# Hyperbolic Functions

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Related to Hyperbolic Functions: Hyperbolic cosine, Inverse hyperbolic functions

## hyperbolic functions

[¦hī·pər¦bäl·ik ′fəŋk·shənz]*x*), cosh (

*x*), tanh (

*x*), coth (

*x*), sech (

*x*), csch (

*x*); they are related to the hyperbola in somewhat the same fashion as the trigonometric functions are related to the circle, and have properties analogous to those of the trigonometric functions.

## Hyperbolic Functions

functions defined by the equations

sinh *x* = (hyperbolic sine of x)

cosh *x* = (hyperbolic cosine of x)

In some cases, the hyperbolic tangent is also considered:

tanh *x* =

(See Figure 1 for graphs of the hyperbolic functions.) Hyperbolic functions are connected by relations similar to those connecting the trigonometric functions:

cosh^{2}x - sinh^{2} = 1

tanh x = sinh x/cosh x

sinh (x±y) = sinh x cosh y ± cosh x sinh y

cosh (x±y) = cosh x cosh y ± sinh x sinh y

Hyperbolic functions may be expressed in terms of trigonometric functions:

Geometrically, the hyperbolic functions are obtained by analysis of the rectangular hyperbola *x*^{2} - *y*^{2} = 1, which may be defined in terms of the parametric equations *x* = cosh *t* and *y* = sinh *t*. The argument *t* represents twice the area of the hyperbolic sector *OAC* (see Figure 2). Inverse hyperbolic functions are defined by the equations

### REFERENCE

Ianpol’skii, A. R.*Giperbolicheskie funktsii*. Moscow, 1960.