WAEC 2011 · Paper 2 · Q1

(a)
Find the truth set of $\sin\theta + \cos2\theta = 0$ , $0^\circ \le \theta \le 360^\circ$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

Use $\cos 2\theta = 1 - 2\sin^2\theta$ , so every term is in $\sin\theta$ : $\sin\theta + 1 - 2\sin^2\theta = 0$ .
Rearrange: $2\sin^2\theta - \sin\theta - 1 = 0$ .
Factorise: $(2\sin\theta + 1)(\sin\theta - 1) = 0$ , so $\sin\theta = 1$ or $\sin\theta = -\frac12$ .
$\sin\theta = 1$ : $\theta = 90^\circ$ .
$\sin\theta = -\frac12$ : the reference angle is $30^\circ$ , and sine is negative in the third and fourth quadrants, so $\theta = 210^\circ$ or $330^\circ$ .
The truth set is $\{90^\circ, 210^\circ, 330^\circ\}$ .