Question 17.2.6: Find the general solution of each equation: (i) √3 cot x+1=0......
Find the general solution of each equation:
(i) \sqrt{3} \cot x+1=0 \quad(ii) \operatorname{cosec} x+\sqrt{2}=0
Step-by-Step
\begin{array}{l}\text{ (i) }\sqrt{3} \cot x+1=0 \\ \\\Rightarrow \cot x=\frac{-1}{\sqrt{3}}\\ \\\Rightarrow \tan x=-\sqrt{3}=-\tan \frac{\pi}{3}=\tan \left(\pi-\frac{\pi}{3}\right)=\tan \frac{2 \pi}{3} \\ \\\Rightarrow \tan x=\tan \frac{2 \pi}{3} \\ \\\Rightarrow x=\left(n \pi+\frac{2 \pi}{3}\right), \text { where } n \in I .\end{array}
Hence, the general solution is x=\left(n \pi+\frac{2 \pi}{3}\right) , where n \in I .
\begin{array}{l}\text{ (ii) }\operatorname{cosec} x+\sqrt{2}=0 \\ \\\Rightarrow \sin x=-\frac{1}{\sqrt{2}}=-\sin \frac{\pi}{4}=\sin \left(\pi+\frac{\pi}{4}\right)=\sin \frac{5 \pi}{4} \\ \\\Rightarrow \sin x=\sin \frac{5 \pi}{4} \\ \\\Rightarrow x=\left\{n \pi+(-1)^{n} \cdot \frac{5 \pi}{4}\right\}, \text { where } n \in I .\end{array}Hence, the general solution is x=\left\{n \pi+(-1)^{n} \cdot \frac{5 \pi}{4}\right\} , where n \in I .
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