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Evaluate sinh(x)
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Hyperbolic cosine calculator

- By Dr. Minas E. Lemonis, PhD - Updated: December 10, 2021

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

x =
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Result:
cosh(x) =
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Definitions

General

The hyperbolic cosine function is defined as:

\cosh{x} = \frac{e^x+e^{-x}}{2}

The graph of the hyperbolic cosine function is shown in the figure below. It is a monotonic function, unlike the trigonometric cosine, which is periodic.

cosh-graph

The points (\cosh{t}, \sinh{t}) form the right wing of an equilateral hyperbola (see figure below), just like the trigonometric cosine, sine pairs form a circle.

The points (\cosh{t}, \sinh{t}) form the right wing of an equilateral hyperbola (see figure below), just like the trigonometric cosine, sine pairs form a circle. Parameter t is the half area between the hyperbola, the x-axis and a ray from origin to the (\cosh{t}, \sinh{t}) point.

sinh-graph

Series

All hyperbolic functions can be defined in an infinite series form. Hyperbolic cosine function can be written as:

\begin{split} \cosh x & = \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!} = \\ & = 1 + \frac{x^2}{2} + \frac{x^4}{24} + \frac{x^6}{720} \cdots \end{split}

Properties

The derivative of the hyperbolic cosine function is the hyperbolic sine:

\left(\cosh{x}\right)' = \sinh{x}

The integral of the hyperbolic cosine is given by:

\int \cosh{x} \, \mathrm{d}x = \sinh{x} +C

Identities

\begin{split} & \cosh{\left(-x\right)} & = \cosh{x} \\ \\ & \cosh{\left(2 x\right)} & = \sinh^2{x}+\cosh^2{x}=2\sinh^2{x}+1 = 2\cosh^2{x}-1 \\ \\ & \cosh{\left(x + y\right)} & = \cosh{x}\cosh{y} + \sinh{x}\sinh{y} \\ \\ & \cosh{x} + \cosh{y} & = 2\cosh{\frac{x+y}{2}} \cosh{\frac{x-y}{2}} \\ \\ & \cosh{\left(\frac{x}{2}\right)} & = \sqrt{\frac{\cosh{x}+1}{2}} \\ \\ & \cosh{x} + \sinh{x} & = e^{x} \\ \\ & \cosh{x} - \sinh{x} & = e^{-x} \\ \\ & \cosh^2{x} - \sinh^2{x} & = 1 \\ \\ \end{split}

See also
Evaluate sinh(x)
Evaluate tanh(x)
Evaluate arcosh(x)
Evaluate exponential
Evaluate cos(x)
All evaluation tools