## Inverse hyperbolic cosine calculator

This tool evaluates the inverse hyperbolic cosine of a number: arcosh(x). Enter the argument x below.

 x = Result: arcosh(x) =

## Definitions

### General

The inverse hyperbolic cosine function, in modern notation written as arcosh(x) or arccosh(x) or cosh-1 x, gives the value t (hyperbolic angle), so that:

$\cosh {t} = x$

The inverse hyperbolic cosine function accepts values not smaller than 1, because $$\cosh{x}\ge 1, \forall x\in \mathbb{R}$$. Since the hyperbolic cosine is defined through the natural exponential function $$\mathrm{e}^x$$, its inverse can be defined through the natural logarithm function, using the following formula, for real x, with x≥1:

$\mathrm{arcosh}\,{x} = \ln\left(x+\sqrt{x^2-1}\right)$

### Properties

The derivative of the inverse hyperbolic cosine function is:

$\left(\mathrm{arcosh}\,{x}\right)' = \frac{1}{\sqrt{x^2-1}}\quad, \left\{x\in\mathbb{R} | x>1\right\}$

The integral of the inverse hyperbolic cosine function is given by:

$\int \mathrm{arcosh}\,{x}\, \mathrm{d}x = x\, \mathrm{arcosh}\,{x} - \sqrt{x^2-1} + C \quad, \left\{x\in\mathbb{R} | x>1\right\}$

### Identities

$\begin{split} & \cosh (\mathrm{arcosh}\,{x}) &= x \\ \\ & \sinh (\mathrm{arcosh}\,{x}) &= \sqrt{x^2-1} \\ \\ & \tanh (\mathrm{arcosh}\,{x}) &= \frac{\sqrt{x^2-1}}{x} \\ \\ & \mathrm{arcosh}\,{x}+\mathrm{arcosh}\,{y} & = \mathrm{arcosh}\,\left(x y + \sqrt{x^2-1}\sqrt{y^2-1} \right) \\ \\ & \mathrm{arcosh}\,{x}-\mathrm{arcosh}\,{y} & = \mathrm{arcosh}\,\left(x y - \sqrt{x^2-1}\sqrt{y^2-1} \right) \\ \\ \end{split}$