What's the relationship between hyperbola, hyperbolic functions and the exponential function?

Question

The hyperbolic functions can be expressed using the exponential function.

However how are these related to "hyperbolas"?

Answer 1

Let $x = cosh(t)$ and $y = sinh(t)$. Then as $t$ varies, the point $(x,y)$ moves along the right branch of the hyperbola $x^2-y^2=1$.

Answer 2

Consider $x=\cosh t$ and $y=\sinh t$.

For $t\in\mathbb R$, The coordinates $(x,y)$ trace the curve $x^2-y^2=1$, which is a hyperbola.

We sometimes call trig functions circular functions for a very similar reason. If $x=\cos t$ and $y=\sin t$, the coordinates $(x,y)$ trace a circle.