Question

In the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ , the equation ${\left(log\right)}_{sin\theta }\left(cos2\theta \right)=2$ has

• A. no solution
• B. a unique solution
• C. two solutions
• D. infinite many solutions

Answer 1

Answer: (B)

Since, $\frac{-\pi }{2}\le \theta \le \frac{\pi }{2}\Rightarrow sin\theta \in \left[-1,1\right]$
Here $sin\theta \in \left(0,1\right)$ for $lo{g}_{sin\theta }cos2\theta =2$ to be defined.
$\Rightarrow cos2\theta =si{n}^2\theta$
$\Rightarrow 1-2si{n}^2\theta =si{n}^2\theta$
$\Rightarrow si{n}^2\theta =\frac{1}{3}\Rightarrow sin\theta =\frac{1}{\sqrt{3}}$ (as $sin\theta \in \left(0,1\right)$ )
$\Rightarrow$ The equation has a unique solution.