The possible values of (0,) such that +4+7=0 are:

Question

The possible values of $	heta\in(0,\pi)$ such that $\sin	heta+\sin4	heta+\sin7	heta=0$ are:

Accepted Answer

Grouping $\sin\theta+\sin7\theta = 2\sin4\theta\cos3\theta$:

$$2\sin4\theta\cos3\theta+\sin4\theta = 0 \implies \sin4\theta(1+2\cos3\theta)=0$$

Case $\sin4\theta=0$: $4\theta=n\pi \implies \theta=\dfrac\pi4,\dfrac\pi2,\dfrac{3\pi}4$ (within $(0,\pi)$).

Case $\cos3\theta=-\dfrac12$: $3\theta=2n\pi\pm\dfrac{2\pi}3 \implies \theta = \dfrac{2\pi}9,\dfrac{4\pi}9,\dfrac{8\pi}9$ (within $(0,\pi)$).

Solutions · 1