Editing Trigonometric functions (section)

==History==
{{Main|History of trigonometry}}
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The [[Chord (geometry)|chord]] function was defined by [[Hipparchus]] of [[İznik|Nicaea]] (180–125&nbsp;BCE) and [[Ptolemy]] of [[Egypt (Roman province)|Roman Egypt]] (90–165&nbsp;CE). The functions of sine and [[versine]] (1 – cosine) are closely related to the [[Jyā, koti-jyā and utkrama-jyā|''jyā'' and ''koti-jyā'']] functions used in [[Gupta period]] [[Indian astronomy]] (''[[Aryabhatiya]]'', ''[[Surya Siddhanta]]''), via translation from Sanskrit to Arabic and then from Arabic to Latin.<ref name="Boyer_1991"/> (See [[Aryabhata's sine table]].)

All six trigonometric functions in current use were known in [[Islamic mathematics]] by the 9th century, as was the [[law of sines]], used in [[solving triangles]].<ref name="Gingerich_1986"/> [[Al-Khwārizmī]] (c. 780–850) produced tables of sines and cosines. Circa 860, [[Habash al-Hasib al-Marwazi]] defined the tangent and the cotangent, and produced their tables.<ref name="Sesiano">Jacques Sesiano, "Islamic mathematics", p. 157, in {{Cite book |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media]] |isbn=978-1-4020-0260-1}}</ref><ref name="Britannica">{{cite web |title=trigonometry |date=17 November 2023 |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |publisher=Encyclopedia Britannica}}</ref> [[Muhammad ibn Jābir al-Harrānī al-Battānī]] (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.<ref name="Britannica"/> The trigonometric functions were later studied by mathematicians including [[Omar Khayyám]], [[Bhāskara II]], [[Nasir al-Din al-Tusi]], [[Jamshīd al-Kāshī]] (14th century), [[Ulugh Beg]] (14th century), [[Regiomontanus]] (1464), [[Georg Joachim Rheticus|Rheticus]], and Rheticus' student [[Valentinus Otho]].

[[Madhava of Sangamagrama]] (c. 1400) made early strides in the [[mathematical analysis|analysis]] of trigonometric functions in terms of [[series (mathematics)|infinite series]].<ref name="mact-biog"/> (See [[Madhava series]] and [[Madhava's sine table]].)

The tangent function was brought to Europe by [[Giovanni Bianchini]] in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.<ref>{{cite journal | url=https://www.jstor.org/stable/45211959 | jstor=45211959 | title=The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates | last1=Van Brummelen | first1=Glen | journal=Archive for History of Exact Sciences | year=2018 | volume=72 | issue=5 | pages=547–563 | doi=10.1007/s00407-018-0214-2 | s2cid=240294796 }}</ref> 

The terms ''tangent'' and ''secant'' were first introduced by the Danish mathematician [[Thomas Fincke]] in his book ''Geometria rotundi'' (1583).<ref name="Fincke"/>

The 17th century French mathematician [[Albert Girard]] made the first published use of the abbreviations ''sin'', ''cos'', and ''tan'' in his book ''Trigonométrie''.<ref name=MacTutor>{{MacTutor|id=Girard_Albert}}</ref>

In a paper published in 1682, [[Gottfried Leibniz]] proved that {{math|sin ''x''}} is not an [[algebraic function]] of {{mvar|x}}.<ref name="Bourbaki_1994"/> Though defined as ratios of sides of a [[right triangle]], and thus appearing to be [[rational function]]s, Leibnitz result established that they are actually [[transcendental function]]s of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his ''[[Introduction to the Analysis of the Infinite]]'' (1748). His method was to show that the sine and cosine functions are [[alternating series]] formed from the even and odd terms respectively of the [[exponential function|exponential series]]. He presented "[[Euler's formula]]", as well as near-modern abbreviations (''sin.'', ''cos.'', ''tang.'', ''cot.'', ''sec.'', and ''cosec.'').<ref name="Boyer_1991"/>

A few functions were common historically, but are now seldom used, such as the [[chord (geometry)|chord]], [[versine]] (which appeared in the earliest tables<ref name="Boyer_1991"/>), [[haversine]], [[coversine]],<ref>{{harvtxt|Nielsen|1966|pp=xxiii–xxiv}}</ref> half-tangent (tangent of half an angle), and [[exsecant]]. [[List of trigonometric identities]] shows more relations between these functions.

: <math>\begin{align}
\operatorname{crd}\theta &= 2 \sin\tfrac12\theta, \\[5mu]
\operatorname{vers}\theta&=1-\cos \theta = 2\sin^2\tfrac12\theta, \\[5mu]
\operatorname{hav}\theta&=\tfrac{1}{2}\operatorname{vers}\theta = \sin^2\tfrac12\theta, \\[5mu]
\operatorname{covers}\theta&=1-\sin\theta = \operatorname{vers}\bigl(\tfrac12\pi - \theta\bigr), \\[5mu]
\operatorname{exsec}\theta &= \sec\theta - 1.
\end{align}</math>

{{anchor|Logarithmic sine|Logarithmic cosine|Logarithmic secant|Logarithmic cosecant|Logarithmic tangent|Logarithmic cotangent}}Historically, trigonometric functions were often combined with [[logarithm]]s in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.<ref name="Hammer_1897">{{cite book |title=Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Gebrauch bei Selbstunterricht und in Schulen, besonders als Vorbereitung auf Geodäsie und sphärische Astronomie |language=de |trans-title= |editor-first=Ernst Hermann Heinrich |editor-last=von Hammer |editor-link=:de:Ernst von Hammer |location=Stuttgart, Germany |publisher=[[J. B. Metzlerscher Verlag]] |date=1897 |edition=2 |url=https://quod.lib.umich.edu/u/umhistmath/ABN6964.0001.001/?view=toc |access-date=2024-02-06}}</ref><ref name="Heß_1916">{{cite book |title=Trigonometrie für Maschinenbauer und Elektrotechniker - Ein Lehr- und Aufgabenbuch für den Unterricht und zum Selbststudium |language=de |trans-title= |author-first=Adolf |author-last=Heß |location=Winterthur, Switzerland |publisher=Springer |edition=6 |date=1926 |orig-date=1916 |doi=10.1007/978-3-662-36585-4 |isbn=978-3-662-35755-2}}</ref><ref name="Lötzbeyer_1950">{{cite book |title=Erläuterungen und Beispiele für den Gebrauch der vierstelligen Tafeln zum praktischen Rechnen |language=de |trans-title= |chapter=§ 14. Erläuterungen u. Beispiele zu T. 13: lg sin X; lg cos X und T. 14: lg tg x; lg ctg X |trans-chapter= |author-first=Philipp |author-last=Lötzbeyer |date=1950 |edition=1 |isbn=978-3-11114038-4 |id=Archive ID 541650 |publication-place=Berlin, Germany |publisher=[[Walter de Gruyter & Co.]] |doi=10.1515/9783111507545 |url=https://www.degruyter.com/document/doi/10.1515/9783111507545/html |chapter-url=https://www.degruyter.com/document/doi/10.1515/9783111507545-015/html |access-date=2024-02-06}}</ref><ref name="Roegel_2016">{{cite book |title=A reconstruction of Peters's table of 7-place logarithms (volume 2, 1940) |date=2016-08-30 |editor-first=Denis |editor-last=Roegel |id=hal-01357842 |url=https://inria.hal.science/hal-01357842/document |location=Vandoeuvre-lès-Nancy, France |publisher=[[Université de Lorraine]] |access-date=2024-02-06 |url-status=live |archive-url=https://web.archive.org/web/20240206211422/https://inria.hal.science/hal-01357842/document |archive-date=2024-02-06}}</ref>