Question

Number of solutions of the equation: $\sin 7\theta =\sin \theta +\sin 3\theta \text{for}\theta \in (0,\pi )$, is equal to

• A. $7$
• B. $6$
• C. $4$
• D. $3$

Answer 1

The correct option is A $7$

$sin7\theta =sin\theta +sin3\theta sin7\theta -sin\theta =sin3\theta 2sin\left(\frac{7\theta -\theta }{2}\right)cos\left(\frac{7\theta +\theta }{2}\right)=sin3\theta 2sin3\theta cos4\theta =sin3\theta sin3\theta (2cos4\theta -1)=0sin3\theta =0and2cos4\theta -1=03\theta =0,\pi ,2\pi ....cos4\theta =1/2\theta =0,\frac{\pi }{3},\frac{2\pi }{3}....4\theta =\frac{\pi }{3},2\pi -\frac{2\pi }{3},2\pi +\frac{\pi }{3}....\theta =\frac{\pi }{12},\frac{\pi }{2}-\frac{\pi }{12},\frac{\pi }{2}+\frac{\pi }{12}..bycheckingtheconditionfordomain,therewillbe7solutions.$