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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Trigonometric Equations

NDA 2018 · Mathematics · Trigonometric Equations

What is/are the solutions of the trigonometric equation cosec x+ x=...

What is/are the solutions of the trigonometric equation cosec⁡x+cot⁡x=3\operatorname{cosec} \mathrm{x}+\cot \mathrm{x}=\sqrt{3}, where 0<x<2π0 < \mathrm{x} < 2 \pi ?

Options

  1. A5π3\frac{5 \pi}{3} only
  2. Bπ3\frac{\pi}{3} only
  3. Cπ\pi only
  4. Dπ,π3,5π3\pi, \frac{\pi}{3}, \frac{5 \pi}{3}

Correct answer

B. π3\frac{\pi}{3} only

Step-by-step solution

Given equation: cosec⁡x+cot⁡x=3cosec⁡x−cot⁡x=13⇒2cosec⁡x=3+13=43⇒cosec⁡x=23⇒sin⁡x=32\begin{aligned} & \operatorname{cosec} x+\cot x=\sqrt{3} \\ & \operatorname{cosec} x-\cot x=\frac{1}{\sqrt{3}} \\ & \Rightarrow \quad 2 \operatorname{cosec} x=\sqrt{3}+\frac{1}{\sqrt{3}}=\frac{4}{\sqrt{3}} \\ & \Rightarrow \quad \operatorname{cosec} x=\frac{2}{\sqrt{3}} \\ & \Rightarrow \quad \sin x=\frac{\sqrt{3}}{2} \end{aligned} Possible values of x=π3,2π3∴x=\frac{\pi}{3}, \frac{2 \pi}{3}\therefore Option (2) is correct.
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