The argument of a hyperbolic function is a hyperbolic angle.

Using the properties of the trigonometric and hyperbolic functions, this may be written in explicitly complex form:

The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine.

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities.

Lambert was the first to introduce hyperbolic functions into trigonometry.

Another example uses hyperbolic functions of a hyperbolic angle φ.

The device could solve line equations involving hyperbolic functions ten times faster than previous methods.

Substitutions of hyperbolic functions can also be used to simplify integrals.

One can play an entirely analogous game with the hyperbolic functions.