## Geometry and Trigonometry## Trigonometry |

## What are the hyperbolic functions (or identities)? |

Similar to the other types of functions and identities above, hyperbolic functions (also called hyperbolic identities) are used to make it easier to find solutions to equations. In fact, there are corresponding trigonometric and hyperbolic identities. For example, the most commonly used trig identity, cos^{2} θ + sin^{2} θ = 1, has a corresponding hyperbolic identity:

In this case, the minus sign is used instead of a plus and the “cos” and “sin” changed to “cosh” and “sinh,” respectively (also changed is the “θ” symbol to *x;* this is because many trig functions use the angle as the argument, while hyperbolic functions generally do not).

But these equations are not exactly the same. In particular, when one has a product of two sines it is replaced by minus a product of the two sinh’s. For example, for the trig term sin^{2}, the hyperbolic identity uses -sinh^{2} (this is called Osborn’s rule). This is not always straightforward because the minus sign is often hidden.