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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2019 · Mathematics · Trigonometric Ratios & Identities

If 2 θ= 3 θ , where 0 < θ < (π)/(2) , then what is θ equal to?

If sin⁡2θ=cos⁡3θ\sin 2 \theta=\cos 3 \theta, where 0<θ<π20 < \theta < \frac{\pi}{2}, then what is sin⁡θ\sin \theta equal to?

Options

  1. A5+14\frac{\sqrt{5}+1}{4}
  2. B5−14\frac{\sqrt{5}-1}{4}
  3. C5+116\frac{\sqrt{5}+1}{16}
  4. D5−116\frac{\sqrt{5}-1}{16}

Correct answer

B. 5−14\frac{\sqrt{5}-1}{4}

Step-by-step solution

sin⁡2θ=cos⁡3θ\sin 2 \theta=\cos 3 \theta⇒sin⁡2θ=sin⁡(90∘−3θ)2θ=90∘−3θ⇒5θ=90∘⇒θ=18∘∴sin⁡θ=sin⁡18∘=5−14\begin{aligned} & \Rightarrow \sin 2 \theta=\sin \left(90^{{}^{\circ}}-3 \theta\right) \\ & 2 \theta=90^{{}^{\circ}}-3 \theta \\ & \Rightarrow 5 \theta=90^{{}^{\circ}} \\ & \Rightarrow \theta=18^{{}^{\circ}} \\ & \therefore \sin \theta=\sin 18^{{}^{\circ}}=\frac{\sqrt{5}-1}{4}\end{aligned}$$
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