The Hyperbolic Functions

For many applications, there are solutions that frequently occur which exhibit properties similar to the trigonometric functions.  These functions occur so frequently that we have abstracted the basic form and given them names to suggest the trigonometric function each resembles.  We wish to list here the definitions and some of the properties which you should be able to easily verify.  We refer to these functions as the hyperbolic functions.

The Hyperbolic Sine Function.

In some applications, is denoted .  A similar convention exists for the other functions as well and will be noted in the definition rather than separately.

You should verify that is an odd function.

The Hyperbolic Cosine Function.

Verify that  is an even function.

As with the trigonometric functions, the other hyperbolic functions are defined in terms of these two functions.

Hyperbolic Identities.

You should verify each if the following identities and become familiar with them.