The principal solutions of the equation $\sqrt{3} \operatorname{cosec} X+2=0$ are

The principal solutions of the equation $\sqrt{3} \operatorname{cosec} X+2=0$ are
  1. $\frac{2 \pi}{3}, \frac{5 \pi}{3}$
  2. $\frac{4 \pi}{3}, \frac{5 \pi}{3}$
  3. $\frac{\pi}{3}, \frac{2 \pi}{3}$
  4. $\frac{2 \pi}{3}, \frac{4 \pi}{3}$

Solution

$\begin{aligned} & \sqrt{3} \operatorname{cosec} x+2 \\ & \Rightarrow \sin x=-\frac{\sqrt{3}}{2}=-\sin \frac{\pi}{3} \\ & \Rightarrow x=\pi+\frac{\pi}{3} \text { or } 2 \pi-\frac{\pi}{3} \\ & \Rightarrow x=\frac{4 \pi}{3} \text { or } \frac{5 \pi}{3}\end{aligned}$

Asked in: MHT CET 2022 (11 Aug Shift 1)

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