Q52 / 90 · KVPY 2017 (19 Nov SB/SX) · Mathematics
The number of solutions of the equation $\sin \theta+\cos \theta=\sin 2 \theta$ in the interval $[-\pi, \pi]$ is
- A1
- Correct answerB2
- C3
- D4
Solution
Squaring $\sin ^{2} \theta+\cos ^{2} \theta+2 \sin 2 \theta=\sin ^{2} 2 \theta$
$$
\begin{array}{l}
\Rightarrow \sin ^{2} 2 \theta-\sin 2 \theta-1=0 \\
\sin 2 \theta=-\left(\frac{\sqrt{5}-1}{2}\right), \sin 2 \theta=\frac{\sqrt{5}+1}{2}(\text { reject })
\end{array}
$$
two solution exist between
$\frac{-\pi}{2}$ to $\frac{-\pi}{4} \& \frac{\pi}{4}$ to 0 (2 solution)
Hence, the correct answer is option: 'B'.