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Trigonometry is a branch of mathematics that studies relationships between angles and side lengths, especially in triangles. It begins with trigonometric ratios in right triangles, including sine, cosine, and tangent, which depend on the measure of an angle. Their reciprocals are cosecant, secant, and cotangent. These functions can also be defined using the unit circle, allowing them to apply to positive and negative angles and to real or complex numbers. The scope of trigonometry extends beyond basic triangle calculations to include inverse functions, identities, laws for solving arbitrary triangles, spherical trigonometry, periodic functions, power series, and connections with complex exponentials. It is widely used in astronomy, navigation, surveying, engineering, physics, acoustics, optics, signal processing, computer graphics, medical imaging, and many other fields.
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Trigonometry developed from ancient efforts to study angles, lengths, and astronomical positions. Sumerian and Babylonian scholars explored angle measurement and ratios in similar triangles, while Greek mathematicians such as Euclid and Archimedes investigated chords and inscribed angles geometrically. In the 2nd century BC, Hipparchus compiled the first known trigonometric table of chords, and Ptolemy later produced detailed chord tables in the Almagest, which remained influential for more than a millennium. Indian mathematicians introduced the sine function and developed early tables of trigonometric ratios. During the medieval period, Islamic mathematicians translated and expanded Greek and Indian work, developing tables for cotangents and all six trigonometric functions. Nasir al-Din al-Tusi established trigonometry as an independent mathematical discipline and significantly advanced spherical trigonometry. Knowledge later reached Western Europe through Latin translations, where scholars such as Regiomontanus, Copernicus, Pitiscus, Viète, Euler, Gregory, Maclaurin, and Taylor helped shape modern trigonometry. Its applications expanded from astronomy to navigation, surveying, geodesy, engineering, physics, and signal analysis.
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