The cosh function is a hyperbolic function. It is also known as the *hyperbolic cosine* function.

## Equation and graph

The cosh function is defined as:

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

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, $e^{x}$ and $e^{-x}$. 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 $e^{x}$ we get:

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

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

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

These alternative forms are sometimes useful.