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

Example: Find an expression for

Ifthen

Multiply both sides byto obtain

Now multiply both sides by 2 obtainingand subtractfrom both sides to obtain

This is a quadratic expression inso 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

Ifthenso

Multiply both sides byto obtain

Now multiply both sides by 2 obtainingand subtractfrom both sides to obtain

This is a quadratic expression inso 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

Ifthenso

Multiply both sides byto obtain

Expand the brackets:and move the term to the left, and theterm to the right, obtaining

Now factorise with

Divide by

Now take the natural logs of both sides and divide by 2.