Hyperbolic functions
mathematical function related with trigonometric functions / From Wikipedia, the free encyclopedia
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined for the unit hyperbola rather than on the unit 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 hyperbola.
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Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations, cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics.
The basic hyperbolic functions are:[1][2]
- hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),[3]
- hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/),[4]
from which are derived:[5]
- hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),[6]
- hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/[4])
- hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/),[7]
- hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/),[8][9]
corresponding to the derived trigonometric functions.
The inverse hyperbolic functions are:[1]
- area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")[10][11]
- area hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh"
- and so on.
The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.[12]