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.
Contents
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 the points (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.
- Universal hyperbolic trigonometry – an algebraic approach based on Rational trigonometry.