Question

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

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (C)

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