Question

Show that hyperbolic cosine and hyperbolic sine functions form a set of parametric equations that translate into the equation for a hyperbola, $x^2-y^2=1$.

Answer 1

Let $x=\cosh t=\frac{e^t+e^{-t}}{2}$

$y=\sinh t=\frac{e^t-e^{-t}}{2}$

$x^2-y^2={\cosh }^{2}t-{\sinh }^{2}t$

$=\frac{(e^t+e^{-t}{)}^2}{4}-\frac{(e^t-e^{-t}{)}^2}{4}$

$=\frac{e^{2t}+e^{-2t}+2-e^{2t}-e^{-2t}+2}{4}=\frac{4}{4}=1$

$\therefore x^2-y^2=1$ is a hyperbola with set of parametric equations $(\cosh t,\sinh t)$