Hyperbolic Functions

The hyperbolic functions are defined as:

The graphs of hyperbolic functions are below:

From these graphs, the domain and range of the hyperbolic functions can be found:

The inverse hyperbolic functions are , , and , also denoted and respectively.

The domains and ranges can be found:

The inverse hyperbolic functions have logarithmic forms that can be proved by using the exponential definition of the hyperbolic functions and solving the hidden quadratic. These are:

The most important hyperbolic identity is:

When proving identities, it is often useful to return to the exponential definitions of , , and . Outside of the syllabus, there is Osborn's Rule for converting normal trigonometric identities into hyperbolic identities.

The derivatives of hyperbolic functions are:

The integrals of hyperbolic functions are:

The following proofs are given to derive the logarithm definitions of inverse hyperbolic functions. Broadly, all follow a similar idea of completing the square on (this can even be used to find logarithmic definitions of inverse trigonometric functions, which has come up before on past papers).