# Hyperbolic Trigonometric Functions

The hyperbolic trigonometric functions are an important class of functions used in engineering. For equivalent results about the traditional trigonometric functions see this page.

Information on derivatives of these functions can be found here and integrals here.

A PDF file containing this information can be found here.

### Definitions

The hyperbolic trigonometric functions are: $\DeclareMathOperator{\cosec}{cosec} \DeclareMathOperator{\cosech}{cosech} \DeclareMathOperator{\sech}{sech}$

 Hyperbolic sine: $\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}$ Hyperbolic cotangent: $\displaystyle \coth x= \frac {\cosh x}{\sinh x}$ Hyperbolic cosine: $\displaystyle{\cosh x=\frac {e^{x}+e^{-x}}{2}}$ Hyperbolic secant: $\displaystyle \sech x= \frac {1}{\cosh x}$ Hyperbolic tangent: $\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}$ Hyperbolic cosecant: $\displaystyle \cosech x= \frac1{\sinh x}$
Note that:
• $\cosech(x)$ is sometimes written as $\operatorname {csch}(x)$
• $\sinh^{-1}(x)$ is sometimes written as $\operatorname{arcsinh}(x)$
• $\sinh^{-1}(x)$ does not mean $\displaystyle{\frac1{\sinh x}}$

### Function Identities

Here are some common identities about the trigonometric functions with their hyperbolic trigonometric equivalents:

Trigonometry Hyperbolic Trigonometry
$\sin(-x)=-\sin x$ $\sinh(-x)=-\sinh x$
$\cos(-x) =\cos x$ $\cosh(-x) =\cosh x$
$\tan(-x)=-\tan x$ $\tanh(-x)=-\tanh x$

### Sum and Difference Identities

The following results show results for angle sums and differences in trigonometry and the equivalent identities in hyperbolic trigonometry:
Trigonometry Hyperbolic Trigonometry
$\sin(x+y)=\sin x\cos y+\cos x\sin y$ $\sinh(x+y)=\sinh x\cosh y+\cosh x\sinh y$
$\cos(x+y)=\cos x\cos y-\sin x\sin y$ $\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y$
$\tan(x+y) = \displaystyle{\frac{\tan x+\tan y}{1-\tan x\tan y}}$ $\tanh(x+y)=\displaystyle{\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}$
$\sin (x-y)=\sin x\cos y-\cos x\sin y$ $\sinh(x-y)=\sinh x\cosh y-\cosh x\sinh y$
$\cos (x-y)=\cos x\cos y+\sin x\sin y$ $\cosh(x-y)=\cosh x\cosh y-\sinh x\sinh y$
$\tan(x-y) = \displaystyle{\frac{\tan x-\tan y}{1+\tan x\tan y}}$$\tanh(x-y)=\displaystyle{\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}$

### Pythagorean Identity

The Pythagorean identity and it's hyperbolic equivalents:

Trigonometry Hyperbolic Trigonometry
$\cos^2x+\sin^2x=1$ $\cosh^2x-\sinh^2x=1$
$1+\tan^2x=\sec^2 x$ $1-\tanh^2x=\sech^2x$
$\cot^2x+1=\cosec^2 x$ $\coth^2x-1=\cosech^2x$