Find the principal solutions of x=-1 2 - Vidyadip

Question

Find the principal solutions of $\sin x=-\frac{1}{2}$

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Answer

$ \sin x=-\frac{1}{2}$
$\therefore \sin x=-\sin \left(\frac{\pi}{6}\right)$
$\therefore \sin x=\sin \left(\pi+\frac{\pi}{6}\right)$
$=\sin \left(\frac{7 \pi}{6}\right) \text { ans } \sin x=\sin \left(2 \pi-\frac{\pi}{6}\right)$
$=\sin \left(\frac{11 \pi}{6}\right) $
such that $0 \leq \frac{7 \pi}{6}<2 \pi$ and $0 \leq \frac{11 \pi}{6}<2 \pi$
$\therefore$ The required principal solutions are $x=\frac{7 \pi}{6}$ and $x=\frac{11 \pi}{6}$.

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