Question

Why are the hyperbolic functions defined using `e`? Why not some other constant?

Answer 1

`e` is the base of the natural logarithm and comes out in many differential equations.

Explanation:

It's primarily because `e` has a very nice property of its derivative being really clean.

All of the trig functions are defined by the differential equation
`f''(x) pm f(x) = 0`
where the minus is hyperbolic and the plus is circular.

There are two solutions for each, both exponentials. Assuming the functions have form `e^(ax)`, we get
`a^2 e^(ax) pm e^(ax) = 0 implies a^2 pm 1 = 0`

so for the circular case, we get `a = pm i` and for the hyperbolic case, we get `a = pm 1`. In either case, we get these exponentials and their sums as the solutions of the differential equation.