History of trigonometry

The history of trigonometric functions may span about 4000 years. There is evidence that the Babylonians first used trigonometric functions, based on a table of numbers written on a Babylonian cuneiform tablet, Plimpton 322 (circa 1900 BC), which can be interpreted as a table of secants. There is, however, much debate on whether it was a trigonometric table. The earliest use of sine appears in the Sulba Sutras written in ancient India from the 8th century BC to the 6th century BC, which correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle), though they had not yet developed the notion of a sine in a general sense.

An artist's rendition of Claudius Ptolemy

Trigonometric functions were later studied by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Claudius Ptolemy of Egypt (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy derived the equivalent of the half-angle formula sin²(A/2) = (1 − cos(A))/2, and created a table of his results. 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.

The next significant developments of trigonometry were in India. 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.

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. 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).

Other Indian mathematicians later expanded Aryabhata's works on trigonometry. Varahamihira developed the formulas sin²x + cos²x = 1, sin x = cos(π/2 − x), and (1 − cos(2x))/2 = sin²x. Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. Brahmagupta developed the formula 1 − sin²x = cos²x = sin²(π/2 − x), and the Brahmagupta interpolation formula for computing sine values, which is a special case of the Newton–Stirling interpolation formula up to second order.

The Indian works were later translated and expanded by Muslim mathematicians. Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī produced tables of sines and tangents, and also contributed to spherical trigonometry. By the 10th century, in the work of Abu'l-Wafa, 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. Abu'l-Wafa also developed the trigonometric formula sin 2x = 2 sin x cos x. Persian mathematician 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 Indian mathematician Bhaskara II and Persian mathematician Nasir al-Din Tusi, who also gave the law of sines and listed the six distinct cases of a right angled triangle in spherical trigonometry. 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.

In the 13th century, Persian mathematician Nasir al-Din Tusi stated the law of sines and provided a proof for it. In the work of Persian mathematician Ghiyath al-Kashi (14th century), there are trigonometric tables giving 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°. Timurid mathematician Ulugh Beg's (14th century) accurate tables of sines and tangents were correct to 8 decimal places.

Madhava (c. 1400) in South India made early strides in the mathematical 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, π/4, the radius, diameter, circumference and angle θ in terms of trigonometric functions. His works were expanded by his followers at the Kerala School upto the 16th century.

The Opus palatinum de triangulis of 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.

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

References

• Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
• 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).