1654:
53:
10636:
446:
7441:
5907:
5545:
733:
10648:
10060:
3614:
3506:
1646:
3727:
10672:
10660:
4086:
3987:
3864:
4896:
5113:
7850:
One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In
Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane
7913:
His major contribution in mathematics (Nasr, 1996, pp. 208–214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental
7311:
596:
has been described as the creator of trigonometry as a mathematical discipline in its own right. He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. He listed the six distinct cases of a
540:
convention we use today. (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in
Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for
1448:
1110:
4678:
4904:
1336:
1253:
6112:
972:
892:
7084:
7851:
and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
6273:
6619:
6704:
1344:
994:
7463:
A treatise on trigonometry, plane and spherical: with its application to navigation and surveying, nautical and practical astronomy and geodesy, with logarithmic, trigonometrical, and nautical tables
6458:
785:
Trigonometric ratios are the ratios between edges of a right triangle. These ratios depend only on one acute angle of the right triangle, since any two right triangles with the same acute angle are
5618:
4909:
4683:
8793:
Elements of Plane and
Spherical Trigonometry: With Their Applications to Heights and Distances Projections of the Sphere, Dialling, Astronomy, the Solution of Equations, and Geodesic Operations
4891:{\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}x^{2n+1}}{(2n+1)!}}\\\end{aligned}}}
5209:
5108:{\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}x^{2n}}{(2n)!}}.\end{aligned}}}
3981:
3858:
6367:
6917:
6853:
6789:
7059:
1464:. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.
1261:
1178:
4080:
3801:
3721:
3685:
3572:
3464:
2929:
6002:
3293:
3100:
3043:
2789:
2706:
2486:
2450:
6478:, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables. It is given by:
3236:
2569:
2332:
2299:
2190:
2157:
2103:
2070:
502:
in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC,
7597:
6991:
6956:
3345:
2890:
2646:
4044:
3924:
3182:
3125:
3007:
2814:
2615:
582:
were used. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also made important innovations in
5650:(in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform,
5280:
In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the
7641:
1739:
1707:
6135:
1961:
1929:
1897:
5515:
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, predicting eclipses, and describing the orbits of the planets.
1865:
1836:
1807:
1778:
3894:
3757:
3536:
3428:
5677:
5648:
4014:
3641:
2414:
1982:
5559:
Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.
2361:
2219:
2037:
3317:
2953:
2862:
2670:
2510:
3257:
3203:
3146:
3064:
2982:
2835:
2753:
2727:
2590:
2536:
2382:
2266:
2240:
2124:
2008:
1494:
ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA:
7524:
3608:
3500:
918:
838:
1453:
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".
7864:
457:
7327:
Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.
622:
197:
7306:{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},\qquad \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \tan x={\frac {i(e^{-ix}-e^{ix})}{e^{ix}+e^{-ix}}}.}
5266:
hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.
475:
triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The
1121:
is the side opposite to the 90-degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle
8870:
The
Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc
9796:
8914:
Plane
Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables
10096:
9802:
6151:
699:
193:
6484:
203:
1443:{\displaystyle \cot A={\frac {1}{\tan A}}={\frac {\textrm {adjacent}}{\textrm {opposite}}}={\frac {\cos A}{\sin A}}={\frac {b}{a}}.}
1105:{\displaystyle \tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{b}}={\frac {a/h}{b/h}}={\frac {\sin A}{\cos A}}.}
6651:
319:
7968:
7838:
575:
1741:. This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:
610:
295:
6378:
4116:
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:
9700:
9660:
9621:
9594:
9567:
9532:
9505:
9457:
9421:
9394:
9361:
9338:
9295:
9268:
9241:
9214:
9187:
9157:
9130:
9096:
9069:
9020:
8993:
8966:
8939:
8895:
8851:
8731:
8669:
8642:
8607:
8580:
8553:
8526:
8472:
8442:
8415:
8388:
8361:
8334:
8304:
8183:
8156:
8129:
8099:
8072:
8045:
8018:
7935:
7817:
7697:
7632:
7588:
7558:
7488:
10459:
10045:
605:
for spherical triangles, and provided proofs for both these laws. Knowledge of trigonometric functions and methods reached
663:
674:
and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.
10604:
10089:
9847:
8632:
10676:
8499:
8235:
7997:
7882:
7515:
5132:
1579:
642:
6289:(known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
336:
10140:
7689:
7357:
7322:
5901:
5487:
3930:
3807:
981:
420:
117:
9820:
6295:
385:, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called
10554:
9821:
Trigonometry, by
Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License
9767:
5232:. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to
4100:
95:
10652:
6861:
6797:
6733:
5937:
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.
906:
10082:
9727:
9717:
1331:{\displaystyle \sec A={\frac {1}{\cos A}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}={\frac {h}{b}},}
1248:{\displaystyle \csc A={\frac {1}{\sin A}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}={\frac {h}{a}},}
567:
8985:
Visual
Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena
8749:
Intel® 64 and IA-32 Architectures
Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C
10664:
8171:
7007:
5712:
523:
382:
6645:
is given by half the product of the lengths of two sides and the sine of the angle between the two sides:
10579:
10135:
9722:
8037:
Groundbreaking
Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance
7414:
5527:
5275:
1653:
618:
91:
5551:
are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a
5214:
This complex exponential function, written in terms of trigonometric functions, is particularly useful.
4050:
3763:
3691:
3647:
3542:
3434:
10150:
7764:
6107:{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R={\frac {abc}{2\Delta }},}
571:
288:
239:
188:
122:
9799:
by Alfred Monroe Kenyon and Louis Ingold, The
Macmillan Company, 1914. In images, full text presented.
7720:
10564:
10536:
10173:
7372:
7337:
5563:
4110:
3382:
The following table summarizes the properties of the graphs of the six main trigonometric functions:
3370:, one can extend the definitions of trigonometric ratios to all positive and negative arguments (see
2896:
1473:
574:
produced the first table of cotangents. By the 10th century AD, in the work of Persian mathematician
467:
astronomers studied angle measure, using a division of circles into 360 degrees. They, and later the
7755:
Moussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations".
3263:
3070:
3013:
2759:
2676:
2456:
2420:
1456:
With these functions, one can answer virtually all questions about arbitrary triangles by using the
10609:
9879:
9746:
9173:
7963:
7833:
7342:
5982:
are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).
5281:
4668:
When considered as functions of a real variable, the trigonometric ratios can be represented by an
3209:
2542:
2305:
2272:
2163:
2130:
2076:
2043:
424:
101:
21:
9480:
6961:
6926:
5592:, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.
3323:
2868:
2624:
541:
performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval
10494:
10484:
10454:
10388:
10123:
10040:
8747:
7377:
7352:
5567:
4020:
3900:
3163:
3106:
2988:
2795:
2596:
579:
440:
386:
358:
79:
67:
62:
9449:
7716:
10592:
10489:
10469:
10464:
10393:
10118:
9941:
9840:
7681:
6720:
5869:
5865:
4333:
3371:
3361:
1157:
797:
793:
727:
675:
583:
515:
416:
312:
9628:
9584:
9557:
9522:
9231:
9177:
9147:
8841:
8721:
8702:
8597:
8570:
8543:
8489:
8405:
8378:
8173:
8146:
8119:
8035:
7898:
7651:
7607:
7578:
7565:
7534:
7505:
7478:
1712:
1680:
10698:
10619:
10549:
10426:
10350:
10289:
10274:
10269:
10246:
10128:
10012:
9411:
9326:
9283:
9258:
9204:
8983:
8956:
8885:
8622:
8516:
8462:
8432:
8351:
8089:
8062:
7807:
7362:
7062:
6120:
5243:
4477:
1935:
1903:
1871:
786:
472:
281:
28:
9384:
8929:
8868:
8380:
Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students
5246:
have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes
1842:
1813:
1784:
1755:
1665:, which is the circle of radius 1 centered at the origin in the plane. In this setting, the
10599:
10479:
10474:
10398:
10299:
9872:
9179:
Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s
7387:
6142:
5499:
5259:
5223:
3876:
3739:
3518:
3410:
707:
626:
593:
511:
453:
427:
with the aim to simplify an expression, to find a more useful form of an expression, or to
208:
127:
72:
17:
8172:
Krystle Rose Forseth; Christopher Burger; Michelle Rose Gilman; Deborah J. Rumsey (2008).
5653:
5624:
3999:
3626:
2388:
1967:
1674:
1666:
8:
10614:
10524:
10446:
10345:
10279:
10236:
10226:
10206:
9981:
9976:
9961:
9956:
9867:
9644:
8545:
Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications
6724:
6475:
5716:
5263:
2338:
2196:
2014:
967:{\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{h}}.}
887:{\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{h}}.}
658:
654:
405:
254:
170:
9474:
3299:
2935:
2844:
2652:
2492:
10640:
10559:
10499:
10431:
10421:
10360:
10335:
10211:
10168:
10163:
10007:
9966:
9889:
9442:
9313:
Study Guide for the Course in Meteorological Mathematics: Latest Revision, Feb. 1, 1943
7776:
7347:
7001:
6642:
5571:
5552:
5247:
5229:
5123:
4106:
3242:
3188:
3131:
3049:
2967:
2820:
2738:
2712:
2575:
2521:
2367:
2251:
2225:
2109:
1993:
646:
559:
546:
370:
52:
9806:
3578:
3470:
909:
leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.
10635:
10355:
10340:
10284:
10231:
10063:
9951:
9909:
9899:
9833:
9743:
9696:
9679:
9675:
9656:
9649:
9617:
9590:
9563:
9528:
9501:
9453:
9417:
9390:
9334:
9291:
9264:
9237:
9210:
9183:
9153:
9126:
9092:
9065:
9016:
8989:
8962:
8935:
8912:
8891:
8847:
8727:
8665:
8638:
8603:
8576:
8549:
8522:
8495:
8468:
8438:
8411:
8384:
8357:
8330:
8300:
8241:
8231:
8200:
8179:
8152:
8125:
8095:
8068:
8041:
8014:
7993:
7931:
7878:
7813:
7780:
7693:
7628:
7584:
7554:
7511:
7484:
5877:
5873:
5853:
5743:
5739:
5735:
5720:
5700:
5612:
1149:
are sometimes used for the opposite and adjacent sides respectively. See below under
483:
259:
163:
9444:
Real-World Flash Game Development: How to Follow Best Practices AND Keep Your Sanity
9284:
Gennady I. Arkhipov; Vladimir N. Chubarikov; Anatoly A. Karatsuba (22 August 2008).
8954:
8825:
8808:
629:. One of the earliest works on trigonometry by a northern European mathematician is
10569:
10544:
10416:
10264:
10201:
9914:
9815:
9781:
9553:
8820:
8460:
7870:
7768:
7620:
5889:
5765:
1544:
649:
with whom he lived for several years. At the same time, another translation of the
533:
495:
428:
5906:
10509:
10436:
10365:
10158:
10033:
9971:
9904:
9611:
9495:
9328:
9311:
9285:
9120:
9086:
9059:
9037:
9010:
8791:
8686:
8659:
8324:
7874:
7548:
7461:
7367:
7067:
6470:
5885:
5817:
5801:
5724:
5544:
4669:
703:
602:
554:
499:
158:
87:
83:
9791:
7409:
10587:
10514:
10221:
9931:
9354:
8628:
8487:
8203:
7072:
6923:
The second and third equations are derived from dividing the first equation by
6285:
5617:
5608:
5251:
5119:
4150:
1657:
Indication of the sign and amount of key angles according to rotation direction
1461:
695:
691:
606:
601:, he stated the law of sines for plane and spherical triangles, discovered the
587:
445:
362:
269:
264:
153:
8064:
From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics
7772:
4113:
the domain of a trigonometric function, however, they can be made invertible.
657:. Trigonometry was still so little known in 16th-century northern Europe that
10692:
10375:
10307:
10259:
8657:
8624:
7793:
Gingerich, Owen. "Islamic astronomy." Scientific American 254.4 (1986): 74–83
7677:
5833:
5519:
715:
687:
683:
634:
249:
8245:
8148:
Mathematical Time Capsules: Historical Modules for the Mathematics Classroom
7928:
The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
985:(denoted tan), defined as the ratio of the opposite leg to the adjacent leg.
10317:
10312:
10216:
5992:
5849:
5805:
5793:
1457:
711:
148:
10074:
9327:
Mary Sears; Daniel Merriman; Woods Hole Oceanographic Institution (1980).
8955:
H. G. ter Morsche; J. C. van den Berg; E. M. van de Vrie (7 August 2003).
825:(denoted sin), defined as the ratio of the side opposite the angle to the
536:
length to define his trigonometric functions, a minor difference from the
10519:
10183:
10106:
8225:
7382:
5881:
5845:
5809:
5777:
5262:
provide function libraries that include the trigonometric functions. The
5233:
5122:. When extended as functions of real or complex variables, the following
3367:
1662:
1640:
1482:
is to remember facts and relationships in trigonometry. For example, the
468:
350:
234:
132:
9741:
7809:
Armchair Algebra: Everything You Need to Know From Integers To Equations
4105:
Because the six main trigonometric functions are periodic, they are not
1541:
One way to remember the letters is to sound them out phonetically (i.e.
732:
558:, and its properties were further documented in the 5th century (AD) by
10504:
10383:
10178:
8117:
5841:
5837:
5825:
5781:
5539:
5449:
5250:
and their inverses). Most allow a choice of angle measurement methods:
5237:
3355:
1118:
826:
671:
503:
491:
449:
409:
244:
224:
9811:
6268:{\displaystyle R={\frac {abc}{\sqrt {(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}}.}
5792:
Other fields that use trigonometry or trigonometric functions include
5118:
With these definitions the trigonometric functions can be defined for
10002:
9751:
9118:
8488:
Martin Brokate; Pammy Manchanda; Abul Hasan Siddiqi (3 August 2019).
8322:
8208:
5861:
5857:
5829:
5769:
5759:
5684:
5680:
5596:
5589:
5583:
5523:
5510:
5383:
5345:
4421:
1649:
Fig. 1a – Sine and cosine of an angle θ defined using the unit circle
1595:). Another method is to expand the letters into a sentence, such as "
800:. Explicitly, they are defined below as functions of the known angle
563:
542:
401:
378:
342:
325:
9035:
8230:. Fiona McGill. Cheltenham: Stanley Thornes Publishers. p. 51.
5555:, the position of the ship can be determined from such measurements.
4672:. For instance, sine and cosine have the following representations:
3613:
522:(from Alexandria, Egypt) constructed detailed trigonometric tables (
10408:
10327:
10254:
9924:
9856:
8927:
7686:
Mathematics Across Cultures: The History of Non-western Mathematics
6614:{\displaystyle {\frac {a-b}{a+b}}={\frac {\tan \left}{\tan \left}}}
5731:
5562:
Trigonometry is still used in navigation through such means as the
5433:
5255:
4583:
1592:
1479:
590:
528:
510:, Asia Minor) gave the first tables of chords, analogous to modern
396:
Throughout history, trigonometry has been applied in areas such as
374:
179:
8883:
8568:
8011:
The Rainbow of Mathematics: A History of the Mathematical Sciences
5996:(also known as the "sine rule") for an arbitrary triangle states:
3505:
1634:
10193:
9919:
8719:
7990:
From Byzantium to Italy. Greek Studies in the Italian Renaissance
5821:
5813:
5797:
5548:
5311:
1645:
519:
397:
229:
9355:"JPEG Standard (JPEG ISO/IEC 10918-1 ITU-T Recommendation T.81)"
9149:
Applications of Digital Signal Processing to Audio and Acoustics
9057:
8616:
8461:
Elizabeth G. Bremigan; Ralph J. Bremigan; John D. Lorch (2011).
9683:
9476:
A System of Crystallography, with Its Application to Mineralogy
9145:
8723:
Programming Challenges: The Programming Contest Training Manual
8118:
James Stewart; Lothar Redlin; Saleem Watson (16 January 2015).
5773:
5755:
5699:
The sine and cosine functions are fundamental to the theory of
4273:
4141:
3726:
901:
637:, who was encouraged to write, and provided with a copy of the
566:. These Greek and Indian works were translated and expanded by
507:
487:
9229:
9172:
8634:
The History of Mathematical Tables: From Sumer to Spreadsheets
8144:
7926:
Berggren, J. Lennart (2007). "Mathematics in Medieval Islam".
7914:
formulas for the solution of spherical right-angled triangles.
816:
refer to the lengths of the sides in the accompanying figure:
10028:
9736:
From Eudoxus to Einstein: A History of Mathematical Astronomy
9256:
8771:
7503:
5708:
5704:
476:
464:
366:
354:
9061:
College Algebra and Trigonometry: Basics Through Precalculus
6699:{\displaystyle {\mbox{Area}}=\Delta ={\frac {1}{2}}ab\sin C}
5784:, and to solve boundary- and transmission-related problems.
4085:
698:
into trigonometry. The works of the Scottish mathematicians
597:
right-angled triangle in spherical trigonometry, and in his
9825:
9119:
E. Richard Heineman; J. Dalton Tarwater (1 November 1992).
9012:
Applications of Fourier Transforms to Generalized Functions
8759:
8326:
Precalculus: A Problems-Oriented Approach, Enhanced Edition
7869:. Vol. 2. Cambridge University Press. pp. 62–83.
4181:
3986:
1571:
1568:
1559:
1553:
821:
706:
in the 18th century were influential in the development of
552:
The modern definition of the sine is first attested in the
537:
390:
8928:
Charles W. J. Withers; Hayden Lorimer (14 December 2015).
8430:
8323:
David Cohen; Lee B. Theodore; David Sklar (17 July 2009).
8839:
8138:
7944:
5228:
Trigonometric functions were among the earliest uses for
3863:
8688:
Instructions for Post-trig and Mannheim-trig Slide Rules
8658:
George S. Donovan; Beverly Beyreuther Gimmestad (1980).
6453:{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
5948:, relate both the sides and angles of a given triangle.
9616:(illustrated ed.). Cengage Learning. p. 856.
9520:
9413:
Foundations of Mathematical and Computational Economics
9233:
Medical Imaging: Analysis of Multimodality 2D/3D Images
9114:
9112:
9110:
9108:
8684:
8456:
8454:
7863:
Berggren, J. L. (October 2013). "Islamic Mathematics".
7738:
7736:
7459:
5526:
to measure the distance to nearby stars, as well as in
1677:
will intersect the unit circle in a point (x,y), where
1661:
Trigonometric ratios can also be represented using the
518:. In the 2nd century AD, the Greco-Egyptian astronomer
9792:
Khan Academy: Trigonometry, free online micro lectures
9277:
9152:. Springer Science & Business Media. p. 404.
9053:
9051:
9049:
8910:
8726:. Springer Science & Business Media. p. 302.
8602:. Springer Science & Business Media. p. 313.
8569:
K. RAJA RAJESWARI; B. VISVESVARA RAO (24 March 2014).
6656:
6577:
6532:
5217:
514:, and used them to solve problems in trigonometry and
9416:. Springer Science & Business Media. p. 46.
8988:. Springer Science & Business Media. p. 15.
8720:
Steven S. Skiena; Miguel A. Revilla (18 April 2006).
7087:
7065:
identities for sine, cosine, and tangent in terms of
7010:
6964:
6929:
6864:
6800:
6736:
6654:
6487:
6381:
6298:
6154:
6123:
6005:
5730:
Even non-periodic functions can be represented as an
5656:
5627:
5135:
4907:
4681:
4053:
4023:
4002:
3933:
3903:
3879:
3810:
3766:
3742:
3694:
3650:
3629:
3581:
3545:
3521:
3473:
3437:
3413:
3326:
3302:
3266:
3245:
3212:
3191:
3166:
3134:
3109:
3073:
3052:
3016:
2991:
2970:
2938:
2899:
2871:
2847:
2823:
2798:
2762:
2741:
2715:
2679:
2655:
2627:
2599:
2578:
2545:
2524:
2495:
2459:
2423:
2391:
2370:
2341:
2308:
2275:
2254:
2228:
2199:
2166:
2133:
2112:
2079:
2046:
2017:
1996:
1970:
1938:
1906:
1874:
1845:
1816:
1787:
1758:
1715:
1683:
1580:
1556:
1550:
1547:
1347:
1264:
1181:
997:
921:
841:
9439:
9139:
9105:
8651:
8541:
8481:
8451:
8407:
Mathematics for Electrical Engineering and Computing
8318:
8316:
8113:
8111:
7733:
7657:
3377:
3356:
Trigonometric functions of real or complex variables
1565:
9672:
Logarithmic and Trigonometric Tables to Five Places
9472:
9166:
9146:Mark Kahrs; Karlheinz Brandenburg (18 April 2006).
9046:
8403:
1562:
9648:
9441:
8921:
8258:A sentence more appropriate for high schools is "'
8198:
7676:Jacques Sesiano (2000). "Islamic mathematics". In
7497:
7476:
7305:
7053:
6985:
6950:
6911:
6847:
6783:
6698:
6613:
6452:
6361:
6267:
6129:
6106:
5671:
5642:
5236:between the values listed to get higher accuracy.
5203:
5107:
4890:
4094:
4074:
4038:
4008:
3975:
3918:
3888:
3852:
3795:
3751:
3715:
3679:
3635:
3602:
3566:
3530:
3494:
3458:
3422:
3339:
3311:
3287:
3251:
3230:
3197:
3176:
3140:
3119:
3094:
3058:
3037:
3001:
2976:
2947:
2923:
2884:
2856:
2829:
2808:
2783:
2747:
2721:
2700:
2664:
2640:
2609:
2584:
2563:
2530:
2504:
2480:
2444:
2408:
2376:
2355:
2326:
2293:
2260:
2234:
2213:
2184:
2151:
2118:
2097:
2064:
2031:
2002:
1976:
1955:
1923:
1891:
1859:
1830:
1801:
1772:
1733:
1701:
1442:
1330:
1247:
1104:
966:
886:
653:from Greek into Latin was completed by the Cretan
357:and side lengths of triangles. In particular, the
9382:
9320:
8431:Ron Larson; Bruce H. Edwards (10 November 2008).
8424:
8349:
8313:
8165:
8108:
7504:Ron Larson; Robert P. Hostetler (10 March 2006).
7316:
10690:
9552:
9287:Trigonometric Sums in Number Theory and Analysis
9029:
8833:
8789:
8562:
8397:
8008:
5269:
5240:had special scales for trigonometric functions.
423:are commonly used for rewriting trigonometrical
9655:(Second ed.). John Wiley & Sons, Inc.
9435:
9433:
9223:
9084:
9036:Lawrence Bornstein; Basic Systems, Inc (1966).
8948:
8709:. Bonnier Corporation. April 1974. p. 125.
8033:
7805:
7675:
7576:
5344:) (which appeared in the earliest tables), the
5204:{\displaystyle e^{x+iy}=e^{x}(\cos y+i\sin y).}
4663:
1635:The unit circle and common trigonometric values
373:during the 3rd century BC from applications of
9487:
9409:
8840:Michael Seeds; Dana Backman (5 January 2009).
8678:
8595:
7801:
7799:
7580:A history of ancient mathematical astronomy. 1
5940:Identities involving only angles are known as
4109:(or, 1 to 1), and thus are not invertible. By
678:was the first to use the word, publishing his
369:of its side lengths. The field emerged in the
10090:
9841:
9376:
8981:
8884:Mordechai Ben-Ari; Francesco Mondada (2018).
8813:Bulletin of the American Mathematical Society
8589:
7711:
7709:
7610:, "Appendix 1: Hipparchus's Table of Chords".
3976:{\displaystyle (-\infty ,-1]\cup [1,\infty )}
3853:{\displaystyle (-\infty ,-1]\cup [1,\infty )}
289:
9548:
9546:
9544:
9430:
7546:
6362:{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}
5776:. In these areas, they are used to describe
5490:for more relations between these functions.
1129:is the other side that is adjacent to angle
10104:
9466:
9403:
9202:
9196:
9058:John J. Schiller; Marie A. Wurster (1988).
8975:
8904:
8866:
8809:"Mathematical methods in ancient astronomy"
8783:
8535:
8383:. Cambridge University Press. p. 175.
8376:
8040:. Greenwood Publishing Group. p. 153.
7930:. Princeton University Press. p. 518.
7796:
7470:
6709:
5595:On a larger scale, trigonometry is used in
905:(denoted cos), defined as the ratio of the
686:described for the first time the method of
27:"Trig" redirects here. For other uses, see
10097:
10083:
9848:
9834:
9582:
9576:
9521:V&S EDITORIAL BOARD (6 January 2015).
9493:
9002:
8961:. Cambridge University Press. p. 61.
8860:
8806:
8514:
8343:
7706:
7654:, "Appendix 3: Ptolemy's Table of Chords".
7553:(4th ed.). Hachette UK. p. 275.
7453:
296:
282:
36:Area of geometry, about angles and lengths
9695:. Springer Science & Business Media.
9541:
9514:
9440:Christopher Griffith (12 November 2012).
9230:Luciano Beolchi; Michael H. Kuhn (1995).
9078:
9008:
8824:
8464:Mathematics for Secondary School Teachers
8370:
8145:Dick Jardine; Amy Shell-Gellasch (2011).
6912:{\displaystyle \cot ^{2}A+1=\csc ^{2}A\ }
6848:{\displaystyle \tan ^{2}A+1=\sec ^{2}A\ }
6784:{\displaystyle \sin ^{2}A+\cos ^{2}A=1\ }
6714:
633:by the 15th century German mathematician
9690:
9609:
9250:
9176:; Michael J. Ostwald (9 February 2015).
8542:Silvia Maria Alessio (9 December 2015).
8508:
7925:
7862:
7647:
7603:
5905:
5616:
5599:to measure distances between landmarks.
5543:
1652:
1644:
731:
444:
9803:Benjamin Banneker's Trigonometry Puzzle
9669:
9263:. Oxford University Press. p. 13.
8931:Geographers: Biobibliographical Studies
8777:
8087:
7969:MacTutor History of Mathematics archive
7839:MacTutor History of Mathematics archive
7669:
7537:, "Greek Trigonometry and Mensuration".
7477:Mary Jane Sterling (24 February 2014).
5749:
721:
549:, and, later, Western European worlds.
10691:
9562:. John Wiley & Sons. p. 435.
9386:The Routledge Linguistics Encyclopedia
9257:Marcus Frederick Charles Ladd (2014).
9039:Trigonometry for the Physical Sciences
8843:Astronomy: The Solar System and Beyond
8353:The Complete Idiot's Guide to Calculus
8294:
8223:
8178:. John Wiley & Sons. p. 218.
7754:
7619:
7613:
7483:. John Wiley & Sons. p. 185.
5951:
5276:Trigonometric functions § History
1137:is the side that is opposite to angle
690:still used today in surveying. It was
10078:
9829:
9643:
9383:Kirsten Malmkjaer (4 December 2009).
9362:International Telecommunication Union
8765:
8491:Calculus for Scientists and Engineers
8060:
7950:
7742:
7663:
7530:
7447:
7054:{\displaystyle e^{ix}=\cos x+i\sin x}
5787:
5602:
471:, studied the ratios of the sides of
353:concerned with relationships between
10659:
10046:List of differential geometry topics
8404:Mary P Attenborough (30 June 2003).
7806:Michael Willers (13 February 2018).
7748:
452:, credited with compiling the first
10671:
9812:Dave's Short Course in Trigonometry
9613:Technical Mathematics with Calculus
9527:. V&S Publishers. p. 288.
8061:Ewald, William Bragg (2005-04-21).
7547:Pimentel, Ric; Wall, Terry (2018).
5218:Calculating trigonometric functions
5126:holds for the complex exponential:
664:De revolutionibus orbium coelestium
570:. In 830 AD, Persian mathematician
415:Trigonometry is known for its many
13:
9710:
9410:Kamran Dadkhah (11 January 2011).
9260:Symmetry of Crystals and Molecules
7407:
6996:
6665:
6463:
6124:
6095:
5518:In modern times, the technique of
5038:
4812:
4075:{\displaystyle (-\infty ,\infty )}
4066:
4060:
3967:
3940:
3844:
3817:
3796:{\displaystyle x\neq \pi /2+n\pi }
3716:{\displaystyle (-\infty ,\infty )}
3707:
3701:
3680:{\displaystyle x\neq \pi /2+n\pi }
3567:{\displaystyle (-\infty ,\infty )}
3558:
3552:
3459:{\displaystyle (-\infty ,\infty )}
3450:
3444:
14:
10710:
9761:
9747:"Trigonometric Addition Formulas"
9589:. Cengage Learning. p. 331.
9524:CONCISE DICTIONARY OF MATHEMATICS
9206:Essential Skills for GCSE Biology
8846:. Cengage Learning. p. 254.
8377:Jenny Olive (18 September 2003).
8297:Memory: A Very Short Introduction
8124:. Cengage Learning. p. 448.
7899:"ṬUSI, NAṢIR-AL-DIN i. Biography"
7510:. Cengage Learning. p. 230.
6278:
5970:are the angles of a triangle and
5922:and respectively opposite angles
5734:of sines and cosines through the
3378:Graphs of trigonometric functions
10670:
10658:
10647:
10646:
10634:
10059:
10058:
9209:. Hodder Education. p. 78.
8685:Ross Raymond Middlemiss (1945).
8664:. Prindle, Weber & Schmidt.
8437:. Cengage Learning. p. 21.
8088:Dempski, Kelly (November 2002).
7866:the cambridge history of science
7583:. Springer-Verlag. p. 744.
7550:Cambridge IGCSE Core Mathematics
7460:Charles William Hackley (1853).
7438:, 2nd Ed., The Gale Group (2002)
7358:List of trigonometric identities
7323:List of trigonometric identities
6637:and the angle between the sides
6137:is the area of the triangle and
5902:List of trigonometric identities
5488:List of trigonometric identities
4084:
3985:
3862:
3725:
3612:
3504:
1543:
349: 'measure') is a branch of
51:
10555:Computational complexity theory
9734:Linton, Christopher M. (2004).
9637:
9603:
9347:
9304:
8911:George Roberts Perkins (1853).
8877:
8826:10.1090/S0002-9904-1948-09089-9
8800:
8740:
8713:
8695:
8596:John Stillwell (23 July 2010).
8252:
8217:
8192:
8081:
8054:
8027:
8009:Grattan-Guinness, Ivor (1997).
8002:
7982:
7956:
7919:
7891:
7856:
7826:
7787:
7690:Springer Science+Business Media
7206:
7149:
5985:
5764:Trigonometry is useful in many
5493:
4147:Range of usual principal value
4138:Range of usual principal value
4101:Inverse trigonometric functions
4095:Inverse trigonometric functions
2924:{\displaystyle -2{\sqrt {3}}/3}
667:to explain its basic concepts.
568:medieval Islamic mathematicians
526:) in Book 1, chapter 11 of his
9583:Ron Larson (29 January 2010).
9085:Dudley H. Towne (5 May 2014).
8958:Fourier and Laplace Transforms
8703:"Calculator keys—what they do"
7757:Arabic Sciences and Philosophy
7627:. Princeton University Press.
7570:
7540:
7428:
7401:
7317:Other trigonometric identities
7260:
7225:
6600:
6588:
6555:
6543:
6256:
6238:
6235:
6217:
6214:
6196:
6193:
6175:
5703:, such as those that describe
5666:
5660:
5637:
5631:
5195:
5168:
5089:
5080:
5056:
5046:
4875:
4860:
4830:
4820:
4069:
4054:
3970:
3958:
3952:
3934:
3847:
3835:
3829:
3811:
3710:
3695:
3597:
3582:
3561:
3546:
3489:
3474:
3453:
3438:
3288:{\displaystyle -{\sqrt {3}}/3}
3095:{\displaystyle 2{\sqrt {3}}/3}
3038:{\displaystyle 2{\sqrt {3}}/3}
2784:{\displaystyle 2{\sqrt {3}}/3}
2701:{\displaystyle -{\sqrt {3}}/3}
2481:{\displaystyle -{\sqrt {3}}/2}
2445:{\displaystyle -{\sqrt {2}}/2}
1160:of these ratios are named the
1150:
796:of this angle that are called
1:
9738:. Cambridge University Press.
8661:Trigonometry with calculators
8575:. PHI Learning. p. 263.
8434:Calculus of a Single Variable
8227:Key Maths : GCSE, Higher
8094:. Premier Press. p. 29.
7964:"Johann Müller Regiomontanus"
7394:
5958:In the following identities,
5895:
5687:, reveals the 6 frequencies (
5533:
5270:Other trigonometric functions
3231:{\displaystyle {\sqrt {3}}/3}
2564:{\displaystyle {\sqrt {3}}/3}
2327:{\displaystyle {\sqrt {2}}/2}
2294:{\displaystyle {\sqrt {3}}/2}
2185:{\displaystyle {\sqrt {2}}/2}
2152:{\displaystyle {\sqrt {3}}/2}
2098:{\displaystyle {\sqrt {3}}/2}
2065:{\displaystyle {\sqrt {2}}/2}
456:, has been described as "the
332: 'triangle' and
9855:
9473:John Joseph Griffin (1841).
9203:Dan Foulder (15 July 2019).
8982:Bernd Thaller (8 May 2007).
8934:. A&C Black. p. 6.
8917:. D. Appleton & Company.
8796:. Baldwin, Cradock, and Joy.
8515:Serge Lang (14 March 2013).
8295:Foster, Jonathan K. (2008).
8091:Focus on Curves and Surfaces
7875:10.1017/CHO9780511974007.004
6986:{\displaystyle \sin ^{2}{A}}
6951:{\displaystyle \cos ^{2}{A}}
6719:The following trigonometric
5944:. Other equations, known as
5727:of trigonometric functions.
5577:
5528:satellite navigation systems
5504:
4664:Power series representations
3340:{\displaystyle -{\sqrt {3}}}
2885:{\displaystyle -{\sqrt {2}}}
2641:{\displaystyle -{\sqrt {3}}}
1467:
710:. Also in the 18th century,
619:Persian and Arab astronomers
381:. The Greeks focused on the
7:
9723:Encyclopedia of Mathematics
9182:. Birkhäuser. p. 260.
8599:Mathematics and Its History
8356:. Alpha Books. p. 45.
7415:Online Etymology Dictionary
7330:
5738:. This has applications to
4039:{\displaystyle x\neq n\pi }
3919:{\displaystyle x\neq n\pi }
3177:{\displaystyle {\sqrt {3}}}
3120:{\displaystyle {\sqrt {2}}}
3002:{\displaystyle {\sqrt {2}}}
2809:{\displaystyle {\sqrt {2}}}
2610:{\displaystyle {\sqrt {3}}}
10:
10715:
10605:Films about mathematicians
9674:(2nd ed.). New York:
9610:Peterson, John C. (2004).
9236:. IOS Press. p. 122.
9042:. Appleton-Century-Crofts.
8350:W. Michael Kelley (2002).
8067:. OUP Oxford. p. 93.
7812:. Book Sales. p. 37.
7765:Cambridge University Press
7320:
5899:
5753:
5606:
5581:
5537:
5508:
5497:
5273:
5221:
4098:
3359:
1638:
1471:
725:
572:Habash al-Hasib al-Marwazi
494:studied the properties of
484:Hellenistic mathematicians
438:
434:
335:
318:
189:Trigonometric substitution
26:
15:
10628:
10578:
10535:
10445:
10407:
10374:
10326:
10298:
10245:
10192:
10174:Philosophy of mathematics
10149:
10114:
10054:
10021:
9990:
9940:
9888:
9863:
9782:Resources in your library
9718:"Trigonometric functions"
8807:Neugebauer, Otto (1948).
8790:Olinthus Gregory (1816).
8691:. Frederick Post Company.
8548:. Springer. p. 339.
8410:. Elsevier. p. 418.
7773:10.1017/S095742391000007X
7373:Small-angle approximation
7061:, produces the following
5723:could be described as an
5679:(in blue), which depicts
5564:Global Positioning System
5254:, radians, and sometimes
1474:Mnemonics in trigonometry
670:Driven by the demands of
524:Ptolemy's table of chords
20:. For the TV series, see
10610:Recreational mathematics
9670:Nielsen, Kaj L. (1966).
9651:A History of Mathematics
9494:Dugopolski (July 2002).
9389:. Routledge. p. 1.
8890:. Springer. p. 16.
8521:. Springer. p. 63.
8175:Pre-Calculus For Dummies
8121:Algebra and Trigonometry
8034:Robert E. Krebs (2004).
7577:Otto Neugebauer (1975).
7480:Trigonometry For Dummies
7343:Generalized trigonometry
6727:and hold for any value:
6710:Trigonometric identities
5942:trigonometric identities
5691:) and their amplitudes (
1734:{\displaystyle y=\sin A}
1702:{\displaystyle x=\cos A}
792:So, these ratios define
736:In this right triangle:
702:in the 17th century and
661:devoted two chapters of
617:as well as the works of
421:trigonometric identities
102:Generalized trigonometry
22:Trigonometry (TV series)
10495:Mathematical statistics
10485:Mathematical psychology
10455:Engineering mathematics
10389:Algebraic number theory
10041:List of geometry topics
9691:Thurston, Hugh (1996).
8623:Martin Campbell-Kelly;
8299:. Oxford. p. 128.
7721:Encyclopædia Britannica
7436:Encyclopedia of Science
7378:Trigonometric functions
7353:List of triangle topics
6624:
6130:{\displaystyle \Delta }
5568:artificial intelligence
1956:{\displaystyle 5\pi /6}
1924:{\displaystyle 3\pi /4}
1892:{\displaystyle 2\pi /3}
798:trigonometric functions
694:who fully incorporated
643:Byzantine Greek scholar
580:trigonometric functions
576:Abū al-Wafā' al-Būzjānī
482:In the 3rd century BC,
479:used a similar method.
441:History of trigonometry
387:trigonometric functions
361:relate the angles of a
359:trigonometric functions
10641:Mathematics portal
10490:Mathematical sociology
10470:Mathematical economics
10465:Mathematical chemistry
10394:Analytic number theory
10275:Differential equations
9479:. R. Griffin. p.
9330:Oceanography, the past
9091:. Dover Publications.
8780:, pp. xxiii–xxiv.
8768:, pp. xxiii–xxiv.
8224:Humble, Chris (2001).
7834:"Nasir al-Din al-Tusi"
7338:Aryabhata's sine table
7307:
7055:
6987:
6952:
6913:
6849:
6785:
6715:Pythagorean identities
6700:
6615:
6454:
6363:
6269:
6131:
6108:
5934:
5870:mechanical engineering
5866:electrical engineering
5746:, among other fields.
5715:discovered that every
5696:
5673:
5644:
5556:
5244:Scientific calculators
5205:
5109:
5042:
4892:
4816:
4076:
4040:
4010:
3977:
3920:
3890:
3854:
3797:
3753:
3717:
3681:
3637:
3604:
3568:
3532:
3496:
3460:
3424:
3372:trigonometric function
3362:Trigonometric function
3341:
3313:
3289:
3253:
3232:
3199:
3178:
3142:
3121:
3096:
3060:
3039:
3003:
2978:
2949:
2925:
2886:
2858:
2831:
2810:
2785:
2749:
2723:
2702:
2666:
2642:
2611:
2586:
2565:
2532:
2506:
2482:
2446:
2410:
2378:
2357:
2328:
2295:
2262:
2236:
2215:
2186:
2153:
2120:
2099:
2066:
2033:
2004:
1978:
1957:
1925:
1893:
1861:
1860:{\displaystyle \pi /2}
1832:
1831:{\displaystyle \pi /3}
1803:
1802:{\displaystyle \pi /4}
1774:
1773:{\displaystyle \pi /6}
1735:
1703:
1658:
1650:
1444:
1332:
1249:
1106:
968:
888:
782:
728:Trigonometric function
676:Bartholomaeus Pitiscus
584:spherical trigonometry
516:spherical trigonometry
461:
10620:Mathematics education
10550:Theory of computation
10270:Hypercomplex analysis
10013:Differential geometry
9448:. CRC Press. p.
9290:. Walter de Gruyter.
7903:Encyclopaedia Iranica
7363:Rational trigonometry
7308:
7056:
6988:
6953:
6914:
6850:
6786:
6701:
6616:
6455:
6364:
6270:
6141:is the radius of the
6132:
6109:
5909:
5674:
5645:
5620:
5547:
5260:programming languages
5206:
5110:
5022:
4893:
4796:
4551:< 90° or 90° <
4077:
4041:
4011:
3978:
3921:
3891:
3889:{\displaystyle 2\pi }
3855:
3798:
3754:
3752:{\displaystyle 2\pi }
3718:
3682:
3638:
3605:
3569:
3533:
3531:{\displaystyle 2\pi }
3497:
3461:
3425:
3423:{\displaystyle 2\pi }
3342:
3314:
3290:
3254:
3233:
3200:
3179:
3143:
3122:
3097:
3061:
3040:
3004:
2979:
2950:
2926:
2887:
2859:
2832:
2811:
2786:
2750:
2724:
2703:
2667:
2643:
2612:
2587:
2566:
2533:
2507:
2483:
2447:
2411:
2379:
2358:
2329:
2296:
2263:
2237:
2216:
2187:
2154:
2121:
2100:
2067:
2034:
2005:
1979:
1958:
1926:
1894:
1862:
1833:
1804:
1775:
1736:
1704:
1656:
1648:
1445:
1333:
1250:
1172:(cot), respectively:
1107:
969:
889:
735:
512:tables of sine values
448:
383:calculation of chords
29:Trig (disambiguation)
10600:Informal mathematics
10480:Mathematical physics
10475:Mathematical finance
10460:Mathematical biology
10399:Diophantine geometry
9497:Trigonometry I/E Sup
8887:Elements of Robotics
8867:John Sabine (1800).
8329:. Cengage Learning.
8151:. MAA. p. 182.
7988:N.G. Wilson (1992).
7953:, pp. 237, 274.
7682:D'Ambrosio, Ubiratan
7388:Uses of trigonometry
7085:
7008:
7004:, which states that
6962:
6927:
6862:
6798:
6734:
6652:
6643:area of the triangle
6485:
6379:
6296:
6152:
6143:circumscribed circle
6121:
6003:
5910:Triangle with sides
5750:Optics and acoustics
5672:{\displaystyle S(f)}
5654:
5643:{\displaystyle s(x)}
5625:
5500:Uses of trigonometry
5224:Trigonometric tables
5133:
4905:
4679:
4051:
4021:
4009:{\displaystyle \pi }
4000:
3931:
3901:
3877:
3808:
3764:
3740:
3692:
3648:
3636:{\displaystyle \pi }
3627:
3579:
3543:
3519:
3471:
3435:
3411:
3324:
3300:
3264:
3243:
3210:
3189:
3164:
3132:
3107:
3071:
3050:
3014:
2989:
2968:
2936:
2897:
2869:
2845:
2821:
2796:
2760:
2739:
2713:
2677:
2653:
2625:
2597:
2576:
2543:
2522:
2493:
2457:
2421:
2409:{\displaystyle -1/2}
2389:
2368:
2339:
2306:
2273:
2252:
2226:
2197:
2164:
2131:
2110:
2077:
2044:
2015:
1994:
1977:{\displaystyle \pi }
1968:
1936:
1904:
1872:
1843:
1814:
1785:
1756:
1713:
1681:
1345:
1262:
1179:
995:
919:
839:
722:Trigonometric ratios
714:defined the general
708:trigonometric series
627:Nasir al-Din al-Tusi
599:On the Sector Figure
594:Nasir al-Din al-Tusi
560:Indian mathematician
379:astronomical studies
209:Trigonometric series
18:Trigonometry (album)
10615:Mathematics and art
10525:Operations research
10280:Functional analysis
9629:Extract of page 856
9556:(19 January 2010).
9333:. Springer-Verlag.
9064:. Scott, Foresman.
8572:SIGNALS AND SYSTEMS
7566:Extract of page 275
6725:Pythagorean theorem
6723:are related to the
5952:Triangle identities
5946:triangle identities
5572:autonomous vehicles
5264:floating point unit
5230:mathematical tables
4653:< 0° or 0° <
2356:{\displaystyle 1/2}
2214:{\displaystyle 1/2}
2032:{\displaystyle 1/2}
659:Nicolaus Copernicus
655:George of Trebizond
613:of Ptolemy's Greek
454:trigonometric table
406:celestial mechanics
171:Pythagorean theorem
16:For the album, see
10560:Numerical analysis
10169:Mathematical logic
10164:Information theory
10008:Algebraic geometry
9814:by David Joyce of
9744:Weisstein, Eric W.
9676:Barnes & Noble
9500:. Addison Wesley.
9122:Plane Trigonometry
9009:M. Rahman (2011).
8631:(2 October 2003).
8201:Weisstein, Eric W.
7625:Ptolemy's Almagest
7303:
7051:
6983:
6948:
6909:
6845:
6781:
6696:
6660:
6611:
6586:
6541:
6450:
6359:
6265:
6127:
6104:
5935:
5788:Other applications
5701:periodic functions
5697:
5669:
5640:
5603:Periodic functions
5557:
5553:marine chronometer
5201:
5105:
5103:
4888:
4886:
4072:
4036:
4006:
3973:
3916:
3886:
3850:
3793:
3749:
3713:
3677:
3633:
3600:
3564:
3528:
3492:
3456:
3420:
3337:
3312:{\displaystyle -1}
3309:
3285:
3249:
3228:
3195:
3174:
3138:
3117:
3092:
3056:
3035:
2999:
2974:
2948:{\displaystyle -1}
2945:
2921:
2882:
2857:{\displaystyle -2}
2854:
2827:
2806:
2781:
2745:
2719:
2698:
2665:{\displaystyle -1}
2662:
2638:
2607:
2582:
2561:
2528:
2505:{\displaystyle -1}
2502:
2478:
2442:
2406:
2374:
2353:
2324:
2291:
2258:
2232:
2211:
2182:
2149:
2116:
2095:
2062:
2029:
2000:
1974:
1953:
1921:
1889:
1857:
1828:
1799:
1770:
1731:
1699:
1659:
1651:
1440:
1328:
1245:
1102:
964:
884:
783:
647:Basilios Bessarion
611:Latin translations
462:
10686:
10685:
10285:Harmonic analysis
10072:
10071:
9768:Library resources
9702:978-0-387-94822-5
9662:978-0-471-54397-8
9623:978-0-7668-6189-3
9596:978-1-4390-4907-5
9569:978-0-471-75684-2
9534:978-93-5057-414-0
9507:978-0-201-78666-8
9459:978-1-136-13702-0
9423:978-3-642-13748-8
9396:978-1-134-10371-3
9340:978-0-387-90497-9
9297:978-3-11-019798-3
9270:978-0-19-967088-8
9243:978-90-5199-210-6
9216:978-1-5104-6003-4
9189:978-3-319-00137-1
9159:978-0-306-47042-4
9132:978-0-07-028187-5
9098:978-0-486-14515-0
9071:978-0-673-18393-4
9022:978-1-84564-564-9
8995:978-0-387-22770-2
8968:978-0-521-53441-3
8941:978-1-4411-0785-5
8897:978-3-319-62533-1
8853:978-0-495-56203-0
8819:(11): 1013–1041.
8733:978-0-387-22081-9
8671:978-0-87150-284-1
8644:978-0-19-850841-0
8627:; Raymond Flood;
8609:978-1-4419-6053-5
8582:978-81-203-4941-4
8555:978-3-319-25468-5
8528:978-3-642-59273-7
8474:978-0-88385-773-1
8444:978-0-547-20998-2
8417:978-0-08-047340-6
8390:978-0-521-01707-7
8363:978-0-02-864365-6
8336:978-1-4390-4460-5
8306:978-0-19-280675-8
8185:978-0-470-16984-1
8158:978-0-88385-984-1
8131:978-1-305-53703-3
8101:978-1-59200-007-4
8074:978-0-19-152309-0
8047:978-0-313-32433-8
8020:978-0-393-32030-5
7937:978-0-691-11485-9
7819:978-0-7858-3595-0
7699:978-1-4020-0260-1
7634:978-0-691-00260-6
7590:978-3-540-06995-9
7560:978-1-5104-2058-8
7490:978-1-118-82741-3
7408:Harper, Douglas.
7298:
7201:
7144:
6908:
6844:
6780:
6679:
6659:
6609:
6585:
6540:
6512:
6445:
6372:or equivalently:
6260:
6259:
6145:of the triangle:
6099:
6064:
6043:
6022:
5878:computer graphics
5874:civil engineering
5854:image compression
5766:physical sciences
5740:quantum mechanics
5736:Fourier transform
5721:periodic function
5613:Fourier transform
5096:
5004:
4979:
4954:
4882:
4778:
4753:
4728:
4661:
4660:
4625:< 0 or 0 <
4432:all real numbers
4092:
4091:
3353:
3352:
3335:
3275:
3252:{\displaystyle 0}
3218:
3198:{\displaystyle 1}
3172:
3141:{\displaystyle 2}
3115:
3082:
3059:{\displaystyle 1}
3025:
2997:
2977:{\displaystyle 2}
2911:
2880:
2830:{\displaystyle 2}
2804:
2771:
2748:{\displaystyle 1}
2722:{\displaystyle 0}
2688:
2636:
2605:
2585:{\displaystyle 1}
2551:
2531:{\displaystyle 0}
2468:
2432:
2377:{\displaystyle 0}
2314:
2281:
2261:{\displaystyle 1}
2235:{\displaystyle 0}
2172:
2139:
2119:{\displaystyle 1}
2085:
2052:
2003:{\displaystyle 0}
1675:standard position
1435:
1422:
1393:
1391:
1386:
1376:
1323:
1310:
1308:
1303:
1293:
1240:
1227:
1225:
1220:
1210:
1097:
1068:
1035:
1022:
1020:
1015:
959:
946:
944:
939:
879:
866:
864:
859:
460:of trigonometry".
429:solve an equation
371:Hellenistic world
306:
305:
198:inverse functions
141:Laws and theorems
10706:
10674:
10673:
10662:
10661:
10650:
10649:
10639:
10638:
10570:Computer algebra
10545:Computer science
10265:Complex analysis
10099:
10092:
10085:
10076:
10075:
10062:
10061:
9850:
9843:
9836:
9827:
9826:
9816:Clark University
9757:
9756:
9731:
9706:
9687:
9666:
9654:
9631:
9627:
9607:
9601:
9600:
9580:
9574:
9573:
9554:Cynthia Y. Young
9550:
9539:
9538:
9518:
9512:
9511:
9491:
9485:
9484:
9470:
9464:
9463:
9447:
9437:
9428:
9427:
9407:
9401:
9400:
9380:
9374:
9373:
9371:
9369:
9359:
9351:
9345:
9344:
9324:
9318:
9317:
9308:
9302:
9301:
9281:
9275:
9274:
9254:
9248:
9247:
9227:
9221:
9220:
9200:
9194:
9193:
9170:
9164:
9163:
9143:
9137:
9136:
9116:
9103:
9102:
9082:
9076:
9075:
9055:
9044:
9043:
9033:
9027:
9026:
9006:
9000:
8999:
8979:
8973:
8972:
8952:
8946:
8945:
8925:
8919:
8918:
8908:
8902:
8901:
8881:
8875:
8874:
8864:
8858:
8857:
8837:
8831:
8830:
8828:
8804:
8798:
8797:
8787:
8781:
8775:
8769:
8763:
8757:
8756:
8754:
8744:
8738:
8737:
8717:
8711:
8710:
8699:
8693:
8692:
8682:
8676:
8675:
8655:
8649:
8648:
8620:
8614:
8613:
8593:
8587:
8586:
8566:
8560:
8559:
8539:
8533:
8532:
8518:Complex Analysis
8512:
8506:
8505:
8485:
8479:
8478:
8458:
8449:
8448:
8428:
8422:
8421:
8401:
8395:
8394:
8374:
8368:
8367:
8347:
8341:
8340:
8320:
8311:
8310:
8256:
8250:
8249:
8221:
8215:
8214:
8213:
8196:
8190:
8189:
8169:
8163:
8162:
8142:
8136:
8135:
8115:
8106:
8105:
8085:
8079:
8078:
8058:
8052:
8051:
8031:
8025:
8024:
8006:
8000:
7986:
7980:
7979:
7977:
7976:
7960:
7954:
7948:
7942:
7941:
7923:
7917:
7916:
7910:
7909:
7895:
7889:
7888:
7860:
7854:
7853:
7847:
7846:
7830:
7824:
7823:
7803:
7794:
7791:
7785:
7784:
7752:
7746:
7740:
7731:
7730:
7728:
7727:
7713:
7704:
7703:
7673:
7667:
7661:
7655:
7645:
7639:
7638:
7617:
7611:
7601:
7595:
7594:
7574:
7568:
7564:
7544:
7538:
7528:
7522:
7521:
7501:
7495:
7494:
7474:
7468:
7467:
7457:
7451:
7445:
7439:
7434:R. Nagel (ed.),
7432:
7426:
7425:
7423:
7422:
7405:
7312:
7310:
7309:
7304:
7299:
7297:
7296:
7295:
7277:
7276:
7263:
7259:
7258:
7243:
7242:
7220:
7202:
7197:
7196:
7195:
7177:
7176:
7163:
7145:
7143:
7135:
7134:
7133:
7115:
7114:
7101:
7060:
7058:
7057:
7052:
7023:
7022:
6993:, respectively.
6992:
6990:
6989:
6984:
6982:
6974:
6973:
6957:
6955:
6954:
6949:
6947:
6939:
6938:
6918:
6916:
6915:
6910:
6906:
6899:
6898:
6874:
6873:
6854:
6852:
6851:
6846:
6842:
6835:
6834:
6810:
6809:
6790:
6788:
6787:
6782:
6778:
6765:
6764:
6746:
6745:
6705:
6703:
6702:
6697:
6680:
6672:
6661:
6657:
6629:Given two sides
6620:
6618:
6617:
6612:
6610:
6608:
6607:
6603:
6587:
6578:
6563:
6562:
6558:
6542:
6533:
6518:
6513:
6511:
6500:
6489:
6459:
6457:
6456:
6451:
6446:
6444:
6433:
6432:
6431:
6419:
6418:
6406:
6405:
6395:
6368:
6366:
6365:
6360:
6334:
6333:
6321:
6320:
6308:
6307:
6274:
6272:
6271:
6266:
6261:
6174:
6173:
6162:
6136:
6134:
6133:
6128:
6113:
6111:
6110:
6105:
6100:
6098:
6090:
6079:
6065:
6063:
6049:
6044:
6042:
6028:
6023:
6021:
6007:
5890:game development
5689:at odd harmonics
5678:
5676:
5675:
5670:
5649:
5647:
5646:
5641:
5485:
5475:
5473:
5472:
5469:
5466:
5465:
5447:
5431:
5429:
5427:
5426:
5423:
5420:
5407:
5405:
5404:
5401:
5398:
5381:
5375:
5373:
5372:
5369:
5366:
5365:
5343:
5341:
5339:
5338:
5335:
5332:
5309:
5307:
5305:
5304:
5301:
5298:
5258:. Most computer
5210:
5208:
5207:
5202:
5167:
5166:
5154:
5153:
5114:
5112:
5111:
5106:
5104:
5097:
5095:
5078:
5077:
5076:
5064:
5063:
5044:
5041:
5036:
5015:
5005:
5003:
4995:
4994:
4985:
4980:
4978:
4970:
4969:
4960:
4955:
4953:
4945:
4944:
4935:
4897:
4895:
4894:
4889:
4887:
4883:
4881:
4858:
4857:
4856:
4838:
4837:
4818:
4815:
4810:
4789:
4779:
4777:
4769:
4768:
4759:
4754:
4752:
4744:
4743:
4734:
4729:
4727:
4719:
4718:
4709:
4646:
4644:
4643:
4640:
4637:
4636:
4620:
4618:
4617:
4614:
4611:
4610:
4591:
4575:
4544:
4536:
4534:
4533:
4530:
4527:
4526:
4518:
4516:
4515:
4512:
4509:
4508:
4485:
4469:
4442:
4429:
4413:
4386:
4384:
4383:
4380:
4377:
4376:
4364:
4362:
4361:
4358:
4355:
4354:
4344:all real numbers
4341:
4325:
4298:
4281:
4265:
4238:
4236:
4235:
4232:
4229:
4228:
4216:
4214:
4213:
4210:
4207:
4206:
4189:
4173:
4135:for real result
4119:
4118:
4088:
4081:
4079:
4078:
4073:
4045:
4043:
4042:
4037:
4015:
4013:
4012:
4007:
3989:
3982:
3980:
3979:
3974:
3925:
3923:
3922:
3917:
3895:
3893:
3892:
3887:
3866:
3859:
3857:
3856:
3851:
3802:
3800:
3799:
3794:
3780:
3758:
3756:
3755:
3750:
3729:
3722:
3720:
3719:
3714:
3686:
3684:
3683:
3678:
3664:
3642:
3640:
3639:
3634:
3616:
3609:
3607:
3606:
3603:{\displaystyle }
3601:
3573:
3571:
3570:
3565:
3537:
3535:
3534:
3529:
3508:
3501:
3499:
3498:
3495:{\displaystyle }
3493:
3465:
3463:
3462:
3457:
3429:
3427:
3426:
3421:
3385:
3384:
3346:
3344:
3343:
3338:
3336:
3331:
3318:
3316:
3315:
3310:
3294:
3292:
3291:
3286:
3281:
3276:
3271:
3258:
3256:
3255:
3250:
3237:
3235:
3234:
3229:
3224:
3219:
3214:
3204:
3202:
3201:
3196:
3183:
3181:
3180:
3175:
3173:
3168:
3147:
3145:
3144:
3139:
3126:
3124:
3123:
3118:
3116:
3111:
3101:
3099:
3098:
3093:
3088:
3083:
3078:
3065:
3063:
3062:
3057:
3044:
3042:
3041:
3036:
3031:
3026:
3021:
3008:
3006:
3005:
3000:
2998:
2993:
2983:
2981:
2980:
2975:
2954:
2952:
2951:
2946:
2930:
2928:
2927:
2922:
2917:
2912:
2907:
2891:
2889:
2888:
2883:
2881:
2876:
2863:
2861:
2860:
2855:
2836:
2834:
2833:
2828:
2815:
2813:
2812:
2807:
2805:
2800:
2790:
2788:
2787:
2782:
2777:
2772:
2767:
2754:
2752:
2751:
2746:
2728:
2726:
2725:
2720:
2707:
2705:
2704:
2699:
2694:
2689:
2684:
2671:
2669:
2668:
2663:
2647:
2645:
2644:
2639:
2637:
2632:
2616:
2614:
2613:
2608:
2606:
2601:
2591:
2589:
2588:
2583:
2570:
2568:
2567:
2562:
2557:
2552:
2547:
2537:
2535:
2534:
2529:
2511:
2509:
2508:
2503:
2487:
2485:
2484:
2479:
2474:
2469:
2464:
2451:
2449:
2448:
2443:
2438:
2433:
2428:
2415:
2413:
2412:
2407:
2402:
2383:
2381:
2380:
2375:
2362:
2360:
2359:
2354:
2349:
2333:
2331:
2330:
2325:
2320:
2315:
2310:
2300:
2298:
2297:
2292:
2287:
2282:
2277:
2267:
2265:
2264:
2259:
2241:
2239:
2238:
2233:
2220:
2218:
2217:
2212:
2207:
2191:
2189:
2188:
2183:
2178:
2173:
2168:
2158:
2156:
2155:
2150:
2145:
2140:
2135:
2125:
2123:
2122:
2117:
2104:
2102:
2101:
2096:
2091:
2086:
2081:
2071:
2069:
2068:
2063:
2058:
2053:
2048:
2038:
2036:
2035:
2030:
2025:
2009:
2007:
2006:
2001:
1983:
1981:
1980:
1975:
1962:
1960:
1959:
1954:
1949:
1930:
1928:
1927:
1922:
1917:
1898:
1896:
1895:
1890:
1885:
1866:
1864:
1863:
1858:
1853:
1837:
1835:
1834:
1829:
1824:
1808:
1806:
1805:
1800:
1795:
1779:
1777:
1776:
1771:
1766:
1744:
1743:
1740:
1738:
1737:
1732:
1708:
1706:
1705:
1700:
1587:
1583:
1578:
1577:
1574:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1478:A common use of
1449:
1447:
1446:
1441:
1436:
1428:
1423:
1421:
1410:
1399:
1394:
1392:
1389:
1387:
1384:
1382:
1377:
1375:
1361:
1337:
1335:
1334:
1329:
1324:
1316:
1311:
1309:
1306:
1304:
1301:
1299:
1294:
1292:
1278:
1254:
1252:
1251:
1246:
1241:
1233:
1228:
1226:
1223:
1221:
1218:
1216:
1211:
1209:
1195:
1111:
1109:
1108:
1103:
1098:
1096:
1085:
1074:
1069:
1067:
1063:
1054:
1050:
1041:
1036:
1028:
1023:
1021:
1018:
1016:
1013:
1011:
973:
971:
970:
965:
960:
952:
947:
945:
942:
940:
937:
935:
893:
891:
890:
885:
880:
872:
867:
865:
862:
860:
857:
855:
781:
766:
751:
500:inscribed angles
346:
339:
329:
322:
298:
291:
284:
55:
41:
40:
10714:
10713:
10709:
10708:
10707:
10705:
10704:
10703:
10689:
10688:
10687:
10682:
10633:
10624:
10574:
10531:
10510:Systems science
10441:
10437:Homotopy theory
10403:
10370:
10322:
10294:
10241:
10188:
10159:Category theory
10145:
10110:
10103:
10073:
10068:
10050:
10017:
9986:
9943:
9936:
9891:
9884:
9859:
9854:
9788:
9787:
9786:
9776:
9775:
9771:
9764:
9742:
9716:
9713:
9711:Further reading
9703:
9693:Early Astronomy
9663:
9640:
9635:
9634:
9624:
9608:
9604:
9597:
9581:
9577:
9570:
9551:
9542:
9535:
9519:
9515:
9508:
9492:
9488:
9471:
9467:
9460:
9438:
9431:
9424:
9408:
9404:
9397:
9381:
9377:
9367:
9365:
9357:
9353:
9352:
9348:
9341:
9325:
9321:
9310:
9309:
9305:
9298:
9282:
9278:
9271:
9255:
9251:
9244:
9228:
9224:
9217:
9201:
9197:
9190:
9171:
9167:
9160:
9144:
9140:
9133:
9125:. McGraw-Hill.
9117:
9106:
9099:
9083:
9079:
9072:
9056:
9047:
9034:
9030:
9023:
9007:
9003:
8996:
8980:
8976:
8969:
8953:
8949:
8942:
8926:
8922:
8909:
8905:
8898:
8882:
8878:
8865:
8861:
8854:
8838:
8834:
8805:
8801:
8788:
8784:
8776:
8772:
8764:
8760:
8752:
8746:
8745:
8741:
8734:
8718:
8714:
8707:Popular Science
8701:
8700:
8696:
8683:
8679:
8672:
8656:
8652:
8645:
8621:
8617:
8610:
8594:
8590:
8583:
8567:
8563:
8556:
8540:
8536:
8529:
8513:
8509:
8502:
8486:
8482:
8475:
8459:
8452:
8445:
8429:
8425:
8418:
8402:
8398:
8391:
8375:
8371:
8364:
8348:
8344:
8337:
8321:
8314:
8307:
8257:
8253:
8238:
8222:
8218:
8199:
8197:
8193:
8186:
8170:
8166:
8159:
8143:
8139:
8132:
8116:
8109:
8102:
8086:
8082:
8075:
8059:
8055:
8048:
8032:
8028:
8021:
8013:. W.W. Norton.
8007:
8003:
7987:
7983:
7974:
7972:
7962:
7961:
7957:
7949:
7945:
7938:
7924:
7920:
7907:
7905:
7897:
7896:
7892:
7885:
7861:
7857:
7844:
7842:
7832:
7831:
7827:
7820:
7804:
7797:
7792:
7788:
7753:
7749:
7741:
7734:
7725:
7723:
7715:
7714:
7707:
7700:
7692:. p. 157.
7674:
7670:
7662:
7658:
7648:Thurston (1996)
7646:
7642:
7635:
7618:
7614:
7604:Thurston (1996)
7602:
7598:
7591:
7575:
7571:
7561:
7545:
7541:
7529:
7525:
7518:
7502:
7498:
7491:
7475:
7471:
7466:. G. P. Putnam.
7458:
7454:
7446:
7442:
7433:
7429:
7420:
7418:
7406:
7402:
7397:
7392:
7368:Skinny triangle
7333:
7325:
7319:
7285:
7281:
7269:
7265:
7264:
7251:
7247:
7232:
7228:
7221:
7219:
7185:
7181:
7169:
7165:
7164:
7162:
7136:
7123:
7119:
7107:
7103:
7102:
7100:
7086:
7083:
7082:
7015:
7011:
7009:
7006:
7005:
7002:Euler's formula
6999:
6997:Euler's formula
6978:
6969:
6965:
6963:
6960:
6959:
6943:
6934:
6930:
6928:
6925:
6924:
6922:
6894:
6890:
6869:
6865:
6863:
6860:
6859:
6830:
6826:
6805:
6801:
6799:
6796:
6795:
6760:
6756:
6741:
6737:
6735:
6732:
6731:
6717:
6712:
6671:
6655:
6653:
6650:
6649:
6627:
6576:
6575:
6571:
6564:
6531:
6530:
6526:
6519:
6517:
6501:
6490:
6488:
6486:
6483:
6482:
6474:, developed by
6471:law of tangents
6466:
6464:Law of tangents
6434:
6427:
6423:
6414:
6410:
6401:
6397:
6396:
6394:
6380:
6377:
6376:
6329:
6325:
6316:
6312:
6303:
6299:
6297:
6294:
6293:
6281:
6163:
6161:
6153:
6150:
6149:
6122:
6119:
6118:
6091:
6080:
6078:
6053:
6048:
6032:
6027:
6011:
6006:
6004:
6001:
6000:
5988:
5954:
5904:
5898:
5886:crystallography
5818:medical imaging
5802:audio synthesis
5790:
5762:
5754:Main articles:
5752:
5655:
5652:
5651:
5626:
5623:
5622:
5615:
5607:Main articles:
5605:
5586:
5580:
5542:
5536:
5513:
5507:
5502:
5496:
5470:
5467:
5463:
5462:
5461:
5459:
5453:
5437:
5424:
5421:
5416:
5415:
5413:
5402:
5399:
5396:
5395:
5393:
5387:
5370:
5367:
5363:
5362:
5361:
5359:
5349:
5336:
5333:
5328:
5327:
5325:
5315:
5302:
5299:
5294:
5293:
5291:
5285:
5278:
5272:
5226:
5220:
5162:
5158:
5140:
5136:
5134:
5131:
5130:
5120:complex numbers
5102:
5101:
5079:
5069:
5065:
5059:
5055:
5045:
5043:
5037:
5026:
5013:
5012:
4996:
4990:
4986:
4984:
4971:
4965:
4961:
4959:
4946:
4940:
4936:
4934:
4921:
4908:
4906:
4903:
4902:
4885:
4884:
4859:
4843:
4839:
4833:
4829:
4819:
4817:
4811:
4800:
4787:
4786:
4770:
4764:
4760:
4758:
4745:
4739:
4735:
4733:
4720:
4714:
4710:
4708:
4695:
4682:
4680:
4677:
4676:
4670:infinite series
4666:
4641:
4638:
4634:
4633:
4632:
4630:
4615:
4612:
4608:
4607:
4606:
4604:
4582:
4569:
4542:
4531:
4528:
4524:
4523:
4522:
4520:
4513:
4510:
4506:
4505:
4504:
4502:
4476:
4463:
4440:
4420:
4407:
4381:
4378:
4374:
4373:
4372:
4370:
4359:
4356:
4352:
4351:
4350:
4348:
4332:
4319:
4296:
4272:
4259:
4233:
4230:
4226:
4225:
4224:
4222:
4211:
4208:
4204:
4203:
4202:
4200:
4180:
4167:
4148:
4139:
4125:Usual notation
4103:
4097:
4052:
4049:
4048:
4022:
4019:
4018:
4001:
3998:
3997:
3932:
3929:
3928:
3902:
3899:
3898:
3878:
3875:
3874:
3809:
3806:
3805:
3776:
3765:
3762:
3761:
3741:
3738:
3737:
3693:
3690:
3689:
3660:
3649:
3646:
3645:
3628:
3625:
3624:
3580:
3577:
3576:
3544:
3541:
3540:
3520:
3517:
3516:
3472:
3469:
3468:
3436:
3433:
3432:
3412:
3409:
3408:
3380:
3364:
3358:
3330:
3325:
3322:
3321:
3301:
3298:
3297:
3277:
3270:
3265:
3262:
3261:
3244:
3241:
3240:
3220:
3213:
3211:
3208:
3207:
3190:
3187:
3186:
3167:
3165:
3162:
3161:
3133:
3130:
3129:
3110:
3108:
3105:
3104:
3084:
3077:
3072:
3069:
3068:
3051:
3048:
3047:
3027:
3020:
3015:
3012:
3011:
2992:
2990:
2987:
2986:
2969:
2966:
2965:
2937:
2934:
2933:
2913:
2906:
2898:
2895:
2894:
2875:
2870:
2867:
2866:
2846:
2843:
2842:
2822:
2819:
2818:
2799:
2797:
2794:
2793:
2773:
2766:
2761:
2758:
2757:
2740:
2737:
2736:
2714:
2711:
2710:
2690:
2683:
2678:
2675:
2674:
2654:
2651:
2650:
2631:
2626:
2623:
2622:
2600:
2598:
2595:
2594:
2577:
2574:
2573:
2553:
2546:
2544:
2541:
2540:
2523:
2520:
2519:
2494:
2491:
2490:
2470:
2463:
2458:
2455:
2454:
2434:
2427:
2422:
2419:
2418:
2398:
2390:
2387:
2386:
2369:
2366:
2365:
2345:
2340:
2337:
2336:
2316:
2309:
2307:
2304:
2303:
2283:
2276:
2274:
2271:
2270:
2253:
2250:
2249:
2227:
2224:
2223:
2203:
2198:
2195:
2194:
2174:
2167:
2165:
2162:
2161:
2141:
2134:
2132:
2129:
2128:
2111:
2108:
2107:
2087:
2080:
2078:
2075:
2074:
2054:
2047:
2045:
2042:
2041:
2021:
2016:
2013:
2012:
1995:
1992:
1991:
1969:
1966:
1965:
1945:
1937:
1934:
1933:
1913:
1905:
1902:
1901:
1881:
1873:
1870:
1869:
1849:
1844:
1841:
1840:
1820:
1815:
1812:
1811:
1791:
1786:
1783:
1782:
1762:
1757:
1754:
1753:
1714:
1711:
1710:
1682:
1679:
1678:
1643:
1637:
1585:
1581:
1546:
1542:
1476:
1470:
1427:
1411:
1400:
1398:
1388:
1383:
1381:
1365:
1360:
1346:
1343:
1342:
1315:
1305:
1300:
1298:
1282:
1277:
1263:
1260:
1259:
1232:
1222:
1217:
1215:
1199:
1194:
1180:
1177:
1176:
1086:
1075:
1073:
1059:
1055:
1046:
1042:
1040:
1027:
1017:
1012:
1010:
996:
993:
992:
951:
941:
936:
934:
920:
917:
916:
871:
861:
856:
854:
840:
837:
836:
767:
752:
737:
730:
724:
704:Colin Maclaurin
696:complex numbers
603:law of tangents
562:and astronomer
555:Surya Siddhanta
532:. Ptolemy used
477:ancient Nubians
443:
437:
302:
123:Exact constants
37:
32:
25:
12:
11:
5:
10712:
10702:
10701:
10684:
10683:
10681:
10680:
10668:
10656:
10644:
10629:
10626:
10625:
10623:
10622:
10617:
10612:
10607:
10602:
10597:
10596:
10595:
10588:Mathematicians
10584:
10582:
10580:Related topics
10576:
10575:
10573:
10572:
10567:
10562:
10557:
10552:
10547:
10541:
10539:
10533:
10532:
10530:
10529:
10528:
10527:
10522:
10517:
10515:Control theory
10507:
10502:
10497:
10492:
10487:
10482:
10477:
10472:
10467:
10462:
10457:
10451:
10449:
10443:
10442:
10440:
10439:
10434:
10429:
10424:
10419:
10413:
10411:
10405:
10404:
10402:
10401:
10396:
10391:
10386:
10380:
10378:
10372:
10371:
10369:
10368:
10363:
10358:
10353:
10348:
10343:
10338:
10332:
10330:
10324:
10323:
10321:
10320:
10315:
10310:
10304:
10302:
10296:
10295:
10293:
10292:
10290:Measure theory
10287:
10282:
10277:
10272:
10267:
10262:
10257:
10251:
10249:
10243:
10242:
10240:
10239:
10234:
10229:
10224:
10219:
10214:
10209:
10204:
10198:
10196:
10190:
10189:
10187:
10186:
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9942:Non-Euclidean
9938:
9937:
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9934:
9932:Solid geometry
9929:
9928:
9927:
9922:
9915:Plane geometry
9912:
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9762:External links
9760:
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9088:Wave Phenomena
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8778:Nielsen (1966)
8770:
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8755:. Intel. 2013.
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8637:. OUP Oxford.
8629:Eleanor Robson
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7795:
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7747:
7745:, p. 238.
7732:
7717:"trigonometry"
7705:
7698:
7678:Selin, Helaine
7668:
7666:, p. 215.
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6476:François Viète
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6343:
6340:
6337:
6332:
6328:
6324:
6319:
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6306:
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6286:law of cosines
6280:
6279:Law of cosines
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2020:
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1989:
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1963:
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1639:Main article:
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1472:Main article:
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1462:law of cosines
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607:Western Europe
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10376:Number theory
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10308:Combinatorics
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10260:Real analysis
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9916:
9913:
9911:
9908:
9906:
9903:
9901:
9900:Combinatorial
9898:
9897:
9895:
9893:
9887:
9881:
9878:
9874:
9871:
9870:
9869:
9866:
9865:
9862:
9858:
9851:
9846:
9844:
9839:
9837:
9832:
9831:
9828:
9822:
9819:
9817:
9813:
9810:
9808:
9804:
9801:
9798:
9795:
9793:
9790:
9789:
9783:
9780:
9779:
9774:
9769:
9754:
9753:
9748:
9745:
9740:
9737:
9733:
9729:
9725:
9724:
9719:
9715:
9714:
9704:
9698:
9694:
9689:
9685:
9681:
9677:
9673:
9668:
9664:
9658:
9653:
9652:
9646:
9642:
9641:
9630:
9625:
9619:
9615:
9614:
9606:
9598:
9592:
9588:
9587:
9579:
9571:
9565:
9561:
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9555:
9549:
9547:
9545:
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9530:
9526:
9525:
9517:
9509:
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9499:
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9490:
9482:
9478:
9477:
9469:
9461:
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9445:
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9434:
9425:
9419:
9415:
9414:
9406:
9398:
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9387:
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9363:
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9350:
9342:
9336:
9332:
9331:
9323:
9315:
9314:
9307:
9299:
9293:
9289:
9288:
9280:
9272:
9266:
9262:
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9253:
9245:
9239:
9235:
9234:
9226:
9218:
9212:
9208:
9207:
9199:
9191:
9185:
9181:
9180:
9175:
9169:
9161:
9155:
9151:
9150:
9142:
9134:
9128:
9124:
9123:
9115:
9113:
9111:
9109:
9100:
9094:
9090:
9089:
9081:
9073:
9067:
9063:
9062:
9054:
9052:
9050:
9041:
9040:
9032:
9024:
9018:
9015:. WIT Press.
9014:
9013:
9005:
8997:
8991:
8987:
8986:
8978:
8970:
8964:
8960:
8959:
8951:
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8933:
8932:
8924:
8916:
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8907:
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8872:
8871:
8863:
8855:
8849:
8845:
8844:
8836:
8827:
8822:
8818:
8814:
8810:
8803:
8795:
8794:
8786:
8779:
8774:
8767:
8762:
8751:
8750:
8743:
8735:
8729:
8725:
8724:
8716:
8708:
8704:
8698:
8690:
8689:
8681:
8673:
8667:
8663:
8662:
8654:
8646:
8640:
8636:
8635:
8630:
8626:
8625:Mary Croarken
8619:
8611:
8605:
8601:
8600:
8592:
8584:
8578:
8574:
8573:
8565:
8557:
8551:
8547:
8546:
8538:
8530:
8524:
8520:
8519:
8511:
8503:
8501:9789811384646
8497:
8493:
8492:
8484:
8476:
8470:
8466:
8465:
8457:
8455:
8446:
8440:
8436:
8435:
8427:
8419:
8413:
8409:
8408:
8400:
8392:
8386:
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8381:
8373:
8365:
8359:
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8346:
8338:
8332:
8328:
8327:
8319:
8317:
8308:
8302:
8298:
8292:
8288:
8284:
8280:
8277:
8275:
8271:
8267:
8263:
8255:
8247:
8243:
8239:
8237:0-7487-3396-5
8233:
8229:
8228:
8220:
8211:
8210:
8205:
8202:
8195:
8187:
8181:
8177:
8176:
8168:
8160:
8154:
8150:
8149:
8141:
8133:
8127:
8123:
8122:
8114:
8112:
8103:
8097:
8093:
8092:
8084:
8076:
8070:
8066:
8065:
8057:
8049:
8043:
8039:
8038:
8030:
8022:
8016:
8012:
8005:
7999:
7998:0-7156-2418-0
7995:
7991:
7985:
7971:
7970:
7965:
7959:
7952:
7947:
7939:
7933:
7929:
7922:
7915:
7904:
7900:
7894:
7886:
7884:9780521594486
7880:
7876:
7872:
7868:
7867:
7859:
7852:
7841:
7840:
7835:
7829:
7821:
7815:
7811:
7810:
7802:
7800:
7790:
7782:
7778:
7774:
7770:
7766:
7762:
7758:
7751:
7744:
7739:
7737:
7722:
7718:
7712:
7710:
7701:
7695:
7691:
7687:
7683:
7679:
7672:
7665:
7660:
7653:
7649:
7644:
7636:
7630:
7626:
7622:
7616:
7609:
7605:
7600:
7592:
7586:
7582:
7581:
7573:
7567:
7562:
7556:
7552:
7551:
7543:
7536:
7532:
7527:
7519:
7517:0-618-64332-X
7513:
7509:
7508:
7500:
7492:
7486:
7482:
7481:
7473:
7465:
7464:
7456:
7449:
7444:
7437:
7431:
7417:
7416:
7411:
7404:
7400:
7389:
7386:
7384:
7381:
7379:
7376:
7374:
7371:
7369:
7366:
7364:
7361:
7359:
7356:
7354:
7351:
7349:
7348:Lénárt sphere
7346:
7344:
7341:
7339:
7336:
7335:
7328:
7324:
7300:
7292:
7289:
7286:
7282:
7278:
7273:
7270:
7266:
7255:
7252:
7248:
7244:
7239:
7236:
7233:
7229:
7222:
7216:
7213:
7210:
7207:
7203:
7198:
7192:
7189:
7186:
7182:
7178:
7173:
7170:
7166:
7159:
7156:
7153:
7150:
7146:
7140:
7137:
7130:
7127:
7124:
7120:
7116:
7111:
7108:
7104:
7097:
7094:
7091:
7088:
7081:
7080:
7079:
7077:
7074:
7070:
7069:
7064:
7048:
7045:
7042:
7039:
7036:
7033:
7030:
7027:
7024:
7019:
7016:
7012:
7003:
6994:
6979:
6975:
6970:
6966:
6944:
6940:
6935:
6931:
6903:
6900:
6895:
6891:
6887:
6884:
6881:
6878:
6875:
6870:
6866:
6858:
6857:
6839:
6836:
6831:
6827:
6823:
6820:
6817:
6814:
6811:
6806:
6802:
6794:
6793:
6775:
6772:
6769:
6766:
6761:
6757:
6753:
6750:
6747:
6742:
6738:
6730:
6729:
6728:
6726:
6722:
6693:
6690:
6687:
6684:
6681:
6676:
6673:
6668:
6662:
6648:
6647:
6646:
6644:
6640:
6636:
6632:
6604:
6597:
6594:
6591:
6582:
6579:
6572:
6568:
6565:
6559:
6552:
6549:
6546:
6537:
6534:
6527:
6523:
6520:
6514:
6508:
6505:
6502:
6497:
6494:
6491:
6481:
6480:
6479:
6477:
6473:
6472:
6447:
6441:
6438:
6435:
6428:
6424:
6420:
6415:
6411:
6407:
6402:
6398:
6391:
6388:
6385:
6382:
6375:
6374:
6373:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6330:
6326:
6322:
6317:
6313:
6309:
6304:
6300:
6292:
6291:
6290:
6288:
6287:
6262:
6253:
6250:
6247:
6244:
6241:
6232:
6229:
6226:
6223:
6220:
6211:
6208:
6205:
6202:
6199:
6190:
6187:
6184:
6181:
6178:
6170:
6167:
6164:
6158:
6155:
6148:
6147:
6146:
6144:
6140:
6101:
6092:
6087:
6084:
6081:
6075:
6072:
6069:
6066:
6060:
6057:
6054:
6050:
6045:
6039:
6036:
6033:
6029:
6024:
6018:
6015:
6012:
6008:
5999:
5998:
5997:
5995:
5994:
5983:
5981:
5977:
5973:
5969:
5965:
5961:
5956:
5949:
5947:
5943:
5938:
5933:
5929:
5925:
5921:
5917:
5913:
5908:
5903:
5893:
5891:
5887:
5883:
5879:
5875:
5871:
5867:
5863:
5859:
5855:
5851:
5847:
5843:
5839:
5835:
5834:number theory
5831:
5827:
5823:
5819:
5815:
5811:
5807:
5803:
5799:
5795:
5785:
5783:
5779:
5775:
5771:
5767:
5761:
5757:
5747:
5745:
5741:
5737:
5733:
5728:
5726:
5722:
5718:
5714:
5710:
5706:
5702:
5694:
5690:
5686:
5682:
5663:
5657:
5634:
5628:
5619:
5614:
5610:
5600:
5598:
5593:
5591:
5585:
5575:
5573:
5569:
5565:
5560:
5554:
5550:
5546:
5541:
5531:
5529:
5525:
5521:
5520:triangulation
5516:
5512:
5501:
5491:
5489:
5483:
5479:
5457:
5451:
5445:
5441:
5435:
5419:
5411:
5391:
5385:
5379:
5357:
5353:
5347:
5331:
5323:
5319:
5313:
5297:
5289:
5283:
5277:
5267:
5265:
5261:
5257:
5253:
5249:
5245:
5241:
5239:
5235:
5231:
5225:
5215:
5198:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5163:
5159:
5155:
5150:
5147:
5144:
5141:
5137:
5129:
5128:
5127:
5125:
5121:
5098:
5092:
5086:
5083:
5073:
5070:
5066:
5060:
5052:
5049:
5033:
5030:
5027:
5023:
5019:
5017:
5009:
5006:
5000:
4997:
4991:
4987:
4981:
4975:
4972:
4966:
4962:
4956:
4950:
4947:
4941:
4937:
4931:
4928:
4925:
4923:
4918:
4915:
4912:
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4878:
4872:
4869:
4866:
4863:
4853:
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4847:
4844:
4840:
4834:
4826:
4823:
4807:
4804:
4801:
4797:
4793:
4791:
4783:
4780:
4774:
4771:
4765:
4761:
4755:
4749:
4746:
4740:
4736:
4730:
4724:
4721:
4715:
4711:
4705:
4702:
4699:
4697:
4692:
4689:
4686:
4675:
4674:
4673:
4671:
4656:
4652:
4648:
4628:
4624:
4602:
4600:
4596:
4593:
4589:
4585:
4580:
4577:
4573:
4567:
4564:
4562:
4559:
4558:
4554:
4550:
4546:
4540:
4500:
4496:
4494:
4490:
4487:
4483:
4479:
4474:
4471:
4467:
4461:
4458:
4456:
4453:
4452:
4448:
4444:
4438:
4434:
4431:
4427:
4423:
4418:
4415:
4411:
4405:
4402:
4400:
4397:
4396:
4392:
4388:
4368:
4346:
4343:
4339:
4335:
4330:
4327:
4323:
4317:
4314:
4312:
4309:
4308:
4304:
4300:
4294:
4290:
4287:
4283:
4279:
4275:
4270:
4267:
4263:
4257:
4254:
4252:
4249:
4248:
4244:
4240:
4220:
4198:
4195:
4191:
4187:
4183:
4178:
4175:
4171:
4165:
4162:
4160:
4157:
4156:
4152:
4146:
4143:
4137:
4134:
4130:
4127:
4124:
4121:
4120:
4117:
4114:
4112:
4108:
4102:
4087:
4083:
4063:
4057:
4047:
4033:
4030:
4027:
4024:
4017:
4003:
3996:
3993:
3992:
3988:
3984:
3964:
3961:
3955:
3949:
3946:
3943:
3937:
3927:
3913:
3910:
3907:
3904:
3897:
3883:
3880:
3873:
3870:
3869:
3865:
3861:
3841:
3838:
3832:
3826:
3823:
3820:
3814:
3804:
3790:
3787:
3784:
3781:
3777:
3773:
3770:
3767:
3760:
3746:
3743:
3736:
3733:
3732:
3728:
3724:
3704:
3698:
3688:
3674:
3671:
3668:
3665:
3661:
3657:
3654:
3651:
3644:
3630:
3623:
3620:
3619:
3615:
3611:
3594:
3591:
3588:
3585:
3575:
3555:
3549:
3539:
3525:
3522:
3515:
3512:
3511:
3507:
3503:
3486:
3483:
3480:
3477:
3467:
3447:
3441:
3431:
3417:
3414:
3407:
3404:
3403:
3399:
3396:
3393:
3390:
3387:
3386:
3383:
3375:
3373:
3369:
3363:
3348:
3332:
3327:
3320:
3306:
3303:
3296:
3282:
3278:
3272:
3267:
3260:
3246:
3239:
3225:
3221:
3215:
3206:
3192:
3185:
3169:
3160:
3157:
3154:
3153:
3149:
3135:
3128:
3112:
3103:
3089:
3085:
3079:
3074:
3067:
3053:
3046:
3032:
3028:
3022:
3017:
3010:
2994:
2985:
2971:
2964:
2961:
2958:
2957:
2942:
2939:
2932:
2918:
2914:
2908:
2903:
2900:
2893:
2877:
2872:
2865:
2851:
2848:
2841:
2838:
2824:
2817:
2801:
2792:
2778:
2774:
2768:
2763:
2756:
2742:
2735:
2732:
2731:
2716:
2709:
2695:
2691:
2685:
2680:
2673:
2659:
2656:
2649:
2633:
2628:
2621:
2618:
2602:
2593:
2579:
2572:
2558:
2554:
2548:
2539:
2525:
2518:
2515:
2514:
2499:
2496:
2489:
2475:
2471:
2465:
2460:
2453:
2439:
2435:
2429:
2424:
2417:
2403:
2399:
2395:
2392:
2385:
2371:
2364:
2350:
2346:
2342:
2335:
2321:
2317:
2311:
2302:
2288:
2284:
2278:
2269:
2255:
2248:
2245:
2244:
2229:
2222:
2208:
2204:
2200:
2193:
2179:
2175:
2169:
2160:
2146:
2142:
2136:
2127:
2113:
2106:
2092:
2088:
2082:
2073:
2059:
2055:
2049:
2040:
2026:
2022:
2018:
2011:
1997:
1990:
1987:
1986:
1971:
1964:
1950:
1946:
1942:
1939:
1932:
1918:
1914:
1910:
1907:
1900:
1886:
1882:
1878:
1875:
1868:
1854:
1850:
1846:
1839:
1825:
1821:
1817:
1810:
1796:
1792:
1788:
1781:
1767:
1763:
1759:
1752:
1749:
1746:
1745:
1742:
1728:
1725:
1722:
1719:
1716:
1696:
1693:
1690:
1687:
1684:
1676:
1672:
1668:
1667:terminal side
1664:
1655:
1647:
1642:
1632:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1591:, similar to
1590:
1589:
1576:
1536:
1532:
1528:
1525:
1522:
1518:
1514:
1511:
1508:
1504:
1500:
1497:
1496:
1495:
1493:
1489:
1485:
1481:
1475:
1465:
1463:
1459:
1454:
1437:
1432:
1429:
1424:
1418:
1415:
1412:
1407:
1404:
1401:
1395:
1378:
1372:
1369:
1366:
1362:
1357:
1354:
1351:
1348:
1341:
1340:
1325:
1320:
1317:
1312:
1295:
1289:
1286:
1283:
1279:
1274:
1271:
1268:
1265:
1258:
1257:
1242:
1237:
1234:
1229:
1212:
1206:
1203:
1200:
1196:
1191:
1188:
1185:
1182:
1175:
1174:
1173:
1171:
1167:
1163:
1159:
1154:
1152:
1148:
1144:
1143:perpendicular
1140:
1136:
1135:opposite side
1132:
1128:
1124:
1120:
1099:
1093:
1090:
1087:
1082:
1079:
1076:
1070:
1064:
1060:
1056:
1051:
1047:
1043:
1037:
1032:
1029:
1024:
1007:
1004:
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760:
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741:
734:
729:
719:
717:
716:Taylor series
713:
709:
705:
701:
700:James Gregory
697:
693:
689:
688:triangulation
685:
684:Gemma Frisius
681:
680:Trigonometria
677:
673:
668:
666:
665:
660:
656:
652:
648:
644:
640:
636:
635:Regiomontanus
632:
631:De Triangulis
628:
624:
620:
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608:
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589:
585:
581:
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348:
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338:
334:
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317:
314:
313:Ancient Greek
310:
299:
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280:
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276:
271:
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250:Regiomontanus
248:
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30:
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10699:Trigonometry
10675:
10663:
10651:
10632:
10565:Optimization
10427:Differential
10351:Differential
10318:Order theory
10313:Graph theory
10217:Group theory
9998:Trigonometry
9997:
9797:Trigonometry
9773:Trigonometry
9772:
9750:
9735:
9721:
9692:
9671:
9650:
9638:Bibliography
9612:
9605:
9586:Trigonometry
9585:
9578:
9558:
9523:
9516:
9496:
9489:
9475:
9468:
9443:
9412:
9405:
9385:
9378:
9366:. Retrieved
9349:
9329:
9322:
9312:
9306:
9286:
9279:
9259:
9252:
9232:
9225:
9205:
9198:
9178:
9174:Kim Williams
9168:
9148:
9141:
9121:
9087:
9080:
9060:
9038:
9031:
9011:
9004:
8984:
8977:
8957:
8950:
8930:
8923:
8913:
8906:
8886:
8879:
8873:. p. 1.
8869:
8862:
8842:
8835:
8816:
8812:
8802:
8792:
8785:
8773:
8766:Boyer (1991)
8761:
8748:
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8697:
8687:
8680:
8660:
8653:
8633:
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8564:
8544:
8537:
8517:
8510:
8494:. Springer.
8490:
8483:
8463:
8433:
8426:
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8399:
8379:
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8325:
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8063:
8056:
8036:
8029:
8010:
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7989:
7984:
7973:. Retrieved
7967:
7958:
7951:Boyer (1991)
7946:
7927:
7921:
7912:
7906:. Retrieved
7902:
7893:
7865:
7858:
7849:
7843:. Retrieved
7837:
7828:
7808:
7789:
7760:
7756:
7750:
7724:. Retrieved
7685:
7671:
7664:Boyer (1991)
7659:
7643:
7624:
7615:
7599:
7579:
7572:
7549:
7542:
7531:Boyer (1991)
7526:
7507:Trigonometry
7506:
7499:
7479:
7472:
7462:
7455:
7448:Boyer (1991)
7443:
7435:
7430:
7419:. Retrieved
7413:
7403:
7326:
7075:
7066:
7000:
6921:
6718:
6638:
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6630:
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6469:
6467:
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6282:
6138:
6116:
5993:law of sines
5991:
5989:
5986:Law of sines
5979:
5975:
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5963:
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5919:
5915:
5911:
5850:oceanography
5806:architecture
5794:music theory
5791:
5768:, including
5763:
5729:
5725:infinite sum
5698:
5693:1/odd number
5692:
5688:
5594:
5587:
5561:
5558:
5517:
5514:
5494:Applications
5481:
5477:
5455:
5443:
5439:
5417:
5409:
5389:
5377:
5355:
5354:) = 1 − sin(
5351:
5329:
5321:
5320:) = 1 − cos(
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5213:
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4667:
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4597:≤ −1 or 1 ≤
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1458:law of sines
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1127:adjacent leg
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340:
333:
326:
323:
316:
309:Trigonometry
308:
307:
45:Trigonometry
44:
38:
33:
10677:WikiProject
10520:Game theory
10500:Probability
10237:Homological
10227:Multilinear
10207:Commutative
10184:Type theory
10151:Foundations
10107:mathematics
9807:Convergence
9559:Precalculus
8204:"SOHCAHTOA"
7650:, pp.
7606:, pp.
7450:, p. .
7383:Unit circle
5882:cartography
5846:meteorology
5836:(and hence
5810:electronics
5782:light waves
5522:is used in
5448:), and the
5358:) = versin(
5238:Slide rules
5234:interpolate
4561:arccosecant
4128:Definition
4111:restricting
3368:unit circle
1663:unit circle
1641:Unit circle
1168:(sec), and
1158:reciprocals
469:Babylonians
425:expressions
351:mathematics
235:Brahmagupta
204:Derivatives
133:Unit circle
10505:Statistics
10384:Arithmetic
10346:Arithmetic
10212:Elementary
10179:Set theory
9982:Riemannian
9977:Projective
9962:Symplectic
9957:Hyperbolic
9890:Euclidean
7992:, London.
7975:2021-01-08
7908:2018-08-05
7845:2021-01-08
7743:Boyer 1991
7726:2008-07-21
7621:Toomer, G.
7533:, p.
7421:2022-03-18
7395:References
7063:analytical
6721:identities
5896:Identities
5842:seismology
5838:cryptology
5826:ultrasound
5717:continuous
5540:Navigation
5534:Navigation
5458:) = exsec(
5450:excosecant
5324:) = 2 sin(
5290:) = 2 sin(
4449:< 180°
4389:−90° <
4311:arctangent
4131:Domain of
3994:cotangent
3366:Using the
3349:undefined
3158:undefined
3155:cotangent
3150:undefined
2962:undefined
2839:undefined
2619:undefined
1673:placed in
1533:pposite ÷
1519:djacent ÷
1505:pposite ÷
1302:hypotenuse
1219:hypotenuse
1119:hypotenuse
943:hypotenuse
863:hypotenuse
827:hypotenuse
672:navigation
623:Al Battani
578:, all six
504:Hipparchus
492:Archimedes
450:Hipparchus
417:identities
410:navigation
389:) such as
311:(from
245:al-Battani
225:Hipparchus
164:Cotangents
118:Identities
10432:Geometric
10422:Algebraic
10361:Euclidean
10336:Algebraic
10232:Universal
10003:Lie group
9967:Spherical
9752:MathWorld
9728:EMS Press
8209:MathWorld
7781:171015175
7287:−
7245:−
7234:−
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7187:−
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7125:−
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6666:Δ
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6550:−
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6351:
6336:−
6251:−
6230:−
6203:−
6125:Δ
6096:Δ
6058:
6037:
6016:
5862:economics
5858:phonetics
5830:chemistry
5770:acoustics
5760:acoustics
5685:frequency
5681:amplitude
5621:Function
5597:geography
5590:surveying
5584:Surveying
5578:Surveying
5524:astronomy
5511:Astronomy
5505:Astronomy
5388:haversin(
5384:haversine
5350:coversin(
5346:coversine
5190:
5175:
5050:−
5039:∞
5024:∑
5010:⋯
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4932:−
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4824:−
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4798:∑
4784:⋯
4756:−
4706:−
4690:
4455:arcsecant
4393:< 90°
4251:arccosine
4107:injective
4067:∞
4061:∞
4058:−
4034:π
4028:≠
4004:π
3968:∞
3956:∪
3947:−
3941:∞
3938:−
3914:π
3908:≠
3884:π
3871:cosecant
3845:∞
3833:∪
3824:−
3818:∞
3815:−
3791:π
3774:π
3771:≠
3747:π
3708:∞
3702:∞
3699:−
3675:π
3658:π
3655:≠
3631:π
3586:−
3559:∞
3553:∞
3550:−
3526:π
3478:−
3451:∞
3445:∞
3442:−
3418:π
3388:Function
3328:−
3304:−
3268:−
2959:cosecant
2940:−
2901:−
2873:−
2849:−
2681:−
2657:−
2629:−
2497:−
2461:−
2425:−
2393:−
1972:π
1943:π
1911:π
1879:π
1847:π
1818:π
1789:π
1760:π
1747:Function
1726:
1694:
1529:angent =
1523:ypotenuse
1509:ypotenuse
1480:mnemonics
1468:Mnemonics
1416:
1405:
1370:
1352:
1287:
1269:
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1186:
1170:cotangent
1151:Mnemonics
1091:
1080:
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926:
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794:functions
682:in 1595.
645:cardinal
641:, by the
564:Aryabhata
543:Byzantine
419:. These
402:surveying
260:de Moivre
194:Integrals
110:Reference
80:Functions
10693:Category
10653:Category
10409:Topology
10356:Discrete
10341:Analytic
10328:Geometry
10300:Discrete
10255:Calculus
10247:Analysis
10202:Abstract
10141:Glossary
10124:Timeline
10064:Category
9952:Elliptic
9944:geometry
9925:Polyform
9910:Discrete
9892:geometry
9873:Timeline
9857:Geometry
9730:. 2001 .
9647:(1991).
8246:47985033
7767:: 1–56.
7684:(eds.).
7623:(1998).
7331:See also
7071:and the
5822:CT scans
5732:integral
5588:In land
5549:Sextants
5480:) = csc(
5442:) = sec(
5434:exsecant
5412:) = sin(
5256:gradians
4445:0° <
3621:tangent
2516:tangent
1623:rippin'
1593:Krakatoa
1515:osine =
1460:and the
1390:opposite
1385:adjacent
1307:adjacent
1224:opposite
1162:cosecant
1019:adjacent
1014:opposite
938:adjacent
907:adjacent
858:opposite
804:, where
651:Almagest
639:Almagest
621:such as
615:Almagest
591:polymath
529:Almagest
486:such as
465:Sumerian
375:geometry
327:trígōnon
320:τρίγωνον
240:al-Hasib
180:Calculus
159:Tangents
10665:Commons
10447:Applied
10417:General
10194:Algebra
10119:History
9920:Polygon
9868:History
9684:61-9103
9368:6 April
9316:. 1943.
8467:. MAA.
8293:lley".
8285:hrough
8281:opping
7652:239–243
7608:235–236
5814:biology
5798:geodesy
5713:Fourier
5711:waves.
5486:). See
5474:
5460:
5432:), the
5428:
5414:
5408:versin(
5406:
5394:
5382:), the
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5312:versine
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5252:degrees
5124:formula
4649:−90° ≤
4645:
4631:
4619:
4605:
4570:arccsc(
4555:≤ 180°
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4521:
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4503:
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4435:0 <
4408:arccot(
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4320:arctan(
4305:≤ 180°
4260:arccos(
4241:−90° ≤
4237:
4223:
4215:
4201:
4168:arcsin(
4159:arcsine
4151:degrees
4142:radians
3734:secant
3513:cosine
3394:Domain
3391:Period
2733:secant
2246:cosine
1615:nother
1537:djacent
1492:tangent
1164:(csc),
982:Tangent
787:similar
588:Persian
547:Islamic
520:Ptolemy
473:similar
435:History
398:geodesy
270:Fourier
230:Ptolemy
196: (
154:Cosines
96:inverse
82: (
68:History
63:Outline
10366:Finite
10222:Linear
10129:Future
10105:Major
9972:Affine
9905:Convex
9770:about
9699:
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9566:
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9420:
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9364:. 1993
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6779:
6641:, the
6117:where
5774:optics
5772:, and
5756:optics
5454:excsc(
5438:exsec(
4657:≤ 90°
4245:≤ 90°
3400:Graph
3397:Range
1631:cid".
1619:ippie
1611:aught
1607:ippie
1501:ine =
1490:, and
1488:cosine
1166:secant
1133:. The
1125:. The
902:Cosine
508:Nicaea
506:(from
496:chords
488:Euclid
458:father
408:, and
367:ratios
355:angles
344:métron
337:μέτρον
128:Tables
10593:lists
10136:Lists
10109:areas
10034:Lists
10029:Shape
10022:Lists
9991:Other
9880:Lists
9358:(PDF)
8753:(PDF)
8270:orse
7777:S2CID
7763:(1).
5778:sound
5709:light
5705:sound
5484:) − 1
5446:) − 1
5282:chord
4547:0° ≤
4537:<
4501:<
4439:<
4369:<
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4301:0° ≤
4284:−1 ≤
4192:−1 ≤
4122:Name
3405:sine
1988:sine
534:chord
365:with
315:
265:Euler
255:Viète
149:Sines
73:Usage
9697:ISBN
9680:LCCN
9657:ISBN
9618:ISBN
9591:ISBN
9564:ISBN
9529:ISBN
9502:ISBN
9454:ISBN
9418:ISBN
9391:ISBN
9370:2019
9335:ISBN
9292:ISBN
9265:ISBN
9238:ISBN
9211:ISBN
9184:ISBN
9154:ISBN
9127:ISBN
9093:ISBN
9066:ISBN
9017:ISBN
8990:ISBN
8963:ISBN
8936:ISBN
8892:ISBN
8848:ISBN
8728:ISBN
8666:ISBN
8639:ISBN
8604:ISBN
8577:ISBN
8550:ISBN
8523:ISBN
8496:ISBN
8469:ISBN
8439:ISBN
8412:ISBN
8385:ISBN
8358:ISBN
8331:ISBN
8301:ISBN
8274:ame
8262:ome
8242:OCLC
8232:ISBN
8180:ISBN
8153:ISBN
8126:ISBN
8096:ISBN
8069:ISBN
8042:ISBN
8015:ISBN
7994:ISBN
7932:ISBN
7879:ISBN
7814:ISBN
7694:ISBN
7629:ISBN
7585:ISBN
7555:ISBN
7512:ISBN
7485:ISBN
6958:and
6658:Area
6633:and
6625:Area
6468:The
6283:The
5990:The
5978:and
5966:and
5888:and
5824:and
5780:and
5758:and
5742:and
5707:and
5611:and
5570:for
5566:and
5392:) =
5286:crd(
4497:0 ≤
4291:0 ≤
1709:and
1599:ome
1584:-kə-
1484:sine
1156:The
1147:base
1145:and
1117:The
822:Sine
812:and
768:tan
753:cos
738:sin
625:and
609:via
586:The
538:sine
498:and
490:and
391:sine
9805:at
9481:119
9450:153
8821:doi
8289:ur
8276:A''
8266:ld
7871:doi
7769:doi
7535:162
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