Generalized trigonometry

Ordinary trigonometry studies triangles in the Euclidean plane ⁠${\displaystyle \mathbb {R} ^{2}}$⁠. 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:
  1. Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions.
  2. 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).
  3. 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

• The Pythagorean theorem in non-Euclidean geometry