Hyperbolic functions and equations

Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.

As
$\cosh^2 \mu - \sinh^2 \mu= 1$
one has for any value of $\mu$ that the point

$x = a\ \cosh\ \mu$

$y = b\ \sinh\ \mu$

satisfies the equation

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

which is the equation of a hyperbola relative its canonical coordinate system.

When μ varies over the interval $-\infty < \mu < \infty$ one gets with this formula all points $(x\ ,\ y)$ on the right branch of the hyperbola.
The left branch for which $x < 0$ is in the same way obtained as