Generalized trigonometry
Study of triangles in other spaces than the Euclidean plane From Wikipedia, the free encyclopedia
Ordinary trigonometry studies triangles in the Euclidean plane . There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions[broken anchor], definitions via differential equations[broken anchor], and 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.
Trigonometry
- 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.
- Hyperbolic trigonometry:
- Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions.
- Hyperbolic functions in Euclidean geometry: The unit circle is parameterized by (cos t, sin t) whereas the equilateral hyperbola is parameterized by (cosh t, sinh t).
- Gyrotrigonometry: A form of trigonometry used in the gyrovector space approach to hyperbolic geometry, with applications to special relativity and quantum computation.
- Trigonometry for taxicab geometry[1]
- Spacetime trigonometries[2]
- Fuzzy qualitative trigonometry[3]
- Operator trigonometry[4]
- Lattice trigonometry[5]
- Trigonometry on symmetric spaces[6][7][8]
Higher dimensions
- Schläfli orthoschemes - right simplexes (right triangles generalized to n dimensions) - studied by Schoute who called the generalized trigonometry of n Euclidean dimensions polygonometry.
- Pythagorean theorems for n-simplices with an "orthogonal corner"
- Trigonometry of a tetrahedron[9]
- De Gua's theorem – a Pythagorean theorem for a tetrahedron with a cube corner
- A law of sines for tetrahedra
- Polar sine
Trigonometric functions
- Trigonometric functions can be defined for fractional differential equations.[10]
- In time scale calculus, differential equations and difference equations are unified into dynamic equations on time scales which also includes q-difference equations. Trigonometric functions can be defined on an arbitrary time scale (a subset of the real numbers).
- The series definitions[broken anchor] of sin and cos define these functions on any algebra where the series converge such as complex numbers[broken anchor], p-adic numbers, matrices, and various Banach algebras.
Other
- Polar/Trigonometric forms of hypercomplex numbers[11][12]
- Polygonometry – trigonometric identities for multiple distinct angles[13]
- The Lemniscate elliptic functions, sinlem and coslem
See also
References
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