The possible values of $θ \in (0, π)$ such that $\sin (θ) + \sin (4 θ) + \sin (7 θ) = 0$ are

Moderate

A

$\frac{π}{4}, \frac{5 π}{12}, \frac{π}{12}, \frac{2 π}{3}, \frac{3 π}{4}, \frac{8 π}{9}$
B

$\frac{2 π}{9}, \frac{π}{4}, \frac{π}{2}, \frac{2 π}{3}, \frac{3 π}{4}, \frac{35 π}{36}$
C

$\frac{2 π}{9}, \frac{π}{4}, \frac{π}{2}, \frac{2 π}{3}, \frac{3 π}{4}, \frac{8 π}{9}$
D

$\frac{2 π}{9}, \frac{π}{4}, \frac{4 π}{9}, \frac{π}{2}, \frac{3 π}{4}, \frac{8 π}{9}$

Solution

$\begin{array}{l} \sin θ + \sin 4 θ + \sin 7 θ = 0 \\ \Rightarrow 2 \sin 4 θ \cos 3 θ + \sin 4 θ = 0 \\ \Rightarrow \sin 4 θ (2 \cos 3 θ + 1) = 0 \end{array}$
$\Rightarrow \sin 4 θ = 0$ or $\cos 3 θ = - 1 / 2 = \cos (2 π / 3)$
$\Rightarrow θ = \frac{π}{4}, \frac{π}{2}, \frac{3 π}{4}$ or $3 θ = 2 π / 3, 2 π 2 π / 3$
$\Rightarrow θ = \frac{2 π}{9}, \frac{π}{4}, \frac{π}{2}, \frac{4 π}{9}, \frac{3 π}{4}, \frac{8 π}{9}$

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