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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

Let 2 θ= 3 θ , where θ is acute angle. What is the value of 1+4 θ ?...

Let sin⁡2θ=cos⁡3θ\sin 2 \theta=\cos 3 \theta, where θ\theta is acute angle. What is the value of 1+4sin⁡θ1+4 \sin \theta ? (\left(\right. given that sin⁡18∘=5−14)\left.\sin 18^{{}^{\circ}}=\frac{\sqrt{5}-1}{4}\right)

Options

  1. A3\sqrt{3}
  2. B2
  3. C5\sqrt{5}
  4. D3

Correct answer

A. 3\sqrt{3}

Step-by-step solution

\Rightarrow sin⁡2θ=cos⁡3θ=sin⁡(90∘−3θ)2θ=90∘−3θ⇒θ=18∘Now, 1+4sin⁡θ=1+4sin⁡18∘\begin{aligned} & \sin 2 \theta=\cos 3 \theta=\sin \left(90^{{}^{\circ}}-3 \theta\right) \\ & 2 \theta=90^{{}^{\circ}}-3 \theta \quad \Rightarrow \quad \theta=18^{{}^{\circ}} \\ & \text{Now, } 1+4 \sin \theta=1+4 \sin 18^{{}^{\circ}}\end{aligned}=1+4 \cdot(3−14)=3\left(\frac{\sqrt{3}-1}{4}\right)=\sqrt{3}
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