History of trigonometry Wikipedia. Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics Rhind Mathematical Papyrus and Babylonian mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. 1 In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata sixth century CE. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, hence it was adopted as a separate subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 1. Isaac Newton and James Stirling and reaching its modern form with Leonhard Euler 1. EtymologyeditThe term trigonometry was derived from Greektrignon, triangle and metron, measure. 2Our modern word sine is derived from the Latin word sinus, which means bay, bosom or fold, translating Arabic jayb. The Arabic term is in origin a corruption of Sanskritjv, or chord. Sanskrit jv in learned usage was a synonym of jy chord, originally the term for bow string. Sanskrit jv was rendered into Arabic as jiba. 34 This term was then transformed into the genuine Arabic word jayb,4 meaning bosom, fold, bay, either by the Arabs or by a mistake of the European translators such as Robert of Chester perhaps because the words were written without vowels, who translated jayb into Latin as sinus. 3 Particularly Fibonaccis sinus rectus arcus proved influential in establishing the term sinus. 5 The words minute and second are derived from the Latin phrases partes minutae primae and partes minutae secundae. 6 These roughly translate to first small parts and second small parts. DevelopmenteditEarly trigonometryeditThe 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. 7The 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. 4 Based on one interpretation of the Plimpton 3. BC, some have even asserted that the ancient Babylonians had a table of secants. 8 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. 4 The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes c. BC, contains the following problem related to trigonometry 4If a pyramid is 2. 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. 4Greek mathematicsedit. The chord of an angle subtends the arc of the angle. 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 chords 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,chord 2sin2,displaystyle mathrm chord theta 2sin frac theta 2,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. 9Although 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. 7 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. 7 To compensate for the lack of a table of chords, mathematicians of Aristarchus time would sometimes use the statement that, in modern notation, sin sin lt lt tan tan whenever 0 lt lt lt 9. Aristarchus inequality. 1. The first trigonometric table was apparently compiled by Hipparchus of Nicaea 1. BCE, who is now consequently known as the father of trigonometry. 1. Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles. 51. Although it is not known when the systematic use of the 3. Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon ca. BC, since he measured an angle in terms of a fraction of a quadrant. 1. It seems that the systematic use of the 3. Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 3. Babylonian astronomy. 1. In ancient astronomy, the zodiac had been divided into twelve signs or thirty six decans. A seasonal cycle of roughly 3. It is due to the Babylonian sexagesimalnumeral system that each degree is divided into sixty minutes and each minute is divided into sixty seconds. 6Menelaus of Alexandria ca. AD wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles. 9 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. 9 Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 1. Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the theorem of Menelaus. 9 He further gave his famous rule of six quantities. 1. Later, Claudius Ptolemy ca. AD expanded upon Hipparchus Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry. Ptolemys table of chords gives the lengths of chords of a circle of diameter 1. ArendtBanal Evil and Use 609 0. pdf. 434 2. Several real climate scientists contacted by Gizmodo felt Smith was not as up to snuff on his research as he claimed to be, however. These comments reflect the. 1 I celebrate myself, and sing myself, And what I assume you shall assume, For every atom belonging to me as good belongs to you. I loafe and invite my soul. The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity. 1. A theorem that was central to Ptolemys calculation of chords was what is still known today as Ptolemys 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 Ptolemys theorem appeared as proposition 9. Euclids Data. Ptolemys theorem leads to the equivalent of the four sum and difference formulas for sine and cosine that are today known as Ptolemys formulas, although Ptolemy himself used chords instead of sine and cosine. 1. Ptolemy further derived the equivalent of the half angle formulasin.
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