Chapter 2.12, Problem 38P

The functions sin t, cos t, …, are called "circular functions" and the functions sinh t, cosh t, …, are called "hyperbolic functions”. To see a reason for this, show that x = cos t, y = sin t, satisfy the equation of a circle $x^2+y^2=1$, while x = cosh t, y = sinh t, satisfy the equation of a hyperbola $x^2-y^2=1$.