Hyperbolic Functions

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Related to Hyperbolic Functions: Hyperbolic cosine, Inverse hyperbolic functions

hyperbolic functions

[¦hī·pər¦bäl·ik ′fəŋk·shənz]
The real or complex functions sinh (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 Functions (hyperbolic sine of x)

cosh x = Hyperbolic Functions (hyperbolic cosine of x)

In some cases, the hyperbolic tangent is also considered:

tanh x = Hyperbolic Functions

(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

Figure 1

Figure 2

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


Ianpol’skii, A. R. Giperbolicheskie funktsii. Moscow, 1960.
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These results show that discount rate estimated by the hyperbolic function in which N is defined by the number of beneficiaries can measure selfishness.
2]) for fits of the hyperbolic function (Equation 2) are also given in Table 1.
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Among the subjects covered by the program include algebra, trigonometry, analytic geometry, probability and statistics, statistical process control (SPC), integrals, differentiation, series, constants, vector analysis, differential equations and hyperbolic functions.