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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Equations

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

In the interval [- (π)/(2), (π)/(2)] , the equation _ sin θ ( 2θ) =...

In the interval [−π2,π2]\left[- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right], the equation log⁡sinθ(cos⁡2θ)=2\log_{sin \theta} \left(\cos 2\theta\right) = 2 has

Options

  1. Ano solution
  2. Ba unique solution
  3. Ctwo solutions
  4. Dinfinite many solutions

Correct answer

B. a unique solution

Step-by-step solution

Since, −π2≤θ≤π2⇒sinθ∈[−1,1]\dfrac{- \pi}{2} \le \theta \le \dfrac{\pi}{2} ⇒ sin \theta \in \left[-1, 1\right]
Here sinθ∈(0,1)sin \theta \in \left(0, 1\right) for log⁡sinθcos⁡2θ=2\log_{sin \theta} \cos 2\theta = 2 to be defined.
⇒cos⁡2θ=sin⁡2θ⇒ \cos 2\theta = \sin^{2} \theta
⇒1−2sin⁡2θ=sin⁡2θ⇒ 1 - 2\sin^{2} \theta = \sin^{2} \theta
⇒sin⁡2θ=13⇒sinθ=13⇒ \sin^{2} \theta = \dfrac{1}{3} ⇒ sin \theta = \dfrac{1}{\sqrt{3}} (as sinθ∈sin \theta \in\left(0, 1\right))<br/>)<br />⇒The equation has a unique solution.
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