First of all, the terminology is "hyperbolic" not "hyperbole."

We have a function

To find the inverse we need to switch the roles of x and y:

and solve for y:

<-- Multiply through by

This is a quadratic in , so

We discard the "-" solution since the arguement of ln cannot be negative. ( is negative everywhere.)

So finally

This is the inverse function to sinh(x). (The dotted line in the graph below is the line y = x.)

-Dan