Question

If $\sin 3\theta =\sin \theta$, then the general value of $\theta$ is

• A. $2n\pi ,(2n+1)\frac{\pi }{3}$
• B. $n\pi ,(2n+1)\frac{\pi }{4}$
• C. $n\pi ,(2n+1)\frac{\pi }{3}$
• D. $Noneofthese$

Answer 1

Answer: (B)

$n\pi ,(2n+1)\frac{\pi }{4}$

Given $\sin 3\theta =\sin \theta$

$⟹3\theta =k\pi +(-1{)}^k\theta$

If $k=2n\pi ,\text{i.e. even}⟹3\theta =2n\pi +\theta ⟹2\theta =2n\pi ⟹\theta =n\pi$

If $k=(2n+1)\pi ,\text{i.e. odd}⟹3\theta =(2n+1)\pi -\theta ⟹4\theta =(2n\pi +1)⟹\theta =(2n+1)\frac{\pi }{4}$

So the general value of $\theta$ is $n\pi ,(2n+1)\frac{\pi }{4}$