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The hyperbolic functions can be expressed using the exponential function.

However how are these related to "hyperbolas"?

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    $\begingroup$ $x^2 - y^2 = 1$ is a hyperbola in the ordinary $x,y$ plane $\endgroup$ – Will Jagy Feb 2 '16 at 21:01
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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.

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

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