Find the principal and general solutions of the equation $\cot x=-\sqrt{3}$
Find the principal and general solutions of the equation $\cot x=-\sqrt{3}$
$\cot x=-\sqrt{3}$
It is known that $\cot \frac{\pi}{6}=\sqrt{3}$
$\therefore \cot \left(\pi-\frac{\pi}{6}\right)=-\cot \frac{\pi}{6}=-\sqrt{3}$ and $\cot \left(2 \pi-\frac{\pi}{6}\right)=-\cot \frac{\pi}{6}=-\sqrt{3}$
i.e., $\cot \frac{5 \pi}{6}=-\sqrt{3}$ and $\cot \frac{11 \pi}{6}=-\sqrt{3}$
Therefore, the principal solutions are $x=\frac{5 \pi}{6}$ and $\frac{11 \pi}{6}$
Now, $\cot x=\cot \frac{5 \pi}{6}$
$\Rightarrow \tan x=\tan \frac{5 \pi}{6}$ $\left[\cot x=\frac{1}{\tan x}\right]$
$\Rightarrow x=n \pi+\frac{5 \pi}{6},$ where $n \in Z$
Therefore, the general solution is $x=n \pi+\frac{5 \pi}{6},$ where $n \in Z$
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