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The 'history of trigonometry and of trigonometric functions' may span about 4000 years.

Contents

Etymology

Development

Early trigonometry

Hellenistic world

Medieval India

Muslim world

Medieval China

Renaissance Europe

Trigonometric analysis

See also

Citations and footnotes

References

Etymology

Our modern word ''sine'' is derived from the Latin word ''sinus'', which means "bay" or "fold", from a mistranslation (via Arabic) of the Sanskrit word ''jiva'', alternatively called ''jya''.^[1] Aryabhata used the term ''ardha-jiva'' ("half-chord"), which was shortened to ''jiva'' and then transliterated by the Arabs as ''jiba'' (جب). European translators like Robert of Chester and Gherardo of Cremona in 12th-century Toledo confused ''jiba'' for ''jaib'' (جب), meaning "bay", probably because ''jiba'' (جب) and ''jaib'' (جب) are written the same in the Arabic script (this writing system, in one of its forms, does not provide the reader with complete information about the vowels). The words "minute" and "second" are derived from the Latin phrases ''partes minutae primar'' and ''partes minutae secundae''.

Development

Trigonometry is not the work of any one man or nation. Its history spans thousands of years and has touched every major civilization.
It should be noted that that from the time of Hipparchus' until modern times there was no such thing as a trigonometric ''ratio''. Instead, the Greeks and after them the Hindus and the Muslims used trigonometric ''lines''. These lines first took the form of chords and later half chords, or sines. These chord and sine lines would then be associated with numerical values, possibly approximations, and listed in trigonometric tables.^[2]

Early trigonometry

The 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".^[3]
Based on one interpretation of the Plimpton 322 cuneiform tablet (circa 1900 BC), some have even asserted that the ancient Babylonians had a table of secants.^[4] There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.

Hellenistic world

The chord of an angle subtends the arc of the angle.

Ancient Greek mathematicians made wide 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, many of the trigonometric identities and theorems that that are known today were also known to the ancient Greeks, but in their equivalent chord form.
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. For instance, propositions twelve and thirteen of book two of the ''Elements'' are the laws of cosine 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. To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the well known theorem that, in modern notation, sin α/ sin β < α/β < tan α/ tan β whenever 0° < β < α < 90°, among other theorems.
The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 - 125 BC), who is now consequently known as "the father of trigonometry."^[5] Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.

A medieval artist's rendition of Claudius Ptolemy

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'' (ca. 260 B.C.), since he measured an angle in terms of a fraction of a quadrant.^[6] 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.^[7] 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. It is due to the Babylonian sexagesimal number system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.
Menelaus of Alexandria (ca. 100 A.D.) wrote in three books his ''Sphaerica''. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles. 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. Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°. 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".^[9]
Later, Claudius Ptolemy (ca. 90 - ca. 168 A.D.) expanded upon Hipparchus' ''Chords in a Circle'' in his ''Almagest'', or the ''Mathematical Syntaxis''. The thirteen book of the ''Almagest'' are the most influential and significant trigonometric work of all antiquity. 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. Ptolemy further derived the equivalent of the half-angle formula . Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.^[10]
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.^[11]

Medieval India

Statue of Aryabhata

The next significant developments of trigonometry were in India. The mathematician-astronomer Aryabhata (476–550), in his work ''Aryabhata-Siddhanta'', 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. His works also contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. He used the words ''jya'' for sine, ''kojya'' for cosine, ''ukramajya'' for versine, and ''otkram jya'' for inverse sine. The words ''jya'' and ''kojya'' eventually became ''sine'' and ''cosine'' respectively after a mistranslation.
Other Indian mathematicians later expanded Aryabhata's works on trigonometry. In the 6th century, Varahamihira used the formulas
$sin^2(x)\; +\; cos^2(x)\; =\; 1$





In the 7th century, 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, Brahmagupta developed the formula as well as the Brahmagupta interpolation formula for computing sine values.

Muslim world

Al-Khwārizmī depicted on a Soviet stamp

The Indian works were later translated and expanded in the Muslim world by Arab and Persian mathematicians. produced tables of sines and tangents, and also developed spherical trigonometry. By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well as tables of tangent values. Abū al-Wafā' also developed the trigonometric formula sin 2''x'' = 2 sin ''x'' cos ''x''. Omar Khayyam solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables.
The Indian works were later translated and expanded in the Islamic world by Arab and Persian Muslim mathematicians. In the 9th century, produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer of spherical trigonometry.
By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions, after discovering the secant, cotangent and cosecant functions. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also developed the following trigonometric formula:
$sin(2x)\; =\; 2\; sin(x)\; cos(x)$
In the 11th century, Omar Khayyam solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables.
All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by Bhaskara II and Nasir al-Din al-Tusi in the 13th century. Nasir al-Din al-Tusi stated the law of sines and provided a proof for it, and also listed the six distinct cases of a right angled triangle in spherical trigonometry.
In the 14th century, Ghiyath al-Kashi gave trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg (14th century) also gives accurate tables of sines and tangents correct to 8 decimal places.

Medieval China

Guo Shoujing (1231-1316)

In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the ''Kaiyuan Zhan Jing'', compiled in 718 AD during the Tang Dynasty.^[12] 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 and then Indian and Islamic worlds.^[13] 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. However, this embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960-1279 AD), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations. For instance, the polymath Chinese scientist Shen Kuo (1031-1095 AD) used trigonometric functions to solve mathematical problems of chords and arcs. As the historians L. Gauchet and Joseph Needham state, the Chinese mathematician Guo Shoujing (1231-1316 AD) used spherical trigonometry in his calculations to improve Chinese astronomy and the calendar system.^[14] Along with a later 17th century Chinese illustration of Guo's mathematical proofs, Needham states:
:''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).''^[15]

Renaissance Europe

Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline, 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 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 Valentin Otho in 1596.
In the 17th century, Isaac Newton and James Stirling developed the general Newton-Stirling interpolation formula for trigonometric functions.

Trigonometric analysis

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 trigonometric series expansions of sine, cosine, tangent and arctangent. 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.^[16][17]
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" ''e''^''ix'' = cos(''x'') + ''i'' sin(''x''). Euler used the near-modern abbreviations ''sin.'', ''cos.'', ''tang.'', ''cot.'', ''sec.'', and ''cosec.''
Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory and Colin Maclaurin were also very influential in the development of trigonometric series.

See also

★ History of mathematics

★ Trigonometric functions

★ Trigonometry

Citations and footnotes

1. O'Connor (1996).
2. , , , Boyer, , 1991,
3. , , , Boyer, , 1991,
4. Joseph, pp. 383–4.
5. , , , Boyer, , 1991,
6. , , , Boyer, , 1991,
7. , , , Boyer, , 1991,
8. , , , Boyer, , 1991,
9. Needham, Volume 3, 108.
10. , , , Boyer, , 1991,
11. Boyer, pp. 158–168.
12. Needham, Volume 3, 109.
13. Needham, Volume 3, 108-109.
14. Gauchet, 151.
15. Needham, Volume 3, 109-110.
16. O'Connor and Robertson (2000).
17. Pearce (2002).

References

★ A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,

★ Gauchet, L. (1917). ''Note Sur La Trigonométrie Sphérique de Kouo Cheou-King''.

★ Joseph, George G., ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.

★ Needham, Joseph (1986). ''Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth''. Taipei: Caves Books, Ltd.

★ O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", ''MacTutor History of Mathematics Archive''. (1996).

★ O'Connor, J.J., and E.F. Robertson, "Madhava of Sangamagramma", ''MacTutor History of Mathematics Archive''. (2000).

★ Pearce, Ian G., "Madhava of Sangamagramma", ''MacTutor History of Mathematics Archive''. (2002).