Hyperbolic Functions

Certain combinations of the exponential functions ex and e-x arise so frequently in mathematics that they have been given special names. These functions have the same relationship to the hyperbola as trigonometric functions have to the circle, therefore they're called hyperbolic functions.


Overview

You can use the following built-in hyperbolic functions.
Function Description
Sinh(x) Hyperbolic sine of x
Defined as: Sinh(x) = (e^x - e^(-x)) / 2

Cosh(x) Hyperbolic cosine of x
Defined as: Cosh(x) = (e^x + e^(-x)) / 2

Tanh(x) Hyperbolic tangent of x
Defined as: Tanh(x) = Sinh(x) / Cosh(x)

ASinh(x) Inverse hyperbolic sine of x or Sinh-1(x)
Defined as: ASinh(x) = Log(x + Sqrt(x^2 + 1))

ACosh(x) Inverse hyperbolic cosine of x or Cosh-1(x)
Defined as: ACosh(x) = Log(x + Sqrt(x^2 - 1))

ATanh(x) Inverse hyperbolic tangent of x or Tanh-1(x)
Defined as: ATanh(x) = 1/2 * Log((1+x) / (1-x))

Csch(x) Hyperbolic cosecant of x
Defined as: Csch(x) = 1/Sinh(x)

Sech(x) Hyperbolic secant of x
Defined as: Sech(x) = 1/Cosh(x)

Coth(x) Hyperbolic cotangent of x
Defined as: Coth(x) = 1/Tanh(x)

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Hyperbolic Identities

The hyperbolic functions satisfy a number of identities that are similar to well-known trigonometric identities. The most important identities are listed below.
Hyperbolic Identities
Sinh(-x) = -Sinh(x)
Cosh(-1) = Cosh(x)

Cosh2(x) - Sinh2(x) = 1
1- Tanh2(x) = Sech2(x)

Sinh(x + y) = Sinh(x)*Cosh(y) + Cosh(x)*Sinh(y)
Cosh(x + y) = Cosh(x)*Cosh(y) + Sinh(x)*Sinh(y)

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Examples

'Examples of using Hyperbolic functions

Sinh(0.5)
       Ans = 0.5210953055
x = 0.5;
(e^x - e^(-x)) / 2
       Ans = 0.5210953055

Cosh(0)
       Ans = 1

Coth(1.5)
       Ans = 1.104791393

Sinh(3)
       Ans = 10.017874927

ASinh(Ans)
       Ans = 3

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See Also

Functions overview, Trigonometric Functions Functionstree