Question

The number of solutions of cos 2$\theta$ = sin$\theta$ in (0, 2$\pi$) is

• A. 1
• B. 2
• C. 3
• D. 5

Answer 1

Answer: (C)

3

$\begin{array}{cc}\text{Solution}-& \text{Given,}\cos 2\theta =\sin \theta \\ & 1-2{\sin }^2\theta =\sin \theta \\ \Rightarrow & {\sin }^{+}\theta -(-2{\sin }^2\theta +1)=0\\ & \Rightarrow \sin \theta +2{\sin }^2\theta -1=0\\ & \Rightarrow (2\sin \theta -1)(\sin \theta +1)=0\\ & \sin \theta =\frac{1}{2};\sin \theta =-1\\ \Rightarrow & \theta \in \{\frac{\pi }{6},\frac{5\pi }{6},\frac{3\pi }{2})\\ \therefore & \text{No. of solutions}=3\\ \text{Hence}& \left(c\right)\text{is the correct option}\end{array}$