Fun with Math!: Hyperbolic Sine & Hyperbolic Cosecant

The hyperbolic sine function is defined as
$\sinh(x)=\frac{e^{x}-e^{-x}}{2}$

or generally,
$\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$

Look at the graphs below, the blue solid curve is the curve of $\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$ , whereas the dotted curves are the graphs of its terms. If you can't remember the graph of $\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$ , you can try to add up the y-values of the two terms (dotted curves) and get the y-value of $\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$ .

Look at the graph below, try to slide the value of a to different values to view the changes on graph.

Now, take a look at the graph of the reciprocal of hyperbolic sine function, i.e. hyperbolic cosecant function, which is defined as
$\mathrm{csch}(ax)=\frac{1}{\sinh(ax)}=\frac{2}{e^{ax}-e^{-ax}}$ .
Observe that when a =1,