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History of trigonometryTrigonometry is a field of mathematics first compiled by 2nd century BCE by the Greek mathematician?Hipparchus. The?history oftrigonometry?and of?trigonometric functions?follows the general lines of the?history of mathematics.Early study of triangles can be traced to the 2nd millennium BC, in?Egyptian mathematics?( HYPERLINK "" \o "Rhind Mathematical Papyrus" Rhind Mathematical Papyrus) andBabylonian mathematics. Systematic study of trigonometric functions began in?Hellenistic mathematics, reaching India as part ofHellenistic astronomy. In?Indian astronomy, the study of trigonometric functions flowered in the?Gupta period, especially due toAryabhata?(6th century). During the Middle Ages, the study of trigonometry continued in?Islamic mathematics, whence it was adopted as a separate 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) and reaching its modern form with?Leonhard Euler?(1748).Contents??[ HYPERLINK "" hide]?1?Etymology2?Development2.1?Early trigonometry2.2?Greek mathematics2.3?Indian mathematics2.4?Islamic mathematics2.5?Chinese mathematics2.6?European mathematics3?See also4?Citations and footnotes5?References[edit]EtymologyThe term "trigonometry" derives from the?Greek?"τριγωνομετρ?α" ("trigonometria"), meaning "triangle measuring", from "τρ?γωνο" (triangle) + "μετρε?ν" (to measure).Our modern word "sine", is derived from the?Latin?word?sinus, which means "bay", "bosom" or "fold", translating Arabic?jayb. The Arabic term is in origin a corruption of?Sanskrit?jīvā?"chord". Sanskrit?jīvā?in learned usage was a synonym of? HYPERLINK "" \o "Jyā" jyā?"chord", originally the term for "bow-string". Sanskrit?jīvā?was loaned into Arabic as?jiba.[1][2][clarification needed]?This term was then transformed[2]?into the genuine Arabic word?jayb, meaning "bosom, fold, bay", either by the Arabs or by a mistake[1]?of the European translators such as?Robert of Chester(perhaps because the words were written without vowels[1]), who translated?jayb?into Latin as?sinus.[3]?Particularly? HYPERLINK "" \o "Fibonacci" Fibonacci's?sinus rectus arcus?proved influential in establishing the term?sinus.[4]The words "minute" and "second" are derived from the Latin phrases?partes minutae primae?and?partes minutae secundae.[5]?These roughly translate to "first small parts" and "second small parts".[edit]Development[edit]Early trigonometryThe ancient?Egyptians?and?Babylonians?had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".[6]The?Babylonian astronomers?kept detailed records on the rising and setting of?stars, the motion of the?planets, and the solar and lunareclipses, all of which required familiarity with?angular?distances measured on the?celestial sphere.[2]?Based on one interpretation of thePlimpton 322?cuneiform?tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants.[7]?There is, however, much debate as to whether it is a table of?Pythagorean triples, a solution of quadratic equations, or a?trigonometric table.The Egyptians, on the other hand, used a primitive form of trigonometry for building?pyramids?in the 2nd millennium BC.[2]?The? HYPERLINK "" \o "Rhind Mathematical Papyrus" Rhind Mathematical Papyrus, written by the Egyptian scribe? HYPERLINK "" \o "Ahmes" Ahmes?(c. 1680–1620 BC), contains the following problem related to trigonometry: HYPERLINK "" \l "cite_note-Maor-20-1" [2]"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its? HYPERLINK "" \o "Seked" 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.[2][edit]Greek mathematicsThe chord of an angle subtends the arc of the angle.Ancient?Greek and Hellenistic mathematicians?made use of the?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 the bisected angle, that is,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 toHellenistic?mathematicians, but in their equivalent chord form.[8]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.[6]?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.[6]?To compensate for the lack of a?table of chords, mathematicians of?Aristarchus'?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 as?Aristarchus' inequality.[9]The first trigonometric table was apparently compiled by?Hipparchus?of?Nicaea?(180 – 125 BC), who is now consequently known as "the father of trigonometry."[10]?Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.[4][10]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?composed?On the Sizes and Distances of the Sun and Moon?(ca. 260 BC), since he measured an angle in terms of a fraction of a quadrant.[9]?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? HYPERLINK "" \o "Hypsicles" Hypsicles?who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.[11]?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.[5]?It is due to the Babylonian? HYPERLINK "" \o "Sexagesimal" sexagesimal?numeral system?that each degree is divided into sixty minutes and each minute is divided into sixty seconds.[5]Menelaus' theoremMenelaus of Alexandria?(ca. 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.[8]?He establishes 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.[8]?Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.[8]?Book II of?Sphaerica?applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".[8]?He further gave his famous "rule of six quantities".[12]Later,?Claudius Ptolemy?(ca. 90 – ca. 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?n?in the corresponding arc of the circle, for?n?ranging from 1/2 to 180 by increments of?1/2.[13]?The thirteen books of the?Almagest?are the most influential and significant trigonometric work of all antiquity.[14]?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.[14]?Ptolemy further derived the equivalent of the half-angle formula[14]Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.[14]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.[15][edit]Indian mathematicsSee also:?Indian Mathematics?and?Indian AstronomyStatue of? HYPERLINK "" \o "Aryabhata" Aryabhata?on the grounds of?IUCAA,? HYPERLINK "" \o "Pune" Pune.The next significant developments of trigonometry were in?India. Influential works from the 4th–5th century, known as the? HYPERLINK "" \o "Siddhanta" Siddhantas?(of which there were five, the most complete survivor of which is the?Surya Siddhanta[16]) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine,? HYPERLINK "" \o "Versine" versine, and inverse sine.[17]?Soon afterwards, another?Indian mathematician?and?astronomer,? HYPERLINK "" \o "Aryabhata" Aryabhata(476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the? HYPERLINK "" \o "Aryabhatiya" Aryabhatiya.[18]?The?Siddhantas?and the?Aryabhatiya?contain the earliest surviving tables of sine values and? HYPERLINK "" \o "Versine" versine?(1???cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.[19]?They used the words? HYPERLINK "" \o "Jya" jya?for sine,? HYPERLINK "" \o "Kojya" kojya?for cosine,? HYPERLINK "" \o "Utkrama-jya" utkrama-jya?for versine, and?otkram jya?for inverse sine. The words? HYPERLINK "" \o "Jya" jya?andkojya?eventually became?sine?and?cosine?respectively after a mistranslation described above.In the 7th century,? HYPERLINK "" \o "Bhaskara I" Bhaskara I?produced a 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?1.9%:Later in the 7th century,? HYPERLINK "" \o "Brahmagupta" Brahmagupta?redeveloped the formula(also derived earlier, as mentioned above) and the? HYPERLINK "" \l "Interpolation_formula" \o "Brahmagupta" Brahmagupta interpolation formula?for computing sine values.[20]Another later Indian author on trigonometry was? HYPERLINK "" \o "Bhaskara II" Bhaskara II?in the 12th century.? HYPERLINK "" \o "Bhaskara II" Bhaskara II?developed?spherical trigonometry, and discovered many trigonometric results.Bhaskara II?was the first to discover??and??trigonometric results like:Madhava?(c. 1400) made early strides in the?analysis?of trigonometric functions and their?infinite series?expansions. He developed the concepts of the?power series?and?Taylor series, and produced the?power series?expansions of sine, cosine, tangent, and arctangent.[21][22]?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?π?and the?θ,?radius,?diameter, and?circumference?of a circle in terms of trigonometric functions. His works were expanded by his followers at the?Kerala School?up to the 16th century.[21][22]No.SeriesNameWestern discoverers of the seriesand approximate dates of discovery HYPERLINK "" \l "cite_note-22" [23]??1sin?x??=??x???x3?/ 3! +?x5?/ 5! ??x7?/ 7! + ... ????? HYPERLINK "" \l "Madhava.27s_sine_series" \o "Madhava series" Madhava's sine series????Isaac Newton (1670) and Wilhelm Leibniz (1676)????2??cos?x??= 1 ??x2?/ 2! +?x4?/ 4! ??x6?/ 6! + ...????? HYPERLINK "" \l "Madhava.27s_cosine_series" \o "Madhava series" Madhava's cosine series????Isaac Newton (1670) and Wilhelm Leibniz (1676)????3??tan?1x??=??x???x3?/ 3 +?x5?/ 5 ??x7?/ 7 + ...????? HYPERLINK "" \l "Madhava.27s_arctangent_series" \o "Madhava series" Madhava's arctangent series????James Gregory (1671) and Wilhelm Leibniz (1676) ??The Indian text the? HYPERLINK "" \o "Yuktibhā?ā" Yuktibhā?ā?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? HYPERLINK "" \o "Yuktibhā?ā" Yuktibhā?ā?also contains rules for finding the sines and the cosines of the sum and difference of two angles.[edit]Islamic mathematicsPage from?The Compendious Book on Calculation by Completion and Balancingby?Muhammad ibn Mūsā al-Khwārizmī?(c.?AD 820)The Indian works were later translated and expanded in the?medieval Islamic world?by?Muslim mathematicians?of mostly?Persian?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 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?or plane?triangle, its sides and?angles."[24]In addition to Indian works, Hellenistic methods dealing with spherical triangles were also known, particularly the method of?Menelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems.[8][25]?However, E. S. Kennedy points out that while it was possible in pre-lslamic 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.[26]?In order to observe holy days on the?Islamic calendar?in which timings were determined by?phases of the moon, astronomers initially used Menalaus' method to calculate the place of the?moon?and?stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting?right 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 the?sun's?altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval?Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.[27]In the early 9th century,?Muhammad ibn Mūsā al-Khwārizmī?produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in?spherical trigonometry. In 830,? HYPERLINK "" \o "Habash al-Hasib al-Marwazi" Habash al-Hasib al-Marwazi?produced the first table of cotangents.[28][29]Muhammad ibn Jābir al-Harrānī al-Battānī?(Albatenius) (853-929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[29]By the 10th century, in the work of? HYPERLINK "" \o "Abū al-Wafā' al-Būzjānī" Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six?trigonometric functions.[30]?Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.[30]?He also developed the following trigonometric formula: HYPERLINK "" \l "cite_note-musa-30" [31]?(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?minutes, and the result is half the sine of the double".[31]?Abū al-Wafā also established the angle addition and difference identities presented with complete proofs: HYPERLINK "" \l "cite_note-musa-30" [31]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".[31]He also discovered the?law of sines?for spherical trigonometry: HYPERLINK "" \l "cite_note-Sesiano-27" [28]Also in the late 10th and early 11th centuries, the Egyptian astronomer? HYPERLINK "" \o "Ibn Yunus" Ibn Yunus?performed many careful trigonometric calculations and demonstrated the following?trigonometric identity[32]: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" in its modern form.[33]?It "contains formulae for?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?ratios?as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influencedRegiomontanus.[33]The method of?triangulation?was first developed by Muslim mathematicians, who applied it to practical uses such as?surveying[34]?and?Islamic geography, as described by?Abu Rayhan Biruni?in the early 11th century. Biruni himself introduced triangulation techniques to?measure the size of the Earth?and the distances between various places.[35]?In the late 11th century,?Omar Khayyám?(1048–1131) solved?cubic equations?using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century,? HYPERLINK "" \o "Nasīr al-Dīn al-Tūsī" Nasīr al-Dīn al-Tūsī?was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.[29]?He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his?On the Sector Figure, 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.[36]In the 15th century,? HYPERLINK "" \o "Jamshīd al-Kāshī" Jamshīd al-Kāshī?provided the first explicit statement of the?law of cosines?in a form suitable for?triangulation.[citation needed]?In?France, the law of cosines is still referred to as the?theorem of Al-Kashi. He also gave trigonometric tables of values of the sine function to four? HYPERLINK "" \o "Sexagesimal" sexagesimal?digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of?1°.[citation needed]?Ulugh Beg?also gives accurate tables of sines and tangents correct to 8 decimal places around the same time.[citation needed][edit]Chinese mathematicsGuo Shoujing?(1231–1316)In?China,? HYPERLINK "" \o "Aryabhata" Aryabhata's table of sines were translated into the?Chinese mathematical?book of the? HYPERLINK "" \o "Treatise on Astrology of the Kaiyuan Era" Kaiyuan Zhanjing, compiled in 718 AD during theTang Dynasty.[37]?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.[38]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.[37]?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.[37]?The?polymath?Chinese scientist, mathematician and official? HYPERLINK "" \o "Shen Kuo" Shen Kuo?(1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[37]?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, sagita?v, and length?c?of the chord subtending the arc, the length of which he approximated as HYPERLINK "" \l "cite_note-katz_308-38" [39]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? HYPERLINK "" \o "Guo Shoujing" Guo Shoujing?(1231–1316).[40]?As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used?spherical trigonometry?in his calculations to improve the?calendar system?and?Chinese astronomy.[37][41]?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 two?meridian arcs, 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).[42]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? HYPERLINK "" \o "Xu Guangqi" Xu Guangqi?(1562–1633) and the Italian Jesuit? HYPERLINK "" \o "Matteo Ricci" Matteo Ricci?(1552–1610).[43][edit]European mathematicsIsaac Newton?in a 1702 portrait byGodfrey Kneller.In 1342, Levi ben Gershon, known as? HYPERLINK "" \o "Gersonides" Gersonides, wrote?On Sines, Chords and Arcs, in particular proving the?sine law?for plane triangles and giving five-figure?sine tables.[44]A simplified trigonometric table, the " HYPERLINK "" \o "Toleta de marteloio" toleta de marteloio", was used by sailors in the?Mediterranean Sea?during the 14th-15th C. to calculate?navigation?courses. It is described by?Ramon Llull?of?Majorca?in 1295, and laid out in the 1436 atlas of?Venetian?captain?Andrea Bianco.Regiomontanus?was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,[45]?in his?De triangulis omnimodus?written in 1464, as well as his later?Tabulae directionum?which included the tangent function, unnamed.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? HYPERLINK "" \o "Valentinus Otho" Valentin Otho?in 1596.In the 17th century,?Isaac Newton?and?James Stirling?developed the general Newton–Stirling interpolation formula for trigonometric functions.In the 18th century,?Leonhard Euler's?Introductio in analysin infinitorum?(1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, defining them as infinite series and presenting "Euler's formula"?eix?=?cos?x?+?i?sin?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).[46]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?in the 17th century and?Colin Maclaurin?in the 18th century were also very influential in the development of trigonometric series. ................
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