Trigonometry
Outline History Usage Functions (sin,cos,tan,inverse) Generalized trigonometry
Reference
Identities Exact constants Tables Unit circle
Laws and theorems
Sines Cosines Tangents Cotangents Pythagorean theorem
Calculus
Trigonometric substitution Integrals (inverse functions) Derivatives Trigonometric series
Mathematicians
Hipparchus Ptolemy Brahmagupta al-Hasib al-Battani Regiomontanus Viète de Moivre Euler Fourier
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Early study of triangles can be traced to the2nd millennium BC, inEgyptian mathematics (Rhind Mathematical Papyrus) andBabylonian mathematics. Trigonometry was also prevalent inKushite mathematics.^[1]Systematic study oftrigonometric functions began inHellenistic mathematics, reachingIndia as part ofHellenistic astronomy.^[2] InIndian astronomy, the study of trigonometric functions flourished in theGupta period, especially due toAryabhata (sixth century AD), who discovered thesine function, cosine function, and versine function.

During theMiddle Ages, the study of trigonometry continued inIslamic mathematics, by mathematicians such asAl-Khwarizmi andAbu al-Wafa. The knowledge of trigonometric functions passed to Arabia from the Indian Subcontinent. It became an independent discipline in theIslamic world, where all sixtrigonometric functions were known.Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in theLatin West beginning in theRenaissance withRegiomontanus.

The development of modern trigonometry shifted during the westernAge of Enlightenment, beginning with 17th-century mathematics (Isaac Newton andJames Stirling) and reaching its modern form withLeonhard Euler (1748).

The term "trigonometry" was derived fromGreekτρίγωνονtrigōnon, "triangle" andμέτρονmetron, "measure".^[3]

The modern words "sine" and "cosine" are derived from theLatin wordsinus via mistranslation fromArabic (seeSine and cosine § Etymology). ParticularlyFibonacci'ssinus rectus arcus proved influential in establishing the term.^[4]

The wordtangent comes from Latintangens meaning "touching", since the linetouches the circle of unit radius, whereassecant stems from Latinsecans "cutting" since the linecuts the circle (see the figure atPythagorean identities).^[5]

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found inEdmund Gunter'sCanon triangulorum (1620), which defines thecosinus as an abbreviation for thesinus complementi (sine of thecomplementary angle) and proceeds to define thecotangens similarly.^[6][7]

The words "minute" and "second" are derived from the Latin phrasespartes minutae primae andpartes minutae secundae.^[8] These roughly translate to "first small parts" and "second small parts".

The ancientEgyptians andBabylonians 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.^[9]

TheBabylonian astronomers kept detailed records on the rising and setting ofstars, the motion of theplanets, and the solar and lunareclipses, all of which required familiarity withangular distances measured on thecelestial sphere.^[10] Based on one interpretation of thePlimpton 322cuneiform 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.^[11] There is, however, much debate as to whether it is a table ofPythagorean triples, a solution of quadratic equations, or atrigonometric table.^[12]

The Egyptians, on the other hand, used a primitive form of trigonometry for buildingpyramids in the 2nd millennium BC.^[10] TheRhind Mathematical Papyrus, written by the Egyptian scribeAhmes (c. 1680–1620 BC), contains the following problem related to trigonometry:^[10]

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is itsseked?"

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 theseked is the cotangent of the angle to the base of the pyramid and its face.^[10]

AncientGreek and Hellenistic mathematicians made use of thechord. 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,^[13]

${\displaystyle \mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}\theta =2r\sin \frac{\theta }{2},}$

and consequently the sine function is also known as thehalf-chord. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known toHellenistic mathematicians, but in their equivalent chord form.^[14][15]

Although there is no trigonometry in the works ofEuclid andArchimedes, 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.^[9] For instance, propositions twelve and thirteen of book two of theElements are thelaws of cosines for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of thelaw of sines. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles.^[9] To compensate for the lack of atable of chords, mathematicians ofAristarchus' time would sometimes use the statement that, in modern notation, sin α/sin β < α/β < tan α/tan β whenever 0° < β < α < 90°, now known asAristarchus's inequality.^[16]

The first trigonometric table was apparently compiled byHipparchus ofNicaea (180 – 125 BC), who is now consequently known as "the father of trigonometry."^[17] Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.^[4][17]

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 afterAristarchus of Samos composedOn 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.^[16] It seems that the systematic use of the 360° circle is largely due to Hipparchus and histable of chords. Hipparchus may have taken the idea of this division fromHypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.^[18] 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.^[8] It is due to the Babyloniansexagesimalnumeral system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.^[8]

Menelaus of Alexandria (c. 100 AD) wrote in three books hisSphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.^[15] 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.^[15] Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.^[15] Book II ofSphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".^[15] He further gave his famous "rule of six quantities".^[19]

Later,Claudius Ptolemy (c. 90 – c. 168 AD) expanded upon Hipparchus'Chords in a Circle in hisAlmagest, or theMathematical 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 n in the corresponding arc of the circle, forn ranging from 1/2 to 180 by increments of 1/2.^[20] The thirteen books of theAlmagest are the most influential and significant trigonometric work of all antiquity.^[21] A theorem that was central to Ptolemy's calculation of chords was what is still known today asPtolemy's theorem, that the sum of the products of the opposite sides of acyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid'sData. 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.^[21] Ptolemy further derived the equivalent of the half-angle formula

${\displaystyle {\sin }^2\left(\frac{x}{2}\right)=\frac{1-\cos (x)}{2}.}$^[21]

Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.^[21]

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.^[22]

Some of the early and very significant developments of trigonometry were inIndia. Influential works from the 4th–5th century AD, known as theSiddhantas (of which there were five, the most important of which is theSurya Siddhanta^[23]) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine,versine, andinverse sine.^[24] Soon afterwards, anotherIndian mathematician andastronomer,Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called theAryabhatiya.^[25] TheSiddhantas and theAryabhatiya contain the earliest surviving tables of sine values andversine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.^[26] They used the wordsjya for sine,kojya for cosine,utkrama-jya for versine, andotkram jya for inverse sine. The wordsjya andkojya eventually becamesine andcosine respectively after a mistranslation described above.

In the 7th century,Bhaskara I produced aformula 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 1.9%:

${\displaystyle \sin x\approx \frac{16x(\pi -x)}{5{\pi }^2-4x(\pi -x)},\left(0\le x\le \pi \right).}$

Later in the 7th century,Brahmagupta redeveloped the formula

${\displaystyle 1-{\sin }^2(x)={\cos }^2(x)={\sin }^2\left(\frac{\pi }{2}-x\right)}$

(also derived earlier, as mentioned above) and theBrahmagupta interpolation formula for computing sine values.^[11]

Madhava (c. 1400) made early strides in theanalysis of trigonometric functions and theirinfinite series expansions. He developed the concepts of thepower series andTaylor series, and produced thepower series expansions of sine, cosine, tangent, and arctangent.^[27][28] 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 ofπ and theangle,radius,diameter, andcircumference of a circle in terms of trigonometric functions. His works were expanded by his followers at theKerala School up to the 16th century.^[27][28]

No.	Series	Name	Western discoverers of the series and approximate dates of discovery[29]
1	$${\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\dots }$$	Madhava's sine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
2	${\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\dots }$	Madhava's cosine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
3	${\displaystyle \arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\dots }$	Madhava's arctangent series	James Gregory (1671) and Wilhelm Leibniz (1676)

The Indian text theYuktibhāṣā contains proof for the expansion of thesine andcosine functions and the derivation and proof of thepower series forinverse tangent, discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles.

InChina,Aryabhata's table of sines were translated into theChinese mathematical book of theKaiyuan Zhanjing, compiled in 718 AD during theTang dynasty.^[30] 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.^[31] Instead, the early Chinese used an empirical substitute known aschong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.^[30] However, this embryonic state of trigonometry in China slowly began to change and advance during theSong dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations.^[30] Thepolymath Chinese scientist, mathematician and officialShen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.^[30] Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc s of a circle given the diameter d,sagitta v, and length c of the chord subtending the arc, the length of which he approximated as^[32]

${\displaystyle s=c+\frac{2v^2}{d}.}$

Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis forspherical trigonometry developed in the 13th century by the mathematician and astronomerGuo Shoujing (1231–1316).^[33] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve thecalendar system andChinese astronomy.^[30][34] Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with twomeridian arcs, one of which passed through thesummer 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).^[35]

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 ofEuclid's Elements by Chinese official and astronomerXu Guangqi (1562–1633) and the Italian JesuitMatteo Ricci (1552–1610).^[36]

Previous works from India and Greece were later translated and expanded in themedieval Islamic world byMuslim mathematicians of mostlyPersian andArab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the completequadrilateral, as was the case in Hellenistic mathematics due to the application ofMenelaus' 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 thespherical or planetriangle, its sides andangles."^[37]

Methods dealing with spherical triangles were also known, particularly the method ofMenelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems.^[15][38] 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.^[39] In order to observe holy days on theIslamic calendar in which timings were determined byphases of the moon, astronomers initially used Menelaus' method to calculate the place of themoon andstars, though this method proved to be clumsy and difficult. It involved setting up two intersectingright triangles; 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 thesun'saltitude, for instance, repeated applications of Menelaus' theorem were required. For medievalIslamic astronomers, there was an obvious challenge to find a simpler trigonometric method.^[40]

In the early 9th century AD,Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables. He was also a pioneer inspherical trigonometry. In 830 AD,Habash al-Hasib al-Marwazi discovered the tangent and the cotangent and produced the first table of thesetrigonometric functions.^[41][42]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°.^[43]

By the 10th century AD, in the work ofAbū al-Wafā' al-Būzjānī, all sixtrigonometric functions were used.^[44] Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.^[44] He also developed the following trigonometric formula:^[45]

${\displaystyle \sin (2x)=2\sin (x)\cos (x)}$ (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 cosineminutes, and the result is half the sine of the double".^[45] Abū al-Wafā also established the angle addition and difference identities presented with complete proofs:^[45]

${\displaystyle \sin (\alpha \pm \beta )=\sqrt{{\sin }^2\alpha -(\sin \alpha \sin \beta {)}^2}\pm \sqrt{{\sin }^2\beta -(\sin \alpha \sin \beta {)}^2}}$

${\displaystyle \sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$

For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the otherminutes. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference".^[45]

He also discovered thelaw of sines for spherical trigonometry:^[41]

${\displaystyle \frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}.}$

Also in the late 10th and early 11th centuries AD, the Egyptian astronomerIbn Yunus performed many careful trigonometric calculations and demonstrated the followingtrigonometric identity:^[46]

${\displaystyle \cos a\cos b=\frac{\cos (a+b)+\cos (a-b)}{2}}$

Al-Jayyani (989–1079) ofal-Andalus wroteThe book of unknown arcs of a sphere, which is considered "the first treatise onspherical trigonometry".^[47] It "contains formulae forright-handed triangles, the general law of sines, and the solution of aspherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition ofratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influencedRegiomontanus.^[47]

The method oftriangulation was first developed by Muslim mathematicians, who applied it to practical uses such assurveying^[48] andIslamic geography, as described byAbu Rayhan Biruni in the early 11th century. Biruni himself introduced triangulation techniques tomeasure the size of the Earth and the distances between various places.^[49] In the late 11th century,Omar Khayyám (1048–1131) solvedcubic equations 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.^[42] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in hisBook on the Complete Quadrilateral, he stated the law of sines for plane and spherical triangles, discovered thelaw of tangents for spherical triangles, and provided proofs for both these laws.^[50]Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.^[51][52][53]

Thelaw of cosines, in geometric form, can be found as propositions II.12–13 inEuclid'sElements (c. 300 BC),^[54] 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'sBook on the Complete Quadrilateral (c. 1250),^[55] and the same method is described in more detail inJamshīd al-Kāshī'sKey of Arithmetic (1427).^[56] Al-Kāshī also computed thesine of 1° accurate to 8sexagesimal digits, and constructed the most accurate trigonometric tables to date, accurate to foursexagesimal places (equivalent to 8 decimal places) for each 1° of arc.^{[citation needed]} Al-Kāshī presumably worked onUlugh Beg's even more comprehensive trigonometric tables, with five-place (sexagesimal) entries for each minute of arc.^{[citation needed]}

In 1342, Levi ben Gershon, known asGersonides, wroteOn Sines, Chords and Arcs, in particular proving thesine law for plane triangles and giving five-figuresine tables.^[57]

A simplified trigonometric table, the "toleta de marteloio", was used by sailors in theMediterranean Sea during the 14th-15th Centuries to calculatenavigation courses. It is described byRamon Llull ofMajorca in 1295, and laid out in the 1436 atlas ofVenetian captainAndrea Bianco.

Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,^[58] in hisDe triangulis omnimodis written in 1464, as well as his laterTabulae directionum which included the tangent function, unnamed.

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TheOpus palatinum de triangulis ofGeorg Joachim Rheticus, a student ofCopernicus, 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' studentValentin Otho in 1596.

In the 17th century,Isaac Newton andJames Stirling developed the general Newton–Stirling interpolation formula for trigonometric functions.

In the 18th century,Leonhard Euler'sIntroduction 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" e^ix = cos x + i sin x. Euler used the near-modern abbreviationssin.,cos.,tang.,cot.,sec., andcosec. Prior to this,Roger Cotes had computed the derivative of sine in hisHarmonia Mensurarum (1722).^[59]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 ofJames Gregory in the 17th century andColin Maclaurin in the 18th century were also very influential in the development of trigonometric series.

• Greek mathematics
• History of mathematics
• Trigonometric functions
• Trigonometry
• Ptolemy's table of chords
• Aryabhata's sine table
• Rational trigonometry

• Boyer, Carl Benjamin (1991).A History of Mathematics (2nd ed.). John Wiley & Sons, Inc.ISBN 978-0-471-54397-8.
• Joseph, George G. (2000).The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.). London:Penguin Books.ISBN 978-0-691-00659-8.
• Katz, Victor J. (1998).A History of Mathematics / An Introduction (2nd ed.). Addison Wesley.ISBN 978-0-321-01618-8.
• Katz, Victor J. (2007).The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press.ISBN 978-0-691-11485-9.
• Needham, Joseph (1986).Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
• Restivo, Sal (1992).Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers.ISBN 1-4020-0039-1.

Wikiquote has quotations related toHistory of trigonometry.

• Braunmühl, Anton von (1900–1903).Vorlesungen über Geschichte der Trigonometrie [Lectures on the History of Trigonometry] (in German). B. G. Teubner.
• Kennedy, Edward S. (1969)."The History of Trigonometry".Historical Topics for the Mathematics Classroom. NCTM Yearbooks. Vol. 31.National Council of Teachers of Mathematics. pp. 333–375.
• Maor, Eli (1998).Trigonometric Delights. Princeton University Press.doi:10.1515/9780691202204.ISBN 0691057540. Archived fromthe original on 2003-07-11.
• Ostermann, Alexander; Wanner, Gerhard (2012). "Trigonometry".Geometry by Its History. Undergraduate Texts in Mathematics. Springer. pp. 113–155.doi:10.1007/978-3-642-29163-0.ISBN 978-3-642-29162-3.
• Van Brummelen, Glen (2009).The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. Princeton University Press.
• Van Brummelen, Glen (2021).The Doctrine of Triangles: A History of Modern Trigonometry. Princeton University Press.

• History of mathematics
• History of geometry
• Trigonometry

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