trigonometry

trigonometry
Chasetaylor1 · CC BY-SA 3.0 · source
Name	Trigonometry
Field	Mathematics
Subfields	Euclidean geometry, Analytic geometry, Spherical astronomy
Notable figures	Hipparchus, Ptolemy, Al‑Battani, Madhava of Sangamagrama, Isaac Newton
First appearance	Antiquity

trigonometry is the branch of mathematics that studies relationships between angles and lengths in triangles and periodic phenomena. It originated in antiquity for astronomical calculations and surveying and later became foundational for calculus, navigation, and signal analysis. Its methods connect geometry, algebra, and analysis and underlie advances in astronomy, cartography, and engineering.

History

Histories of angle measurement and chord tables trace back to ancient Babylonian and Egyptian surveys, while the first surviving systematic tables appear in works by Hipparchus and Ptolemy in Alexandria. Developments in India by figures such as Aryabhata and Brahmagupta introduced sine concepts later refined by Madhava of Sangamagrama and the Kerala school of astronomy and mathematics. Islamic scholars including Al‑Battani and Al‑Khwarizmi transmitted and expanded classical tables to Cordoba and Baghdad, influencing European scholars in Renaissance centers like Florence and Venice. The invention of analytic trigonometry and series expansions involved contributors such as Johannes Kepler, Isaac Newton, Girolamo Cardano, and Leonhard Euler, propelling applications in navigation for expeditions sponsored by states like Portugal and Spain and scientific institutions including the Royal Society and the Académie des Sciences.

Definitions and Basic Functions

Basic definitions arise from a right triangle: given vertices associated with points used in surveys like those by Jacques Cartier or James Cook, the primary ratios are sine, cosine, and tangent defined as opposite, adjacent, and opposite over adjacent relative to a chosen angle. In analytic contexts developed by Descartes and Pierre de Fermat, these functions are extended to the unit circle, connecting to exponential functions formalized by Niels Henrik Abel and Évariste Galois. Complementary relationships were used by astronomers such as Tycho Brahe and Johannes Hevelius to relate sine and cosine values, while the cotangent, secant, and cosecant appear in classical works of Niccolò Tartaglia and tables compiled in the libraries of Vatican Library and Bodleian Library.

Identities and Formulas

Key identities used by mathematicians in proofs—such as the Pythagorean identity, addition and subtraction formulas, double‑angle and half‑angle formulas—were employed in computations by Ptolemy and proved in the era of Euclid and later generalized by Joseph Fourier in harmonic analysis. Series expansions (Taylor and Maclaurin series) developed by Brook Taylor and Colin Maclaurin enabled computation of trigonometric values and were instrumental for James Watt and engineers of the Industrial Revolution. Product‑to‑sum and sum‑to‑product identities influenced transform methods used by Guglielmo Marconi and refined by Heinrich Hertz and Oliver Heaviside in electromagnetic theory. Complex exponential representations, attributed to work by Euler and explored further by Carl Friedrich Gauss, connect trigonometric identities to algebraic structures studied in Élie Cartan’s and Sophus Lie’s works.

Applications

Practical applications span navigation and geodesy used by explorers like Ferdinand Magellan and naval institutions such as the Royal Navy, to astronomical ephemerides compiled by observatories like Greenwich Observatory and Paris Observatory. Surveying and cartography for projects by Ferdinand von Richthofen and urban planners in cities like London and Vienna used trigonometric triangulation methods formalized by surveyors such as George Everest. In physics and engineering, wave phenomena analyzed by Christian Doppler, James Clerk Maxwell, and Ludwig Boltzmann rely on trigonometric modeling; signal processing techniques used by Claude Shannon and Norbert Wiener employ Fourier methods rooted in trigonometric series. Structural analysis for bridges by engineers like Isambard Kingdom Brunel and aerodynamics calculations in projects by Wright brothers incorporated trigonometric computations, while modern applications in digital imaging and acoustics tie to work at institutions like Bell Labs and MIT.

Extensions and Generalizations

Extensions include spherical and hyperbolic analogues developed for celestial mechanics in works by Simon Newcomb and geodesy by Friedrich Robert Helmert, and non‑Euclidean adaptations influenced by Janos Bolyai and Nikolai Lobachevsky. Multivariable and vector generalizations integrate with linear algebra advanced by Arthur Cayley and Hermann Grassmann, while complex analysis expansions by Augustin‑Louis Cauchy and Bernhard Riemann yield complex‑valued trigonometric functions. Modern generalizations underpin harmonic analysis, representation theory pursued by David Hilbert and Hermann Weyl, and computational algorithms implemented at research centers like Los Alamos National Laboratory and companies such as IBM and Bell Labs.

Category:Mathematics