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

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

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