## Inverting the Hyperbolic Trigonometric Formulae

We often have to find analytic expressions for the trigonometric formulae The method is illustrated in the following examples.

Example: Find an expression for If then Multiply both sides by to obtain Now multiply both sides by 2 obtaining and subtract from both sides to obtain This is a quadratic expression in so we can solve it using the ordinary quadratic formula. - remember that our quadratic is in terms of  Now take the natural logarithms of both sides to obtain Since Example Find an expression for If then so Multiply both sides by to obtain Now multiply both sides by 2 obtaining and subtract from both sides to obtain This is a quadratic expression in so we can solve it using the ordinary quadratic formula. - remember that our quadratic is in terms of  Now take the natural logarithms of both sides to obtain Since Example Find an expression for If then so Multiply both sides by to obtain Expand the brackets: and move the term to the left, and the term to the right, obtaining Now factorise with Divide by Now take the natural logs of both sides and divide by 2.  