The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry."[3] Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees.[4]They and their successors the Babylonians studied the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar methodology.[5] The ancient Greeks transformed trigonometry into an ordered science.[6] Classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chordsand inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemyexpanded upon Hipparchus' Chords in a Circle in his Almagest.[7] The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th centuryIndian mathematician and astronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.[citation needed] At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[9] One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematicianRegiomontanus. Trigonometry was still so little known in 16th century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explaining its basic concepts. Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics.[10] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[11] Gemma Frisiusdescribed for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.[12] Also in the 18th century, Brook Taylor defined the general Taylor series.[13] [edit]Overview

In this right triangle: sin A = a/c; cos A = b/c;tan A = a/b. If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. Theshape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

Sine function (sin), defined as the ratio of the side opposite the angle to thehypotenuse.

Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The...

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