Hyperbolic functions
| (Math.) | certain functions which have relations to the hyperbola corresponding to those which sines, cosines, tangents, etc., have to the circle; and hence, called hyperbolic sines, hyperbolic cosines, etc. |
See also: Hyperbolic
Hyperbolic Functions
hyperbolic functions
[¦hī·pər¦bäl·ik ′fəŋk·shənz]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:
cosh2x - sinh2 = 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 x2 - y2 = 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
