# Cosh function

Level: A Level Maths

The cosh function is a hyperbolic function.

## Equation and graph

The cosh function is defined as:

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

Here is a graph of the function:

## cosh as average of two exponentials

The cosh function can be interpreted as the average of two functions, ${\displaystyle e^{x}}$ and ${\displaystyle e^{-x}}$. This animation illustrates this:

## Other forms of the equation

If we multiply the top and bottom of the original equation for the cosh function by ${\displaystyle e^{x}}$ we get

${\displaystyle \cosh x={\frac {e^{2x}+1}{2e^{x}}}}$

Alternatively, if we multiply the top and bottom of the original equation for the cosh function by ${\displaystyle e^{-x}}$ we get

${\displaystyle \cosh x={\frac {1+e^{-2x}}{2e^{-x}}}}$