Question

The number of values of $\theta \in \left[\frac{- 3 \pi }{2} , \frac{4 \pi }{3}\right]$ which satisfies the system of equations $2sin^{2}\theta +sin^{2}2\theta =2$ and $sin2\theta +cos2\theta =tan\theta $ is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (C)

First equation is $4sin^{2}\theta cos^{2}\theta =2cos^{2}\theta $
$\Rightarrow \, cos^{2} \theta =0$ or $sin^{2}\theta =\frac{1}{2}$ $=\left(\frac{1}{\sqrt{2}}\right)^{2}=\left(sin\right)^{2}\frac{\pi }{4}$
$\Rightarrow \, \theta =\left(2 n + 1\right)\frac{\pi }{2}, \, n\in I \, $ or $ \, \, \theta =n\pi \pm\frac{\pi }{4}, \, n\in I$
Second equation is not satisfied by $\theta =\left(2 n + 1\right)\frac{\pi }{2}, \, n\in I$ but satisfied by $\theta =n\pi \pm\frac{\pi }{4}, \, n\in I \, $ So $\theta =n\pi \pm \frac{\pi }{4} , n \in I$
$\therefore $ In $\left[\right. \frac{- 3 \pi }{2} , \frac{4 \pi }{3} \left]\right.$ , the values of $\theta $ are
$\frac{- 5 \pi }{4} , \frac{- 3 \pi }{4} , \frac{- \pi }{4} , \frac{\pi }{4} , \frac{3 \pi }{4} , \frac{5 \pi }{4}$