Why hyperbolic functions are basic

[Dirk Laurie <dirk.laurie@...>]

1. Reason for the name

The points (x,y) that satisfy the equation x^2 + y^2 = 1 lie on a circle,
and are parametrically given by x = cos(t), y = sin(t).

The points (x,y) that satisfy the equation x^2 - y^2 = 1 lie on a pair of
recatangular hyperboles, and are parametrically given by x = cosh(t),
y = sinh(t).

2. Identities in terms of complex numbers

Let i^2 = -1, then cos(x) = cosh(ix),  sin(x) = sinh(ix)/i.

In elementary calculus, these identities can be used to cut the
number of formulas you need to memorize by half. (E.g. you know
how to integrate 1/sqrt(1+x^2)? Then you can also integrate
1/sqrt(1-x^2).)