How many solutions does $2 \cos^2x + 5\sin x = 4$ have?
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Ref: R9491

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Solution

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

$2$,
$4$,
$6$,
$8$.

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).

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Trigonometry: Triangles to Functions

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

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How many solutions does $2 \cos^2x + 5\sin x = 4$ have?
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