NECO 2023 · Paper 1 · Q21

If $\sin\theta = \cos2\theta$ and $0^\circ < \theta < 90^\circ$ , find the value of $4\theta$ .

Worked solution (try it first)

Write $\cos 2\theta$ in terms of $\sin\theta$ : $\cos 2\theta = 1 - 2\sin^2\theta$ , so $\sin\theta = 1 - 2\sin^2\theta$ .
Bring everything to one side: $2\sin^2\theta + \sin\theta - 1 = 0$ , which factorises as $(2\sin\theta - 1)(\sin\theta + 1) = 0$ .
$\theta$ is acute, so $\sin\theta$ is positive: $\sin\theta = \frac12$ and $\theta = 30^\circ$ .
So $4\theta = 120^\circ$ , option D.