Review question

# How many solutions does $2 \cos^2x + 5\sin x = 4$ have? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9491

## Solution

How many values of $x$ satisfy the equation $2 \cos^2x + 5\sin x = 4$ in the range $0 \leq x < 2 \pi$?

1. $2$,

2. $4$,

3. $6$,

4. $8$.

We have \begin{align*} & 2 \cos^2x + 5\sin x = 4 \\ \iff & 2 - 2\sin^2 x + 5 \sin x = 4 \\ \iff & 2\sin^2 x - 5 \sin x + 2 = 0 \\ \iff & (2\sin x - 1)(\sin x - 2) = 0, \end{align*}

so $\sin x = \frac{1}{2}$ (since $\sin x = 2$ is impossible).

This occurs at $x = \dfrac{\pi}{6}$ and $x = \dfrac{5\pi}{6}$ in the given range, so there are $2$ such values of $x$ and the answer is (a).