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  4. In the interval [- (π /2) , (π /2)] , the equation (log)sin θ ⁡ (cos ⁡ 2 θ )=2 has

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

NTA AbhyasNTA Abhyas 2020

Solution:

Since, $\frac{- \pi }{2}\leq \theta \leq \frac{\pi }{2}\Rightarrow sin \theta \in \left[- 1 , 1\right]$
Here $sin \theta \in \left(0 , 1\right)$ for $log_{sin \theta } ⁡ cos ⁡ 2 \theta =2$ to be defined.
$\Rightarrow cos 2 \theta =sin^{2} ⁡ \theta $
$\Rightarrow 1-2sin^{2} \theta =sin^{2} ⁡ \theta $
$\Rightarrow sin^{2} \theta =\frac{1}{3}\Rightarrow sin ⁡ \theta =\frac{1}{\sqrt{3}}$ (as $sin \theta \in \left(0 , 1\right)$ )
$\Rightarrow $ The equation has a unique solution.