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In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t , sin t ) form a circle with a unit radius , the points (cosh t , sinh t ) form the right half of the unit hyperbola .
The argument to the hyperbolic functions is a hyperbolic angle measure. In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant ...
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals . In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration .
tanh is the hyperbolic tangent function. This equation was obtained in 1955 by Yu. I. Shimansky, at first empirically , and later derived theoretically. The Shimansky equation does not contain any arbitrary constants , since the value of T C can be determined experimentally and L 0 can be calculated if L has been measured experimentally for at ...
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices . Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane.
Similar expressions can be written for tanh x, coth x, sech x, and csch x. Geometrically, this change of variables is a one-dimensional stereographic projection of the hyperbolic line onto the real interval, analogous to the Poincaré disk model of the hyperbolic plane.
Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane. In the descriptions below the constant Gaussian curvature of the plane is −1. Sinh, cosh and tanh are hyperbolic functions.
Soboleva modified hyperbolic tangent; T. ... Tanh This page was last edited on 5 March 2020, at 10:32 (UTC). Text is available under the Creative Commons ...