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Evaluate sinh(x)
Evaluate cosh(x)
Evaluate arsinh(x)
Evaluate exponential
Evaluate arccos(x)
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Inverse hyperbolic cosine calculator

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019

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

x =
Result:
arcosh(x) =
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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

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

See also
Evaluate sinh(x)
Evaluate cosh(x)
Evaluate arsinh(x)
Evaluate exponential
Evaluate arccos(x)
All evaluation tools