Question

The possible values of $\theta ϵ(0,\pi )$ such that $\sin \left(\theta \right)+\sin \left(4\theta \right)+\sin \left(7\theta \right)=0$ are:

• A. $\frac{2\pi }{9},\frac{\pi }{4},\frac{4\pi }{9},\frac{\pi }{2},\frac{3\pi }{4},\frac{8\pi }{9}$
• B. $\frac{\pi }{4},\frac{5\pi }{12},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{8\pi }{9}$
• C. $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{35\pi }{36}$
• D. $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{8\pi }{9}$

Answer 1

Answer: (A)

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

$\theta ϵ(0,\pi )$

$\sin \theta +\sin 4\theta +\sin 7\theta =0$

$\sin (4\theta -3\theta )+\sin 4\theta +\sin (4\theta +3\theta )=0$

$2\sin 4\theta \cos 3\theta +\sin 4\theta =0$

$\sin 4\theta (2\cos 3\theta +1)=0$

$\therefore \sin 4\theta =0$ or $2\cos 3\theta +1=0\Rightarrow \cos 3\theta =\frac{-1}{2}$

$\therefore 4\theta =n\pi ;\therefore 3\theta =(2n+1)\pi \pm \frac{\pi }{3}$

$\theta =\frac{n\pi }{4};\theta =\left(\frac{2n+1}{3}\right)\pi \pm \frac{\pi }{9}$

$0<\frac{n\pi }{4}<\pi ;\therefore \theta =\frac{2\pi }{9},\frac{4\pi }{9},\frac{8\pi }{9}$

$0<n<4$

$\therefore \theta =\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}$