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Trigonometry
- the term originates from the Greek trigōnon "triangle" + metron
"measure" is a branch of mathematics that studies triangles and the
relationships between the lengths of their sides and the angles between those
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.
There
are an enormous number of uses of trigonometry and trigonometric functions. For
instance, the technique of triangulation is used in astronomy to measure the
distance to nearby stars, in geography to measure distances between landmarks,
and in satellite navigation systems. The sine and cosine functions are
fundamental to the theory of periodic functions such as those that describe sound
and light waves.
Fields
that use trigonometry or trigonometric functions include astronomy especially
for locating apparent positions of celestial objects, in which spherical
trigonometry is essential and hence navigation on the oceans, in aircraft, and
in space, music theory, acoustics, optics, analysis of financial markets,
electronics, probability theory, statistics, biology, medical imaging CAT scans
and ultrasound, pharmacy, chemistry, number theory and hence cryptology,
seismology, meteorology, oceanography, many physical sciences, land surveying
and geodesy, architecture, phonetics, economics, electrical engineering,
mechanical engineering, civil engineering, computer graphics, cartography,
crystallography and game development.
Ordinary
trigonometry studies triangles in the Euclidean plane R2. There are a number of
ways of defining the ordinary Euclidean geometric trigonometric functions on
real numbers: right-angled triangle definitions, unit-circle definitions,
series definitions, definitions via differential equations, definitions using
functional equations. Generalizations of trigonometric functions are often
developed by starting with one of the above methods and adapting it to a
situation other than the real numbers of Euclidean geometry. Generally,
trigonometry can be the study of triples of points in any kind of geometry or
space. A triangle is the polygon with the smallest number of vertices, so one
direction to generalize is to study higher-dimensional analogs of angles and
polygons: solid angles and polytopes such as tetrahedrons and n-simplices.
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In spherical trigonometry, triangles on
the surface of a sphere are studied. The spherical triangle identities are
written in terms of the ordinary trigonometric functions but differ from the
plane triangle identities.
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Hyperbolic trigonometry
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Rational trigonometry – a reformulation
of trigonometry in terms of spread and quadrance rather than angle and length.
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Trigonometry in Galois fields
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Trigonometry for taxicab geometry
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Spacetime trigonometries
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Fuzzy qualitative trigonometry
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Operator trigonometry
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Lattice trigonometry
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Trigonometry on symmetric spaces
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