967:
13897:
13847:. See
13059:Here,
12706:, and
12645:where
12526:Python
12522:math.h
12504:, and
12492:double
12486:math.h
12453:CORDIC
12354:cosec.
12352:, and
12273:, and
12192:Nicaea
12120:cosine
12046:Arabic
11806:, sin
10087:where
8196:. Let
7284:, the
6693:arccos
6615:arcsin
6570:arccos
6536:arcsin
6479:arccos
6430:arccos
6365:arcsin
6313:arcsin
6211:arccos
6184:arcsin
6042:arcsin
5977:arcsin
5939:arcsin
5901:arcsin
5628:where
5613:7.6404
5404:where
5217:where
3980:Angle
2049:where
1860:, and
1800:, and
1266:Exact
1260:Exact
208:of an
202:cosine
186:, etc.
14834:Angle
14813:atan2
14791:Other
14663:(PDF)
14390:sinus
14347:(PDF)
14214:(PDF)
13895:JSTOR
13805:sinus
12601:cospi
12597:sinpi
12573:turns
12542:cmath
12506:sinl(
12500:float
12498:sinf(
12342:tang.
12184:chord
12132:' as
12104:sinus
12033:χορδή
12015:word
11802:As a
9491:When
3983:Sine
3155:range
1256:Turns
352:sound
230:ratio
224:(the
214:angle
210:angle
14644:ISBN
14624:ISBN
14601:ISBN
14573:ISBN
14549:ISBN
14519:ISBN
14494:ISBN
14427:ISBN
14384:jaib
14372:jiba
14366:jīvá
14328:ISBN
14306:ISBN
14269:ISBN
14241:ISBN
14011:ISBN
13965:ISBN
13843:See
13568:2019
13467:help
13016:and
12808:2048
12599:and
12557:math
12538:math
12532:and
12490:sin(
12476:and
12462:and
12451:The
12350:sec.
12346:cot.
12338:cos.
12334:sin.
12233:and
12212:and
12113:sine
12109:toga
12089:and
12078:jayb
12051:jība
12024:jīvá
12009:sine
11562:sin(
11072:sinh
11036:cosh
10892:sinh
10856:cosh
10671:cosh
10486:sinh
10442:sinh
10107:and
9796:and
9729:sinh
9705:cosh
9664:sinh
9640:cosh
9603:as:
9546:and
9372:as:
8928:and
8926:real
8481:and
8450:The
8223:and
7320:are
7276:, a
6603:and
5708:The
5696:and
5652:and
5265:and
5042:and
4955:and
4867:and
4777:and
4554:<
4548:<
4502:<
4496:<
4391:<
4385:<
4351:<
4345:<
4252:<
4246:<
4205:<
4199:<
4099:<
4093:<
4059:<
4053:<
4017:Sign
4002:Sign
3408:even
3027:>
2949:<
2943:<
2887:<
2881:<
2770:and
2715:and
2346:and
2282:and
2248:and
2244:The
2076:The
1730:The
1716:and
1708:Laws
1647:90°
1642:0.5
1530:60°
1417:45°
1379:0.5
1300:30°
1231:cos(
1224:sin(
1190:<
1184:<
1022:and
555:The
528:The
501:The
365:and
354:and
325:real
288:and
204:are
200:and
198:sine
40:Sign
34:and
14593:doi
14419:doi
14381:as
14360:jyā
14298:doi
14261:doi
14053:by
13887:doi
13831:of
13818:of
13722:257
13496:162
13413:294
13358:165
13278:632
13262:634
13161:'s
13124:sin
13106:sin
13068:sin
13025:cos
12995:sin
12657:sin
12615:sin
12478:cos
12474:sin
12464:cos
12460:sin
12389:by
12324:'s
12308:of
12285:of
12275:tan
12271:cos
12267:sin
12209:jyā
12198:of
12190:of
12140:'s
12136:in
12083:جيب
12048:as
12044:in
12018:jyā
11740:sin
11624:sin
11455:sin
11348:sin
11296:sin
11057:sin
11021:cos
10993:sin
10978:sin
10957:cos
10942:cos
10911:cos
10877:cos
10841:sin
10813:sin
10798:cos
10777:cos
10762:sin
10731:sin
10512:cos
10263:sin
10056:sin
10009:cos
9942:sin
9921:cos
9804:sin
9772:cos
9720:sin
9696:cos
9680:cos
9655:cos
9631:sin
9615:sin
9432:cos
9384:sin
9251:sin
9230:cos
9185:sin
9164:cos
9047:cos
8972:sin
8883:by
8809:sin
8720:cos
8601:sin
8570:cos
8418:sin
8366:cos
7973:cos
7744:sin
7501:sin
7430:sin
7398:cos
7363:sin
7328:cos
7296:sin
7222:cos
7189:cos
7163:cos
7130:sin
7066:sin
7019:cos
6984:sin
6959:cos
6933:cos
6911:cos
6896:sin
6868:sin
6818:cos
6793:sin
6730:for
6702:cos
6652:for
6624:sin
6561:cos
6527:sin
6396:cos
6279:sin
6147:to
6100:to
5860:sin
5852:,
5822:sin
5784:sin
5751:cos
5721:sin
5318:cos
5285:is
5184:sin
5159:cos
5134:cos
5106:sin
4654:sin
4642:cos
4506:360
4487:270
4355:270
4336:180
4209:180
3885:sin
3864:cos
3830:cos
3812:sin
3729:cos
3687:sin
3679:is
3583:sin
3561:cos
3522:cos
3500:sin
3461:by
3410:if
3356:if
3354:odd
3303:cos
3276:cos
3257:sin
3230:sin
2835:cos
2810:sin
2778:sin
2746:cos
2466:cos
2380:sin
2256:in
2202:cos
2123:cos
2016:sin
1995:sin
1974:sin
1933:sin
1912:sin
1891:sin
1671:100
1274:0°
1133:cos
1114:sin
1080:leg
1030:cos
998:sin
932:cos
904:sec
866:sin
838:csc
800:tan
772:cot
734:cos
717:sin
692:tan
624:cos
592:sin
362:jyā
296:cos
264:sin
192:In
127:cos
90:sin
14830::
14687:MR
14685:,
14599:,
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14529:MR
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7237:)
7231:2
7228:(
7219:+
7216:1
7210:=
7207:)
7201:(
7193:2
7182:2
7178:)
7172:2
7169:(
7157:1
7151:=
7148:)
7142:(
7134:2
7115:-
7112:x
7084:)
7078:(
7070:2
7062:2
7056:1
7053:=
7043:1
7037:)
7031:(
7023:2
7015:2
7012:=
7002:)
6996:(
6988:2
6977:)
6971:(
6963:2
6955:=
6948:)
6942:2
6939:(
6926:,
6923:)
6917:(
6908:)
6902:(
6893:2
6890:=
6883:)
6877:2
6874:(
6839:=
6836:)
6830:(
6822:2
6814:+
6811:)
6805:(
6797:2
6735:0
6720:=
6717:)
6714:)
6708:(
6699:(
6684:2
6665:2
6642:=
6639:)
6636:)
6630:(
6621:(
6591:x
6588:=
6585:)
6582:)
6579:x
6576:(
6567:(
6557:x
6554:=
6551:)
6548:)
6545:x
6542:(
6533:(
6503:k
6497:2
6494:+
6491:)
6488:x
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6473:=
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6457:,
6454:k
6448:2
6445:+
6442:)
6439:x
6436:(
6427:=
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6414:x
6411:=
6408:)
6405:y
6402:(
6389:k
6383:2
6380:+
6377:)
6374:x
6371:(
6356:=
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6340:,
6337:k
6331:2
6328:+
6325:)
6322:x
6319:(
6310:=
6307:y
6297:x
6294:=
6291:)
6288:y
6285:(
6255:k
6235:,
6231:)
6218:(
6208:=
6204:)
6191:(
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6113:2
6086:2
6054:)
6051:x
6048:(
6022:x
5995:2
5992:=
5989:)
5986:0
5983:(
5954:=
5951:)
5948:0
5945:(
5919:0
5916:=
5913:)
5910:0
5907:(
5881:0
5878:=
5875:)
5869:2
5866:(
5840:0
5837:=
5834:)
5828:(
5802:0
5799:=
5796:)
5793:0
5790:(
5758:1
5728:1
5702:)
5700:x
5694:)
5692:x
5604:2
5601:+
5589:2
5583:=
5574:2
5567:2
5563:)
5559:4
5555:/
5551:1
5548:(
5539:+
5531:2
5527:)
5523:4
5519:/
5515:1
5512:(
5500:3
5492:2
5487:4
5481:=
5478:L
5454:k
5430:)
5427:k
5424:,
5418:(
5412:E
5392:,
5388:)
5381:2
5377:1
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5358:E
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5348:=
5345:x
5342:d
5336:)
5333:x
5330:(
5322:2
5314:+
5311:1
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5225:C
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5202:C
5199:+
5196:)
5193:x
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5178:x
5175:d
5171:)
5168:x
5165:(
5152:C
5149:+
5146:)
5143:x
5140:(
5128:=
5125:x
5122:d
5118:)
5115:x
5112:(
5065:1
5062:=
5059:)
5056:0
5053:(
5050:x
5030:0
5027:=
5024:)
5021:0
5018:(
5015:y
4995:)
4989:(
4986:y
4980:=
4977:)
4971:(
4964:x
4943:)
4937:(
4934:x
4931:=
4928:)
4922:(
4915:y
4890:1
4887:=
4884:)
4881:0
4878:(
4875:x
4855:0
4852:=
4849:)
4846:0
4843:(
4840:y
4817:)
4811:(
4808:y
4802:=
4799:)
4793:(
4786:x
4765:)
4759:(
4756:x
4753:=
4750:)
4744:(
4737:y
4713:)
4710:)
4704:(
4701:y
4698:,
4695:)
4689:(
4686:x
4683:(
4663:)
4651:,
4639:(
4608:+
4557:2
4551:x
4543:2
4536:3
4499:x
4404:2
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4388:x
4348:x
4276:+
4249:x
4241:2
4202:x
4157:+
4130:+
4107:2
4096:x
4090:0
4056:x
4044:0
3937:+
3929:(
3900:.
3897:)
3894:x
3891:(
3879:=
3876:)
3873:x
3870:(
3858:x
3855:d
3851:d
3845:,
3842:)
3839:x
3836:(
3827:=
3824:)
3821:x
3818:(
3806:x
3803:d
3799:d
3785:.
3779:x
3747:x
3744:=
3741:)
3738:x
3735:(
3705:0
3702:=
3699:)
3696:0
3693:(
3663:0
3660:x
3655:)
3652:n
3650:x
3644:n
3640:x
3615:.
3611:)
3599:2
3590:(
3580:=
3573:)
3567:(
3554:,
3550:)
3538:2
3529:(
3519:=
3512:)
3506:(
3474:2
3445:)
3442:x
3439:(
3436:f
3433:=
3430:)
3427:x
3421:(
3418:f
3394:)
3391:x
3388:(
3385:f
3379:=
3376:)
3373:x
3367:(
3364:f
3340:f
3318:.
3315:)
3309:(
3300:=
3297:)
3291:2
3288:+
3282:(
3272:,
3269:)
3263:(
3254:=
3251:)
3245:2
3242:+
3236:(
3216:-
3204:y
3180:1
3174:y
3168:1
3138:2
3135:=
3109:=
3096:-
3084:y
3061:2
3053:=
3030:0
3009:-
3006:x
2999:θ
2990:-
2987:y
2957:2
2940:0
2930:-
2918:y
2895:2
2878:0
2856:.
2853:x
2850:=
2847:)
2841:(
2831:,
2828:y
2825:=
2822:)
2816:(
2790:)
2784:(
2758:)
2752:(
2736:-
2724:y
2713:-
2701:x
2690:-
2678:x
2633:1
2630:=
2625:2
2621:y
2617:+
2612:2
2608:x
2587:)
2584:0
2581:,
2578:0
2575:(
2535:.
2528:|
2524:b
2520:|
2515:|
2511:a
2507:|
2500:b
2492:a
2485:=
2478:)
2472:(
2459:,
2452:|
2448:b
2444:|
2439:|
2435:a
2431:|
2424:|
2419:b
2411:a
2406:|
2399:=
2392:)
2386:(
2355:b
2333:a
2291:b
2269:a
2220:0
2217:=
2214:)
2208:(
2182:2
2178:/
2171:=
2146:2
2142:c
2138:=
2135:)
2129:(
2120:b
2117:a
2114:2
2106:2
2102:b
2098:+
2093:2
2089:a
2057:R
2037:,
2034:R
2031:2
2028:=
2012:c
2007:=
1991:b
1986:=
1970:a
1948:.
1943:c
1927:=
1922:b
1906:=
1901:a
1808:c
1788:b
1768:a
1748:C
1745:B
1742:A
1685:4
1682:/
1679:1
1666:π
1660:2
1657:/
1654:1
1635:2
1632:/
1629:1
1606:2
1602:3
1585:6
1582:/
1579:1
1567:3
1564:/
1561:2
1557:+
1549:π
1543:3
1540:/
1537:1
1509:2
1505:2
1476:2
1472:2
1455:8
1452:/
1449:1
1436:π
1430:4
1427:/
1424:1
1396:2
1392:3
1372:2
1369:/
1366:1
1352:/
1349:1
1337:3
1334:/
1331:1
1327:+
1319:π
1313:6
1310:/
1307:1
1233:x
1226:x
1220:x
1198:2
1181:0
1159:2
1155:2
1149:=
1130:=
1092:2
1042:)
1036:(
1010:)
1004:(
967:.
950:=
944:)
938:(
928:1
923:=
916:)
910:(
897:,
884:=
878:)
872:(
862:1
857:=
850:)
844:(
831:,
818:=
812:)
806:(
796:1
791:=
784:)
778:(
765:,
752:=
746:)
740:(
729:)
723:(
711:=
704:)
698:(
652:.
639:=
636:)
630:(
620:,
607:=
604:)
598:(
567:b
540:h
525:.
513:a
485:C
482:B
479:A
400:α
308:)
302:(
276:)
270:(
142:=
139:)
133:(
105:=
102:)
96:(
50:.
20:)
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