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Evaluate sinh(x)
Evaluate cosh(x)
Evaluate arcosh(x)
Evaluate exponential
Evaluate arcsin(x)
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Inverse hyperbolic sine calculator

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

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

x =
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Result:
arsinh(x) =
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Definitions

General

The inverse hyperbolic sine function, in modern notation written as arsinh(t) or arcsinh(t) or sinh-1t, gives the value x (hyperbolic angle), so that:

\sinh {x} = t

The inverse hyperbolic sine function accepts arguments from the whole real space. Since the hyperbolic sine 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 any real x:

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

Properties

The derivative of the inverse hyperbolic sine function is:

\left(\mathrm{arsinh}\,{x}\right)' = \frac{1}{\sqrt{1+x^2}}

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

\int \mathrm{arsinh}\,{x}\, \mathrm{d}x = x\, \mathrm{arsinh}\,{x} - \sqrt{1+x^2} + C

Identities

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

See also
Evaluate sinh(x)
Evaluate cosh(x)
Evaluate arcosh(x)
Evaluate exponential
Evaluate arcsin(x)
All evaluation tools