A) \[8\]
B) \[9\]
C) \[5\]
D) \[7\]
Correct Answer: D
Solution :
We have, \[\sin \,3\theta =\sin \theta \] \[\Rightarrow \] \[\sin \,3\theta -\sin \theta =0\] \[\Rightarrow \] \[2\cos \left( \frac{3\theta +\theta }{2} \right)\sin \left( \frac{3\theta -\theta }{2} \right)=0\] \[\Rightarrow \] \[\cos 2\theta .\sin \theta =0\] \[\Rightarrow \] \[\cos \,2\,\theta =0\] or \[\sin \theta =0,\] \[\pi ,\]\[2\pi \] \[\Rightarrow \] \[\cos \,2\,\theta =\cos \left( \frac{\pi }{2} \right)\] or \[\theta =0,\] \[\pi ,\]\[2\pi \] \[\Rightarrow \] \[2\theta =\frac{\pi }{2}\] or \[\theta =0,\] \[\pi ,\]\[2\pi \] \[\Rightarrow \] \[\theta =\frac{n\,\pi }{4}\] or \[\theta =0,\] \[\pi ,\]\[2\pi \] \[\because \] \[-2\pi <\theta <2\pi \] \[\therefore \] \[\theta =\frac{\pi }{4},\]\[\frac{3\pi }{4},\]\[\frac{5\pi }{4},\]\[\frac{7\pi }{4}\] or \[\theta =0,\] \[\pi ,\]\[2\pi \] Thus, total number of solutions =7
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