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{{Trigonometry}}
Early study of triangles can be traced to the [[2nd millennium BC]], in [[Egyptian mathematics]] ([[Rhind Mathematical Papyrus]]) and [[Babylonian mathematics]]. Trigonometry was also prevalent in [[Kingdom of Kush|Kushite]] mathematics.<ref>{{cite book|author=Otto Neugebauer |title=A history of ancient mathematical astronomy. 1 |url=https://books.google.com/books?id=vO5FCVIxz2YC&pg=PA744 |year=1975 |publisher=Springer-Verlag |isbn=978-3-540-06995-9 |page=744}}</ref>
Systematic study of [[trigonometric]] functions began in [[Hellenistic mathematics]], reaching [[India]] as part of [[Hellenistic astronomy]].{{sfn|Katz|1998|p=212}} In [[Indian astronomy]], the study of trigonometric functions flourished in the [[Gupta period]], especially due to [[Aryabhata]] (sixth century AD), who discovered the [[sine function|sine function, cosine function, and versine function.]] 

When during the [[Middle Ages]], the study of trigonometry continued in [[Islamic mathematics]], by mathematicians such as [[Al-Khwarizmi]] and [[Abu al-Wafa]]. It became an independent discipline in the [[Islamic world]], where all six [[trigonometric functions]] were known. [[Latin translations of the 12th century|Translations of Arabic and Greek texts]] led to trigonometry being adopted as a subject in the [[Latin West]] beginning in the [[Renaissance]] with [[Regiomontanus]]. 

The development of modern trigonometry shifted during the western [[Age of Enlightenment]], beginning with 17th-century mathematics ([[Isaac Newton]] and [[James Stirling (mathematician)|James Stirling]]) and reaching its modern form with [[Leonhard Euler]] (1748).

== Etymology ==
The term "trigonometry" was derived from [[Ancient Greek|Greek]] [[wikt:τρίγωνον|τρίγωνον]] ''trigōnon'', "triangle" and [[wikt:μέτρον|μέτρον]] ''metron'', "measure".<ref>{{cite dictionary|url=http://www.etymonline.com/index.php?term=trigonometry|title=trigonometry|dictionary=Online Etymology Dictionary}}</ref>

The modern words "sine" and "cosine" are derived from the [[Latin]] word {{lang|la|sinus}} via mistranslation from [[Arabic language|Arabic]] (see {{slink|Sine and cosine#Etymology}}). Particularly [[Fibonacci]]'s ''[[sinus rectus arcus]]'' proved influential in establishing the term.<ref name="oconnor1996">{{cite web |last1=O'Connor |first1=J.J. |last2=Robertson |first2=E.F. |title=Trigonometric functions |work=[[MacTutor History of Mathematics Archive]] |year=1996 |url=http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html |archive-url=https://web.archive.org/web/20070604020939/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html |url-status=dead |archive-date=2007-06-04 }}</ref>

The word ''tangent'' comes from Latin {{lang|la|tangens}} meaning "touching", since the line ''touches'' the circle of unit radius, whereas ''secant'' stems from Latin {{lang|la|secans}} "cutting" since the line ''cuts'' the circle (see the figure at [[List_of_trigonometric_identities#Pythagorean_identities|Pythagorean identities]]).<ref>Oxford English Dictionary</ref>

The prefix "[[co (function prefix)|co]]-" (in "cosine", "cotangent", "cosecant") is found in [[Edmund Gunter]]'s ''Canon triangulorum'' (1620), which defines the ''cosinus'' as an abbreviation for the ''sinus complementi'' (sine of the [[complementary angle]]) and proceeds to define the ''cotangens'' similarly.<ref name="Gunter_1620">{{cite book |author-first=Edmund |author-last=Gunter |author-link=Edmund Gunter |title=Canon triangulorum |date=1620}}</ref><ref name="Roegel_2010">{{cite web |title=A reconstruction of Gunter's Canon triangulorum (1620) |editor-first=Denis |editor-last=Roegel |type=Research report |publisher=HAL |date=6 December 2010 |id=inria-00543938 |url=https://hal.inria.fr/inria-00543938/document |access-date=28 July 2017 |url-status=live |archive-url=https://web.archive.org/web/20170728192238/https://hal.inria.fr/inria-00543938/document |archive-date=28 July 2017}}</ref>

The words "minute" and "second" are derived from the Latin phrases ''[[partes minutae primae]]'' and ''[[partes minutae secundae]]''.<ref name="Boyer The 360-circle">{{harvnb|Boyer|1991|loc=Greek Trigonometry and Mensuration |pp=[https://archive.org/details/historyofmathema00boye/page/166 166]–167}}: "It should be recalled that form the days of Hipparchus until modern times there were no such things as trigonometric ''ratios''. The Greeks, and after them the Hindus and the Arabs, used trigonometric ''lines''. These at first took the form, as we have seen, of chords in a circle, and it became incumbent upon Ptolemy to associate numerical values (or approximations) with the chords. [...] It is not unlikely that the 260-degree measure was carried over from astronomy, where the zodiac had been divided into twelve "signs" or 36 "decans". A cycle of the seasons of roughly 360 days could readily be made to correspond to the system of zodiacal signs and decans by subdividing each sign into thirty parts and each decan into ten parts. Our common system of angle measure may stem from this correspondence. Moreover since the Babylonian position system for fractions was so obviously superior to the Egyptians unit fractions and the Greek common fractions, it was natural for Ptolemy to subdivide his degrees into sixty ''[[partes minutae primae]]'', each of these latter into sixty ''[[partes minutae secundae]]'', and so on. It is from the Latin phrases that translators used in this connection that our words "minute" and "second" have been derived. It undoubtedly was the sexagesimal system that led Ptolemy to subdivide the diameter of his trigonometric circle into 120 parts; each of these he further subdivided into sixty minutes and each minute of length sixty seconds."</ref> These roughly translate to "first small parts" and "second small parts".

== Ancient ==

=== Ancient Near East ===
The ancient [[Egyptian mathematics|Egyptians]] and [[Babylonian mathematics|Babylonians]] had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.<ref name="Boyer Early Trigonometry">{{harvnb|Boyer|1991|loc=Greek Trigonometry and Mensuration |pp=[https://archive.org/details/historyofmathema00boye/page/158 158]–159}}: "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry", or the measure of three sided polygons (trilaterals), than "trigonometry", the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the ''Elements'', for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles."</ref>

The [[Babylonian astronomer]]s kept detailed records on the rising and setting of [[star]]s, the motion of the [[planet]]s, and the solar and lunar [[eclipse]]s, all of which required familiarity with [[Angle|angular]] distances measured on the [[celestial sphere]].<ref name="Maor-20">{{Cite book |title=Trigonometric Delights |url=https://archive.org/details/trigonometricdel00maor_444 |url-access=limited |first=Eli |last=Maor |year=1998 |publisher=[[Princeton University Press]] |isbn=978-0-691-09541-7 |page=[https://archive.org/details/trigonometricdel00maor_444/page/n27 20]}}</ref> Based on one interpretation of the [[Plimpton 322]] [[cuneiform script|cuneiform]] tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants but does not work in this context as without using circles and angles in the situation modern trigonometric notations will not apply.{{sfn|Joseph|2000|pp=383–384}} There is, however, much debate as to whether it is a table of [[Pythagorean triple]]s, a solution of quadratic equations, or a [[Generating trigonometric tables|trigonometric table]].<ref>{{Cite journal |last=Mansfield |first=Daniel F. |last2=Wildberger |first2=N.J. |date=November 2017 |title=Plimpton 322 is Babylonian exact sexagesimal trigonometry |url=https://linkinghub.elsevier.com/retrieve/pii/S0315086017300691 |journal=Historia Mathematica |language=en |volume=44 |issue=4 |pages=395–419 |doi=10.1016/j.hm.2017.08.001}}</ref>

The Egyptians, on the other hand, used a primitive form of trigonometry for building [[pyramid]]s in the 2nd millennium BC.<ref name="Maor-20" /> The [[Rhind Mathematical Papyrus]], written by the Egyptian scribe [[Ahmes]] (c. 1680–1620 BC), contains the following problem related to trigonometry:<ref name="Maor-20" />

{{blockquote|"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its ''[[seked]]''?"}}
Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the ''seked'' is the cotangent of the angle to the base of the pyramid and its face.<ref name="Maor-20" />

=== Classical antiquity ===
[[File:Trigonometric Chord.svg|thumb|The chord of an angle subtends the arc of the angle.]]
Ancient [[Greek mathematics|Greek and Hellenistic mathematicians]] made use of the [[chord (geometry)|chord]]. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of one half the bisected angle, that is,{{sfn|Katz|1998|p=143}}

: <math>\mathrm{chord}\ \theta = 2 r\sin \frac{\theta}{2}, </math>

and consequently the sine function is also known as the ''half-chord''. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to [[Hellenistic civilization|Hellenistic]] mathematicians, but in their equivalent chord form.<ref>As these historical calculations did not make use of a unit circle, the length of the radius was needed in the formula. Contrast this with the modern use of the ''crd function'' that assumes a unit circle in its definition.</ref><ref name="Boyer Menelaus of Alexandria" />

Although there is no trigonometry in the works of [[Euclid]] and [[Archimedes]], in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas.<ref name="Boyer Early Trigonometry" /> For instance, propositions twelve and thirteen of book two of the ''Elements'' are the [[laws of cosines]] for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the [[law of sines]]. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles.<ref name="Boyer Early Trigonometry" /> To compensate for the lack of a [[table of chords]], mathematicians of [[Aristarchus of Samos|Aristarchus]]' time would sometimes use the statement that, in modern notation, sin&nbsp;''α''/sin&nbsp;''β''&nbsp;<&nbsp;''α''/''β''&nbsp;<&nbsp;tan&nbsp;''α''/tan&nbsp;''β'' whenever 0°&nbsp;<&nbsp;β&nbsp;<&nbsp;α&nbsp;<&nbsp;90°, now known as [[Aristarchus's inequality]].<ref name="Boyer Aristarchus of Samos" />

The first trigonometric table was apparently compiled by [[Hipparchus]] of [[Nicaea]] (180 – 125 BC), who is now consequently known as "the father of trigonometry."<ref name="Boyer Hipparchus of Nicaea quote">{{harvnb|Boyer|1991|loc=Greek Trigonometry and Mensuration|p=[https://archive.org/details/historyofmathema00boye/page/162 162]}}: "For some two and a half centuries, from Hippocrates to Eratosthenes, Greek mathematicians had studied relationships between lines and circles and had applied these in a variety of astronomical problems, but no systematic trigonometry had resulted. Then, presumably during the second half of the 2nd century BC, the first trigonometric table apparently was compiled by the astronomer Hipparchus of Nicaea (ca. 180–ca. 125 BC), who thus earned the right to be known as "the father of trigonometry". Aristarchus had known that in a given circle the ratio of arc to chord decreases as the arc decreases from 180° to 0°, tending toward a limit of 1. However, it appears that not until Hipparchus undertook the task had anyone tabulated corresponding values of arc and chord for a whole series of angles."</ref> Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.<ref name="oconnor1996" /><ref name="Boyer Hipparchus of Nicaea quote" />

Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after [[Aristarchus of Samos]] composed ''[[On the Sizes and Distances of the Sun and Moon]]'' (c. 260 BC), since he measured an angle in terms of a fraction of a quadrant.<ref name="Boyer Aristarchus of Samos">{{harvnb|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/159 159] |loc=Greek Trigonometry and Mensuration}}: "Instead we have an treatise, perhaps composed earlier (ca. 260 BC), ''On the Sizes and Distances of the Sun and Moon'', which assumes a geocentric universe. In this work Aristarchus made the observation that when the moon is just half-full, the angle between the lines of sight to the sun and the moon is less than a right angle by one thirtieth of a quadrant. (The systematic introduction of the 360° circle came a little later. In trigonometric language of today this would mean that the ratio of the distance of the moon to that of the sun (the ration ME to SE in Fig. 10.1) is sin(3°). Trigonometric tables not having been developed yet, Aristarchus fell back upon a well-known geometric theorem of the time which now would be expressed in the [[inequality (mathematics)|inequalities]] sin&nbsp;α/&nbsp;sin&nbsp;β&nbsp;<&nbsp;α/β&nbsp;<&nbsp;tan&nbsp;α/&nbsp;tan&nbsp;β, for 0°&nbsp;<&nbsp;β&nbsp;<&nbsp;α&nbsp;<&nbsp;90°.)"</ref> It seems that the systematic use of the 360° circle is largely due to Hipparchus and his [[table of chords]].  Hipparchus may have taken the idea of this division from [[Hypsicles]] who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.<ref>{{harvnb|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/162 162] |loc=Greek Trigonometry and Mensuration}}: "It is not known just when the systematic use of the 360° circle came into mathematics, but it seems to be due largely to Hipparchus in connection with his table of chords. It is possible that he took over from Hypsicles, who earlier had divided the day into parts, a subdivision that may have been suggested by Babylonian astronomy."</ref> In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts.<ref name="Boyer The 360-circle" /> It is due to the Babylonian [[sexagesimal]] [[numeral system]] that each degree is divided into sixty minutes and each minute is divided into sixty seconds.<ref name="Boyer The 360-circle" />

[[File:Menelaus-thm.png|thumb|[[Menelaus' theorem]]]]
[[Menelaus of Alexandria]] (c. 100 AD) wrote in three books his ''Sphaerica''. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.<ref name="Boyer Menelaus of Alexandria" /> He established a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles.<ref name="Boyer Menelaus of Alexandria" /> Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.<ref name="Boyer Menelaus of Alexandria" /> Book II of ''Sphaerica'' applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".<ref name="Boyer Menelaus of Alexandria">{{harvnb|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/163 163] |loc=Greek Trigonometry and Mensuration}}: "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue – that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem ''A''&nbsp;+&nbsp;''B''&nbsp;+&nbsp;''C''&nbsp;>&nbsp;180° is established. The second book of the ''Sphaerica'' describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form – a geometry or trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB (multiplied by the radius of the circle)."</ref> He further gave his famous "rule of six quantities".{{sfn|Needham|1986|p=108}}

Later, [[Ptolemy|Claudius Ptolemy]] (c. 90 – c. 168 AD) expanded upon Hipparchus' ''Chords in a Circle'' in his ''[[Almagest]]'', or the ''Mathematical Syntaxis''.  The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry. [[Ptolemy's table of chords]] gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees&nbsp;''n'' in the corresponding arc of the circle, for ''n'' ranging from 1/2 to 180 by increments of&nbsp;1/2.<ref name=toomer>{{cite book |title=Ptolemy's Almagest |last=Toomer |first=Gerald J. |author-link=Gerald J. Toomer |publisher=Princeton University Press |year=1998 |isbn=978-0-691-00260-6}}</ref> The thirteen books of the ''Almagest'' are the most influential and significant trigonometric work of all antiquity.<ref name="Boyer Ptolemy" /> A theorem that was central to Ptolemy's calculation of chords was what is still known today as [[Ptolemy's theorem]], that the sum of the products of the opposite sides of a [[cyclic quadrilateral]] is equal to the product of the diagonals.  A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's ''Data''.  Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine.<ref name="Boyer Ptolemy" /> Ptolemy further derived the equivalent of the half-angle formula

: <math>\sin^2\left(\frac{x}{2}\right) = \frac{1 - \cos(x)}{2}.</math><ref name="Boyer Ptolemy" />

Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.<ref name="Boyer Ptolemy">{{harvnb|Boyer|1991|loc=Greek Trigonometry and Mensuration|pp=[https://archive.org/details/historyofmathema00boye/page/164 164]–166}}: "The theorem of Menelaus played a fundamental role in spherical trigonometry and astronomy, but by far the most influential and significant trigonometric work of all antiquity was composed by Ptolemy of Alexandria about half a century after Menelaus. [...] Of the life of the author we are as little informed as we are of that of the author of the Elements. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from AD. 127 to 151 and, therefore, assume that he was born at the end of the 1st century. Suidas, a writer who lived in the 10th century, reported that Ptolemy was alive under Marcus Aurelius (emperor from AD 161 to 180).<br />Ptolemy's ''Almagest'' is presumed to be heavily indebted for its methods to the ''Chords in a Circle'' of Hipparchus, but the extent of the indebtedness cannot be reliably assessed. It is clear that in astronomy Ptolemy made use of the catalog of star positions bequeathed by Hipparchus, but whether or not Ptolemy's trigonometric tables were derived in large part from his distinguished predecessor cannot be determined. [...] Central to the calculation of Ptolemy's chords was a geometric proposition still known as "Ptolemy's theorem": [...] that is, the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. [...] A special case of Ptolemy's theorem had appeared in Euclid's ''Data'' (Proposition 93): [...] Ptolemy's theorem, therefore, leads to the result sin(''α''&nbsp;−&nbsp;''β'') =&nbsp;sin&nbsp;''α''&nbsp;cos&nbsp;''β''&nbsp;−&nbsp;cos&nbsp;''α''&nbsp;sin&nbsp;''Β''. Similar reasoning leads to the formula [...] These four sum-and-difference formulas consequently are often known today as Ptolemy's formulas.<br />It was the formula for sine of the difference – or, more accurately, chord of the difference – that Ptolemy found especially useful in building up his tables. Another formula that served him effectively was the equivalent of our half-angle formula."</ref>

Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.{{sfn|Boyer|1991|pp=158–168}}

=== Indian mathematics ===
{{See also|l1=Indian Mathematics|Indian_mathematics|l2=Indian astronomy|Indian astronomy}}
Some of the early and very significant developments of trigonometry were in [[History of India|India]]. Influential works from the 4th–5th century AD, known as the [[Siddhanta]]s (of which there were five, the most important of which is the [[Surya Siddhanta]]{{sfn|Boyer|1991|p=208}}) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, [[versine]], and [[inverse sine]].{{sfn|Boyer|1991|p=209}}  Soon afterwards, another [[Indian mathematics|Indian mathematician]] and [[Indian astronomy|astronomer]], [[Aryabhata]] (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the ''[[Aryabhatiya]]''.{{sfn|Boyer|1991|p=210}} The ''Siddhantas'' and the ''Aryabhatiya''  contain the earliest surviving tables of sine values and [[versine]] (1&nbsp;−&nbsp;cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.{{sfn|Boyer|1991|p=215}} They used the words ''[[jya]]'' for sine, ''[[kojya]]'' for cosine, ''[[utkrama-jya]]'' for versine, and ''otkram jya'' for inverse sine. The words ''[[jya]]'' and ''[[kojya]]'' eventually became ''sine'' and ''cosine'' respectively after a mistranslation described above.

In the 7th century, [[Bhaskara I]] produced a [[Bhaskara I's sine approximation formula|formula]] for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(''x''), which had a relative error of less than&nbsp;1.9%:

: <math>\sin x \approx \frac{16x (\pi - x)}{5 \pi^2 - 4x (\pi - x)}, \qquad \left(0\leq x\leq\pi\right).</math>

Later in the 7th century, [[Brahmagupta]] redeveloped the formula

: <math>\ 1 - \sin^2(x) = \cos^2(x) = \sin^2\left (\frac{\pi}{2} - x\right )</math>

(also derived earlier, as mentioned above) and the [[Brahmagupta interpolation formula]] for computing sine values.{{sfn|Joseph|2000|pp=383–384}}

[[Madhava of Sangamagrama|Madhava]] (c. 1400) made early strides in the [[mathematical analysis|analysis]] of trigonometric functions and their [[series (mathematics)|infinite series]] expansions. He developed the concepts of the [[power series]] and [[Taylor series]], and produced the [[Madhava series|power series]] expansions of sine, cosine, tangent, and arctangent.<ref name="Robertson" /><ref name="Pearce" /> Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of [[Pi|π]] and the [[angle]], [[radius]], [[diameter]], and [[circumference]] of a circle in terms of trigonometric functions. His works were expanded by his followers at the [[Kerala school of astronomy and mathematics|Kerala School]] up to the 16th century.<ref name="Robertson">{{cite web |last1=O'Connor |first1=J.J. |last2=Robertson |first2=E.F. |title=Madhava of Sangamagramma |work=[[MacTutor History of Mathematics Archive]] |year=2000 |url=http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html}}</ref><ref name=Pearce>{{cite web |last=Pearce |first=Ian G. |title=Madhava of Sangamagramma |work=[[MacTutor History of Mathematics Archive]] |year=2002 |url=http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_3.html}}</ref>

{| class="wikitable" style="margin:1em auto;"
! No.
! Series
! Name
! Western discoverers of the series<br />and approximate dates of discovery<ref>{{Cite book |last=Charles Henry Edwards |title=The historical development of the calculus |publisher=Springer |year=1994 |edition=3 |series=Springer Study Edition Series |pages=205 |isbn=978-0-387-94313-8}}</ref>
|-
| &nbsp;&nbsp;1
| {{nowrap|1=sin ''x'' &nbsp;=&nbsp; ''x'' − ''x''<sup>3</sup> / 3! + ''x''<sup>5</sup> / 5! − ''x''<sup>7</sup> / 7! + ...}} &nbsp;&nbsp;
| &nbsp;&nbsp;[[Madhava series#Madhava's sine series|Madhava's sine series]]&nbsp;&nbsp;
|&nbsp;&nbsp;Isaac Newton (1670) and Wilhelm Leibniz (1676)&nbsp;&nbsp;
|-
| &nbsp;&nbsp;2&nbsp;&nbsp;
| {{nowrap|1=cos ''x'' &nbsp;= 1 − ''x''<sup>2</sup> / 2! + ''x''<sup>4</sup> / 4! − ''x''<sup>6</sup> / 6! + ...}}&nbsp;&nbsp;
| &nbsp;&nbsp;[[Madhava series#Madhava's cosine series|Madhava's cosine series]]&nbsp;&nbsp;
|&nbsp;&nbsp;Isaac Newton (1670) and Wilhelm Leibniz (1676)&nbsp;&nbsp;
|-
| &nbsp;&nbsp;3&nbsp;&nbsp;
| {{nowrap|1=tan<sup>−1</sup>''x'' &nbsp;=&nbsp; ''x'' − ''x''<sup>3</sup> / 3 + ''x''<sup>5</sup> / 5 − ''x''<sup>7</sup> / 7 + ...}}&nbsp;&nbsp;
| &nbsp;&nbsp;[[Madhava's arctangent series]]&nbsp;&nbsp;
| &nbsp;&nbsp;James Gregory (1671) and Wilhelm Leibniz (1676) &nbsp;&nbsp;
|}

The Indian text the [[Yuktibhāṣā]] contains proof for the expansion of the [[sine]] and [[cosine]] functions and the derivation and proof of the [[power series]] for [[inverse tangent]], discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles.

=== Chinese mathematics ===
[[File:Guo Shoujing-beijing.JPG|thumb|upright=0.8|[[Guo Shoujing]] (1231–1316)]]
In [[China]], [[Aryabhata]]'s table of sines were translated into the [[Chinese mathematics|Chinese mathematical]] book of the ''[[Treatise on Astrology of the Kaiyuan Era|Kaiyuan Zhanjing]]'', compiled in 718 AD during the [[Tang dynasty]].{{sfn|Needham|1986|p=109}} Although the Chinese excelled in other fields of mathematics such as solid geometry, [[binomial theorem]], and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Greek, Hellenistic, Indian and Islamic worlds.{{sfn|Needham|1986|pp=108–109}} Instead, the early Chinese used an empirical substitute known as ''chong cha'', while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.{{sfn|Needham|1986|p=109}} However, this embryonic state of trigonometry in China slowly began to change and advance during the [[Song dynasty]] (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations.{{sfn|Needham|1986|p=109}} The [[polymath]] Chinese scientist, mathematician and official [[Shen Kuo]] (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.{{sfn|Needham|1986|p=109}} Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc&nbsp;''s'' of a circle given the diameter&nbsp;''d'', [[Sagitta (geometry)|sagitta]]&nbsp;''v'', and length&nbsp;''c'' of the chord subtending the arc, the length of which he approximated as{{sfn|Katz|2007|p=308}}

: <math>s = c + \frac{2v^2}{d}.</math>

Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for [[spherical trigonometry]] developed in the 13th century by the mathematician and astronomer [[Guo Shoujing]] (1231–1316).{{sfn|Restivo|1992|p=32}} As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the [[Chinese calendar|calendar system]] and [[Chinese astronomy]].{{sfn|Needham|1986|p=109}}<ref>{{cite book |last=Gauchet |first=L. |year=1917 |title=Note Sur La Trigonométrie Sphérique de Kouo Cheou-King |page=151}}</ref> Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:

<blockquote>
Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two [[meridian arc]]s, one of which passed through the [[summer solstice]] point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).{{sfn|Needham|1986|pp=109–110}}
</blockquote>

Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of ''[[Euclid's Elements]]'' by Chinese official and astronomer [[Xu Guangqi]] (1562–1633) and the Italian Jesuit [[Matteo Ricci]] (1552–1610).{{sfn|Needham|1986|p=110}}

== Medieval ==
=== Islamic world ===
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|Page from ''[[The Compendious Book on Calculation by Completion and Balancing]]'' by [[Muhammad ibn Mūsā al-Khwārizmī]] (c.&nbsp;AD 820)]]
Previous works from India and Greece were later translated and expanded in the [[Islamic Golden Age|medieval Islamic world]] by [[Mathematics in medieval Islam|Muslim mathematicians]] of mostly [[List of Iranian scientists and scholars|Persian]] and [[List of Arab scientists and scholars|Arab descent]], who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete [[quadrilateral]], as was the case in Hellenistic mathematics due to the application of [[Menelaus' theorem]]. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the [[Spherical trigonometry|spherical]] or plane [[triangle]], its sides and [[angle]]s."<ref>{{Cite journal |first=E. S. |last=Kennedy |title=The History of Trigonometry |journal=31st Yearbook |publisher=National Council of Teachers of Mathematics |location= Washington DC |year=1969}} ([[cf.]] {{Cite book |first=Syed Nomanul |last=Haq |chapter=The Indian and Persian background |title=History of Islamic Philosophy |editor2=Oliver Leaman |editor1=Seyyed Hossein Nasr |editor1-link=Seyyed Hossein Nasr |year=1996 |publisher=[[Routledge]] |isbn=978-0-415-13159-9 |pages=52–70 [60–63]}})</ref>

Methods dealing with spherical triangles were also known, particularly the method of [[Menelaus of Alexandria]], who developed "Menelaus' theorem" to deal with spherical problems.<ref name="Boyer Menelaus of Alexandria" /><ref>{{MacTutor |id=Menelaus |title=Menelaus of Alexandria}} "Book 3 deals with spherical trigonometry and includes Menelaus's theorem".</ref> However, E. S. Kennedy points out that while it was possible in pre-Islamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice.<ref>{{Cite journal |first=E. S. |last=Kennedy |title=The History of Trigonometry |journal=31st Yearbook |publisher=National Council of Teachers of Mathematics |location= Washington DC |year=1969 |page=337}} ([[cf.]] {{Cite book |first=Syed Nomanul |last=Haq |chapter=The Indian and Persian background |title=History of Islamic Philosophy |editor2=Oliver Leaman |editor1=Seyyed Hossein Nasr |editor1-link=Seyyed Hossein Nasr |year=1996 |publisher=[[Routledge]] |isbn=978-0-415-13159-9 |pages=52–70 [68]}})</ref> In order to observe holy days on the [[Islamic calendar]] in which timings were determined by [[phases of the moon]], astronomers initially used Menelaus' method to calculate the place of the [[moon]] and [[star]]s, though this method proved to be clumsy and difficult. It involved setting up two intersecting [[right triangle]]s; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the [[sun]]'s [[altitude]], for instance, repeated applications of Menelaus' theorem were required. For medieval [[Astronomy in medieval Islam|Islamic astronomers]], there was an obvious challenge to find a simpler trigonometric method.<ref name=Gingerich>{{Cite journal|last=Gingerich |first=Owen |date=April 1986 |title=Islamic astronomy |journal=[[Scientific American]] |volume=254 |issue=10 |page=74 |doi=10.1038/scientificamerican0486-74 |url=http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |access-date=2008-05-18 |archive-url=https://web.archive.org/web/20110101211150/http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |archive-date=2011-01-01 |url-status=dead |bibcode=1986SciAm.254d..74G}}</ref>

In the early 9th century AD, [[Muhammad ibn Mūsā al-Khwārizmī]] produced accurate sine and cosine tables. He was also a pioneer in [[spherical trigonometry]]. In 830 AD, [[Habash al-Hasib al-Marwazi]] discovered the tangent and the cotangent and produced the first table of these [[Trigonometric functions|trigonometric functions]].<ref name="Sesiano" /><ref name="Britannica" /> [[Muhammad ibn Jābir al-Harrānī al-Battānī]] (Albatenius) (853–929 AD) discovered the secant and the cosecant, and produced the first table of cosecants for each degree from 1° to 90°.<ref>{{cite book |last=Maor |first=Eli |date=1998 |title=Trigonometric Delights |location=Princeton |publisher=Princeton University Press |page=38 |isbn=978-0-691-15820-4}}</ref>

By the 10th century AD, in the work of [[Abū al-Wafā' al-Būzjānī]], all six [[trigonometric functions]] were used.{{sfn|Boyer|1991|p=238}} Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.{{sfn|Boyer|1991|p=238}} He also developed the following trigonometric formula:<ref name="musa">{{cite journal |last=Moussa |first=Ali |title=Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations |journal=Arabic Sciences and Philosophy |year=2011 |volume=21 |issue=1 |pages=1–56 |publisher=[[Cambridge University Press]] |doi=10.1017/S095742391000007X|s2cid=171015175 }}</ref>

:<math>\ \sin(2x) = 2 \sin(x) \cos(x) </math> (a special case of Ptolemy's angle-addition formula; see above)

In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine [[minute of arc|minutes]], and the result is half the sine of the double".<ref name="musa" /> Abū al-Wafā also established the angle addition and difference identities presented with complete proofs:<ref name="musa" />

:<math>\sin(\alpha \pm \beta) = \sqrt{\sin^2 \alpha - (\sin \alpha \sin \beta)^2} \pm \sqrt{\sin^2 \beta- (\sin \alpha\sin \beta)^2}</math>

:<math>\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta </math>

For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other ''minutes''. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference".<ref name="musa" />

He also discovered the [[Law of sines#Curvature|law of sines]] for spherical trigonometry:<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>

:<math>\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.</math>

Also in the late 10th and early 11th centuries AD, the Egyptian astronomer [[Ibn Yunus]] performed many careful trigonometric calculations and demonstrated the following [[trigonometric identity]]:<ref>William Charles Brice, '[https://books.google.com/books?id=6DYVAAAAIAAJ&pg=PA413 An Historical atlas of Islam]', p.413</ref>

:<math>\cos a \cos b = \frac{\cos(a+b) + \cos(a-b)}{2}</math>

[[Al-Jayyani]] (989–1079) of [[al-Andalus]] wrote ''The book of unknown arcs of a sphere'', which is considered "the first treatise on [[spherical trigonometry]]".<ref name="MacTutor Al-Jayyani">{{MacTutor|id=Al-Jayyani|title=Abu Abd Allah Muhammad ibn Muadh Al-Jayyani}}</ref> It "contains formulae for [[Special right triangles|right-handed triangles]], the general law of sines, and the solution of a [[spherical triangle]] by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of [[ratio]]s as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced [[Regiomontanus]].<ref name="MacTutor Al-Jayyani" />

The method of [[triangulation]] was first developed by Muslim mathematicians, who applied it to practical uses such as [[surveying]]<ref>[[Donald Routledge Hill]] (1996), "Engineering", in Roshdi Rashed, ''Encyclopedia of the History of Arabic Science'', Vol. 3, p. 751–795 [769].</ref> and [[Islamic geography]], as described by [[Abu Rayhan Biruni]] in the early 11th century. Biruni himself introduced triangulation techniques to [[Abu Rayhan Biruni#Geodesy and geography|measure the size of the Earth]] and the distances between various places.<ref>{{MacTutor|id=Al-Biruni|title=Abu Arrayhan Muhammad ibn Ahmad al-Biruni}}</ref> In the late 11th century, [[Omar Khayyám]] (1048–1131) solved [[cubic equation]]s using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century, [[Naṣīr al-Dīn al-Ṭūsī]] was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.<ref name="Britannica">{{cite encyclopedia |title=trigonometry |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |encyclopedia=[[Encyclopædia Britannica]] |access-date=2008-07-21}}</ref> He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his ''Book on the Complete Quadrilateral'', he stated the law of sines for plane and spherical triangles, discovered the [[law of tangents]] for spherical triangles, and provided proofs for both these laws.<ref>{{cite book |first=J. Lennart |last=Berggren |title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook |chapter=Mathematics in Medieval Islam |publisher=Princeton University Press |year=2007 |isbn=978-0-691-11485-9 |page=518}}</ref> [[Nasir al-Din al-Tusi]] has been described as the creator of trigonometry as a mathematical discipline in its own right.<ref>{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir.html|title=Al-Tusi_Nasir biography|website=www-history.mcs.st-andrews.ac.uk|access-date=2018-08-05|quote=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 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.}}</ref><ref>{{Cite book |chapter-url=https://www.cambridge.org/core/books/the-cambridge-history-of-science/islamic-mathematics/4BF4D143150C0013552902EE270AF9C2 |chapter-url-access=subscription |last=Berggren |first=J. L. |title=The Cambridge History of Science |chapter=Islamic Mathematics |publisher=Cambridge University Press |date=October 2013 |isbn=978-0-511-97400-7 |doi=10.1017/CHO9780511974007.004 | pages=62–83}}</ref><ref>{{Cite web|url=http://www.iranicaonline.org/articles/tusi-nasir-al-din-bio|title=ṬUSI, NAṢIR-AL-DIN i. Biography – Encyclopaedia Iranica|last=electricpulp.com|website=www.iranicaonline.org|language=en|access-date=2018-08-05|quote=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 formulas for the solution of spherical right-angled triangles.}}</ref>

The [[law of cosines]], in geometric form, can be found as propositions II.12–13 in [[Euclid's Elements|Euclid's ''Elements'']] (c. 300 BC),<ref>{{cite web|last=Euclid | title = Elements | editor = [[Thomas L. Heath]] | translator = Thomas L. Heath | url = http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0086%3Abook%3D2%3Atype%3DProp
|access-date=24 January 2023 |quote=In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.}}</ref> but was not used for the solution of triangles per se. Medieval Islamic mathematicians developed a method for finding the third side of an arbitrary triangle given two sides and the included angle based on the same concept but more similar to the modern formulation of the law of cosines. A sketch of the method can be found in Naṣīr al-Dīn al-Ṭūsī's ''Book on the Complete Quadrilateral'' (c. 1250),<ref>{{cite book |title=Traité du quadrilatère attribué a Nassiruddinel-Toussy |chapter=Ch. 3.2: Sur la manière de calculer les côtés et les angles d'un triangle les uns par les autres |page=69 |language=fr |author=Naṣīr al-Dīn al-Ṭūsī |translator-last=Caratheodory |translator-first=Alexandre Pacha |translator-link=Alexander Karatheodori Pasha |chapter-url=https://books.google.com/books?id=gVE7AQAAIAAJ&pg=PA69 |publisher=Typographie et Lithographie Osmanié |year=1891 |quote=On donne deux côtés et un angle. [...] Que si l'angle donné est compris entre les deux côtés donnés, comme l'angle A est compris entre les deux côtés AB AC, abaissez de B sur AC la perpendiculaire BE. Vous aurez ainsi le triangle rectangle [BEA] dont nous connaissons le côté AB et l'angle A; on en tirera BE, EA, et l'on retombera ainsi dans un des cas précédents; c. à. d. dans le cas où BE, CE sont connus; on connaîtra dès lors BC et l'angle C, comme nous l'avons expliqué |trans-quote=Given [...] the angle A is included between the two sides AB AC, drop from B to AC the perpendicular BE. You will thus have the right triangle [BEA] of which we know the side AB and the angle A; in that triangle compute BE, EA, and the problem is reduced to one of the preceding cases; that is, to the case where BE, CE are known; we will thus know BC and the angle C, as we have explained.}}</ref> and the same method is described in more detail in [[Jamshīd al-Kāshī]]'s ''Key of Arithmetic'' (1427).<ref>{{cite journal |last=Azarian |first=Mohammad K. |year=2000 |title=Meftab Al-Hesab: A Summary |journal=Missouri Journal of Mathematical Sciences |volume=12 |number=2 |pages=75–95 |doi=10.35834/2000/1202075 |doi-access=free|url=https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-12/issue-2/Meftab-Al-Hesab-A-Summary/10.35834/2000/1202075.pdf }} {{pb}} {{cite book |editor1-last=Aydin |editor1-first=Nuh |editor2-last=Hammoudi |editor2-first=Lakhdar |editor3-last=Bakbouk |editor3-first=Ghada |year=2020 |page=31 |title=Al-Kashi's Miftah al-Hisab, Volume II: Geometry |publisher=Birkhäuser |doi=10.1007/978-3-030-61330-3|isbn=978-3-030-61329-7 |quote='''Another case''' is when two sides and the angle between them are known and the rest are unknown. We multiply one of the sides by the sine of the [known] angle one time and by the sine of its complement the other time converted and we subtract the second result from the other side if the angle is acute and add it if the angle is obtuse. We then square the result and add to it the square of the first result. We take the square root of the sum to get the remaining side.... }}</ref> Al-Kāshī also computed the [[sine of 1°]] accurate to 8 [[sexagesimal]] digits, and constructed the most accurate trigonometric tables to date, accurate to four [[sexagesimal]] places (equivalent to 8 decimal places) for each 1° of arc.{{Citation needed|date=May 2011}} Al-Kāshī presumably worked on [[Ulugh Beg]]'s even more comprehensive trigonometric tables, with five-place (sexagesimal) entries for each minute of arc.{{Citation needed|date=May 2011}}

=== European renaissance ===
In 1342, Levi ben Gershon, known as [[Gersonides]], wrote ''On Sines, Chords and Arcs'', in particular proving the [[sine law]] for plane triangles and giving five-figure [[sine table]]s.<ref>{{cite journal | url=http://web.stonehill.edu/compsci/Shai_papers/MathofLevi.pdf | author=Charles G. Simonson | title=The Mathematics of Levi ben Gershon, the Ralbag | journal=Bekhol Derakhekha Daehu | publisher=Bar-Ilan University Press | volume=10 | pages=5–21 | date=Winter 2000 }}</ref>

A simplified trigonometric table, the "''[[toleta de marteloio]]''", was used by sailors in the [[Mediterranean Sea]] during the 14th-15th Centuries to calculate [[navigation]] courses.  It is described by [[Ramon Llull]] of [[Majorca]] in 1295, and laid out in the 1436 atlas of [[Venice|Venetian]] captain [[Andrea Bianco]].

[[Regiomontanus]] was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,{{sfn|Boyer|1991|p=274}} in his ''De triangulis omnimodis'' written in 1464, as well as his later ''Tabulae directionum'' which included the tangent function, unnamed.

== Modern ==
{{Expand section|date=March 2024}}

The ''Opus palatinum de triangulis'' of [[Georg Joachim Rheticus]], a student of [[Copernicus]], was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student [[Valentinus Otho|Valentin Otho]] in 1596.

In the 17th century, [[Isaac Newton]] and [[James Stirling (mathematician)|James Stirling]] developed the general Newton–Stirling interpolation formula for trigonometric functions.

In the 18th century, [[Leonhard Euler]]'s ''Introduction in analysin infinitorum'' (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their infinite series and presenting "[[Euler's formula]]"&nbsp;''e''<sup>''ix''</sup>&nbsp;=&nbsp;cos&nbsp;''x''&nbsp;+&nbsp;''i''&nbsp;sin&nbsp;''x''.  Euler used the near-modern abbreviations ''sin.'', ''cos.'', ''tang.'', ''cot.'', ''sec.'', and ''cosec.'' Prior to this, [[Roger Cotes]] had computed the derivative of sine in his ''Harmonia Mensurarum'' (1722).<ref>{{cite journal |doi=10.1016/0315-0860(87)90064-4 |title=The calculus of the trigonometric functions |journal=Historia Mathematica |volume=14|issue=4|pages=311–324|date=November 1987 |last1=Katz |first1=Victor J. |doi-access=free}}. The proof of Cotes is mentioned on p. 315.</ref>
Also in the 18th century, [[Brook Taylor]] defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of [[James Gregory (astronomer and mathematician)|James Gregory]] in the 17th century and [[Colin Maclaurin]] in the 18th century were also very influential in the development of trigonometric series.

== See also ==
{{Portal|History of science|Mathematics}}
* [[Greek mathematics]]
* [[History of mathematics]]
* [[Trigonometric functions]]
* [[Trigonometry]]
* [[Ptolemy's table of chords]]
* [[Aryabhata's sine table]]
* [[Rational trigonometry]]

== Citations and footnotes ==
{{Reflist}}

== References ==
*{{cite book |first=Carl Benjamin |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=2nd |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=978-0-471-54397-8 |url=https://archive.org/details/historyofmathema00boye }}
*{{cite book |first=George G. |last=Joseph |location=London |edition=2nd |title=The Crest of the Peacock: Non-European Roots of Mathematics |publisher=[[Penguin Books]] |year=2000 |isbn=978-0-691-00659-8 |url=https://archive.org/details/crestofpeacockno00jose }}
*{{cite book |last=Katz|first=Victor J.|title=A History of Mathematics / An Introduction |year=1998 |edition=2nd |publisher=Addison Wesley |isbn=978-0-321-01618-8 |url=https://archive.org/details/historyofmathema00katz}}
*{{cite book |last=Katz|first=Victor J.|year=2007|title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook|place= Princeton|publisher= Princeton University Press|isbn= 978-0-691-11485-9}}
*{{cite book |last=Needham |first=Joseph |year=1986 |title=Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth |location=Taipei |publisher=Caves Books, Ltd.}}
*{{cite book |last=Restivo |first=Sal |year=1992 |title=Mathematics in Society and History: Sociological Inquiries |location=Dordrecht |publisher=Kluwer Academic Publishers |isbn=1-4020-0039-1}}

== Further reading ==
{{wikiquote}}
* {{cite book |last=Braunmühl |first=Anton von |author-link=Anton von Braunmühl |year=1900–1903 |title=Vorlesungen über Geschichte der Trigonometrie |trans-title=Lectures on the History of Trigonometry |language=de |publisher=B. G. Teubner |url=https://archive.org/details/vorlesungenber00brauuoft/ }}
* {{cite book |last=Kennedy |first=Edward S. | authorlink= Edward Stewart Kennedy |year=1969 |chapter=The History of Trigonometry |title=Historical Topics for the Mathematics Classroom |publisher=[[National Council of Teachers of Mathematics]] |chapter-url=https://archive.org/details/historicaltopics0000unse_x0v6/page/333 |series=NCTM Yearbooks |volume=31 |pages=333–375 |chapter-url-access=limited }}
* {{cite book |last=Maor |first=Eli |author-link=Eli Maor |year= 1998 |title=Trigonometric Delights |publisher=Princeton University Press |doi=10.1515/9780691202204 |archive-url=https://web.archive.org/web/20030711040044/http://www.pupress.princeton.edu/books/maor/ |url-status=dead |url=http://www.pupress.princeton.edu/books/maor/ |archive-date=2003-07-11 |isbn=0691057540 }}
* {{cite book |last1=Ostermann |first1=Alexander |last2=Wanner |first2=Gerhard |year=2012 |title=Geometry by Its History |chapter=Trigonometry |series=Undergraduate Texts in Mathematics |publisher=Springer |doi=10.1007/978-3-642-29163-0 |pages=113–155 |isbn=978-3-642-29162-3 }}
* {{cite book |last=Van Brummelen |first=Glen |author-link=Glen Van Brummelen |year=2009 |title=The Mathematics of the Heavens and the Earth: The Early History of Trigonometry |publisher=Princeton University Press }}
* {{cite book |last=Van Brummelen |first=Glen |author-link=Glen Van Brummelen |year=2021 |title=The Doctrine of Triangles: A History of Modern Trigonometry |publisher=Princeton University Press }}


{{History of science}}

{{History of mathematics}}
{{Islamic mathematics}}

[[Category:History of mathematics|Trigonometry]]
[[Category:History of geometry|Trigonometry]]
[[Category:Trigonometry]]