Trigonometry (from Greek trigōnon
“triangle” + metron “measure”) is a branch of mathematics that studies
triangles and the relationships between their sides and the angles
between these sides.
Trigonometry defines the trigonometric
functions, which describe those relationships and have applicability to
cyclical phenomena, such as waves.
The field evolved during the third
century BC as a branch of geometry used extensively for astronomical
studies. It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate
course or as part of a precalculus course.
The trigonometric functions are pervasive
in parts of pure mathematics and applied mathematics such as Fourier
analysis and the wave equation, which are in turn essential to many
branches of science and technology.
Spherical trigonometry studies triangles
on spheres, surfaces of constant positive curvature, in elliptic
geometry. It is fundamental to astronomy and navigation. Trigonometry on
surfaces of negative curvature is part of Hyperbolic geometry.
History
Sumerian astronomers introduced angle
measure, using a division of circles into 360 degrees. 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. The ancient Greeks transformed trigonometry into an ordered
science. Classical Greek mathematicians (such as Euclid and
Archimedes) studied the properties of chords and inscribed angles in
circles, and proved theorems that are equivalent to modern trigonometric
formulae, although they presented them geometrically rather than
algebraically. Claudius Ptolemy expanded upon Hipparchus’ Chords in a
Circle in his Almagest.
The modern sine function was first
defined in the Surya Siddhanta, and its properties were further
documented by the 5th century Indian mathematician and astronomer
Aryabhata.
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.
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.
One of the earliest works on trigonometry
by a European mathematician is De Triangulis by the 15th century German
mathematician Regiomontanus. 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.
