Use inverse functions where needed to find all solutions of the equation in the interval 2 cos2 x + 9 sin x = 6

You can use the identity to write the equation as

Now, from the fundamental equation of trigonometry, deduce an expression for in terms of :

The equation becomes

Simplify the left hand side and move all terms to left hand side:

Now, if you let , this equation becomes a quadratic equation:

The two solutions of this equations are

We must be careful, because we have to remember that was actually . This means that can only assume values between -1 and 1. The second solution exceeds 1, so we reject it. So, we have