# cosh function

By Martin McBride, 2020-09-10
Tags: cosh hyperbolic function
Categories: hyperbolic functions pure mathematics

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

Here is a video that explains sinh, cosh and tanh:

## 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:

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

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

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

## Derivation of other forms

To see how the two formulae above were derived, we start with the original definition of cosh:

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

Multiplying top and bottom by $e^{x}$ gives:

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

Remember that $e^{x}e^{x}$ is $e^{2x}$. Also $e^{x}e^{-x}$ is 1. This gives:

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

which is the second form of the cosh equation. The other alternative form is derived in a similar way.