Trigonometric function

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

History

The earliest systematic study of trigonometric functions and tabulation of their values was performed 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, Ptolemy (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 also derived the equivalent of the half-angle formula sin(A/2)2 = (1 − cos(A))/2, allowing him to create tables with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have survived to the present day.

Related Topics:
Hipparchus - Nicaea - 180 - 125 BC - Ptolemy - 2nd century - Almagest

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The next significant development of trigonometry was in India, in the works known as the Siddhantas (4th–5th century), which first defined the sine as the modern relationship between half an angle and half a chord. The Siddhantas also contained the earliest surviving tables of sine values (along with 1 − cos values), in 3.75-degree intervals from 0 to 90 degrees.

Related Topics:
India - Siddhantas - 4th - 5th century

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The Hindu works were later translated and expanded by the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all six trigonometric functions, and had sine tables in 0.25-degree increments, to 8 decimal places of accuracy, as well as tables of tangent values.

Related Topics:
Hindu - Arab - 10th century - Abu'l-Wefa

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Our modern word sine comes, via sinus ("bay" or "fold") in Latin, from a mistranslation of the Sanskrit jiva (or jya). jiva (originally called ardha-jiva, "half-chord", in the 6th century Aryabhatiya) was transliterated by the Arabs as jiba (جب), but was confused for another word, jaib (جب) ("bay"), by European translators such as Robert of Chester and Gherardo of Cremona in Toledo in the 12th century, probably because jiba (جب) and jaib (جب) are written the same in Arabic (many vowels are excluded from words written in the Arabic alphabet).

Related Topics:
Latin - Sanskrit - 6th century - Aryabhatiya - Robert of Chester - Gherardo of Cremona - Toledo - 12th century - Arabic alphabet

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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 the De triangulis omnimodus (1464) of Regiomontanus (1436–1476), as well as his later Tabulae directionum (which included the tangent function, unnamed).

Related Topics:
De triangulis omnimodus - 1464 - Regiomontanus - 1436 - 1476 - Tabulae directionum

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The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was 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.

Related Topics:
Opus palatinum de triangulis - Rheticus - Copernicus - Valentin Otho - 1596

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The Introductio in analysin infinitorum (1748) of Euler was primarily responsible for establishing the analytic treatment of trigonometric functions, 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..

Related Topics:
Introductio in analysin infinitorum - 1748 - Euler - Euler's formula

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