The hyperbolic functions can be expressed using the exponential function.
However how are these related to "hyperbolas"?
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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.