History of trigonometry
Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus.
[9] The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.
[10] Based on one interpretation of the Plimpton 322 cuneiform 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.
Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form.
[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.
[24] Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya.
He also gave the power series of π and the angle, radius, diameter, and circumference of a circle in terms of trigonometric functions.
In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang 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 as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.
[30] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.
[33] 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.
[36] Previous works were later translated and expanded in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, 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.
[40] In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables.
[45] Abū al-Wafā also established the angle addition and difference identities presented with complete proofs:[45] For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other minutes.
[45] He also discovered the law of sines for spherical trigonometry:[41] Also in the late 10th and early 11th centuries AD, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity:[46] 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".
[47] The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying[48] and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century.
[49] In the late 11th century, Omar Khayyám (1048–1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables.
[51][52][53] The law of cosines, in geometric form, can be found as propositions II.12–13 in Euclid's Elements (c. 300 BC),[54] but was not used for the solution of triangles per se.
[citation needed] Al-Kāshī presumably worked on Ulugh Beg's even more comprehensive trigonometric tables, with five-place (sexagesimal) entries for each minute of arc.
[57] A simplified trigonometric table, the "toleta de marteloio", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate navigation courses.
Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,[58] in his De triangulis omnimodis 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 Valentin Otho in 1596.
