Definitions of Hyperbolic Functions

The Hyperbolic Functions

Recall that the trigonometric functions could be defined in terms of the exponential function by using Euler's Theorem:

These two definitions inspire two more. Let's define the hyperbolic cosine and hyperbolic sine as

and

the parallels with the circular trigonometric functions are obvious. Consider

and

verify for yourself that

so that the two hyperbolic functions satisfy the equation of a hyperbola:

Hence the name!!!

We can also define, by extension, the other hyperbolic functions:

Red Question: Use the definitions of the hyperbolic functions to find identities based on

, and .

Green Question: Find a power series representations for

, and . Do these remind you of anything?

Orange Question: What do the graphs of

, and look like? Specify their domains and ranges.

Blue Question: Using the graphs of the hyperbolic functions to guide you in finding appropriate definitions for the inverse hyperbolic functions ,

and based on logarithms.