## Cosh function

The $cosh$ function is defined as:

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

Here is the graph of the function:

## cosh as average of two exponentials

The formula for $cosh$ can be interpreted as the *average* of two terms, $e^x$ and $e^{-x}$.

This animation draws the first term in blue, the second term in green, and the averages (ie the $cosh$ function) in red:

## Alternative equations

If we take the equation above and multiply top and bottom by $e^x$ we get an alternative form of the equation:

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

or alternatively, multiplying top and bottom by $e^{-x}$

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

These are alternative ways of writing the same formula.