Early study of triangles can be traced to Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics during the 2nd millennium BC. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (6th century AD), who discovered the sine function, cosine function, and versine function.
During the Middle Ages, the study of trigonometry continued in Islamic mathematics, building on earlier foundations from Indian (notably Aryabhata's sine and cosine concepts) and Greek (Hipparchus' chord table) sources.Pingree, D. (1970). "The Greek Influence on Early Islamic Mathematics". Isis, 61(1): 92–98. Mathematicians such as al-Khwarizmi and Abu al-Wafa made important contributions, but the field was fundamentally transformed by Nasir al-Din al-Tusi (1201–1274).
Al-Tusi was the first to treat trigonometry as an independent mathematical discipline, separate from astronomy.Berggren, J. L. (2003). Episodes in the Mathematics of Medieval Islam. Springer. pp. 123–145. He fully established all six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) with complete proofs, and formulated the plane and spherical law of sines, the form still taught in schools and universities today.Rosenfeld, B. A. (1960). "Nasir al-Din al-Tusi's work on trigonometry". Journal of the History of Science, 12(3): 215–228. He also rewrote and corrected Ptolemy's Almagest, replacing chords with sine and tangent tables and adding practical, real-world examples.Kennedy, E. S. (1956). A Survey of Islamic Astronomical Tables. Transactions of the American Philosophical Society. pp. 40–45.
Through translations of Arabic and Greek texts, this knowledge reached the Latin West. However, during the Renaissance, Regiomontanus (c. 1464) translated al-Tusi's work into Latin without crediting him, effectively erasing al-Tusi's name from European historiography.Cardano, G. (1570). De proportionibus. Basel. (Cardano's criticism of Regiomontanus for copying Jabir ibn Aflah). Consequently, for centuries, European textbooks mistakenly attributed the development of trigonometry to Regiomontanus and earlier Greek sources, sidelining al-Tusi's central role.
Modern scholarship has since restored al-Tusi's credit: he is now recognized in academic literature as the "Master of Trigonometry", the one who gave the field its modern, independent form.Saliba, G. (2007). Islamic Science and the Making of the European Renaissance. MIT Press. pp. 215–220.
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).
Etymology The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure".
The word sine is derived, indirectly, from the Sanskrit word jya-ardha used by Indian mathematicians, was shortened to jiva and was later adopted into Arabic as jiba, a phonetically similar but meaningless word, before being translated into Latin as sinus and was finally imported as sine in English. (See Jyā, koti-jyā and utkrama-jyā and .) Particularly Fibonacci's sinus rectus arcus proved influential in establishing the term.
The word tangent comes from Latin meaning "touching", since the lineSee the figure at Pythagorean identities touches the circle of unit radius, whereas secant stems from Latin "cutting" since the line cuts the circle (see the figure at Pythagorean identities).Oxford English Dictionary
The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.
The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae.: "It should be recalled that form the days of Hipparchus until modern times there were no such things as trigonometric ratios. The Greeks, and after them the Hindus and the Arabs, used trigonometric lines. These at first took the form, as we have seen, of chords in a circle, and it became incumbent upon Ptolemy to associate numerical values (or approximations) with the chords. [...] It is not unlikely that the 360-degree measure was carried over from astronomy, where the zodiac had been divided into twelve "signs" or 36 "decans". A cycle of the seasons of roughly 360 days could readily be made to correspond to the system of zodiacal signs and decans by subdividing each sign into thirty parts and each decan into ten parts. Our common system of angle measure may stem from this correspondence. Moreover since the Babylonian position system for fractions was so obviously superior to the Egyptians unit fractions and the Greek common fractions, it was natural for Ptolemy to subdivide his degrees into sixty partes minutae primae, each of these latter into sixty partes minutae secundae, and so on. It is from the Latin phrases that translators used in this connection that our words "minute" and "second" have been derived. It undoubtedly was the sexagesimal system that led Ptolemy to subdivide the diameter of his trigonometric circle into 120 parts; each of these he further subdivided into sixty minutes and each minute of length sixty seconds." These roughly translate to "first small parts" and "second small parts".
Ancient
Ancient Near East The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.: "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry", or the measure of three sided polygons (trilaterals), than "trigonometry", the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the Elements, for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles."
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. 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. 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. The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c. 1680–1620 BC), contains the following problem related to trigonometry:
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.
Classical antiquity
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 one half 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 to Hellenistic mathematicians, but in their equivalent chord form.As these historical calculations did not make use of a unit circle, the length of the radius was needed in the formula. Contrast this with the modern use of the crd function that assumes a unit circle in its definition.
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 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. 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's inequality.
The first-known trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BC), who is now consequently known as "the father of trigonometry.": "For some two and a half centuries, from Hippocrates to Eratosthenes, Greek mathematicians had studied relationships between lines and circles and had applied these in a variety of astronomical problems, but no systematic trigonometry had resulted. Then, presumably during the second half of the 2nd century BC, the first trigonometric table apparently was compiled by the astronomer Hipparchus of Nicaea (ca. 180–ca. 125 BC), who thus earned the right to be known as "the father of trigonometry". Aristarchus had known that in a given circle the ratio of arc to chord decreases as the arc decreases from 180° to 0°, tending toward a limit of 1. However, it appears that not until Hipparchus undertook the task had anyone tabulated corresponding values of arc and chord for a whole series of angles."His table of chords, analogous to modern tables of sine values, was the first to tabulate the corresponding arc and chord values for a series of angles.