Question

if sinθ+sin5θ=sin3θ then what is the value of θ?

Answer 1

Given the equation $\sin \theta +\sin 5\theta =\sin 3\theta$.

Use the sum formula,

$\sin A+\sin B=2\sin(\frac{A+B}{2}) \cos (\frac{A-B}{2})$.

Using the above formula the equation becomes,

$2 \sin3 \theta\cos 2\theta =\sin 3\theta\\ \sin 3\theta=0,2\cos 2\theta=1\\ \sin 3\theta=0,\cos \theta=\frac{1}{2}$

The general solution of the trigonometric equation is

$3\theta=n\pi, \theta =2n\pi \pm \frac{\pi}{3} ,n=0,1,2,3,...$

or

$\theta=\frac{n\pi}{3}, \theta =2n\pi \pm \frac{\pi}{3} ,n=0,1,2,3,...$