# cosh 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.

## See also

## Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

## Popular tags

angle cartesian equation chord circle combinations cosh cosine cosine rule cube diagonal directrix ellipse equilateral triangle exponential exterior angle focus horizontal hyperbola hyperbolic function interior angle inverse hyperbolic function isosceles triangle locus major axis minor axis normal parabola parametric equation permutations power quadrilateral radius root sine rule sinh sloping lines solving equations solving triangles square standard curves straight line graphs tangent tanh transformations triangle vertical