Question

Find the principle solution of the equation .
$\sin x=-\frac{1}{2}$

Answer 1

We have,

$\sin x=\frac{-1}{2}$

$\Rightarrow \sin x=\sin \frac{\pi }{6}$

$\Rightarrow \sin x=\sin (\pi +\frac{\pi }{6})\because \sin (\pi +\theta )=-\sin \theta$

$\Rightarrow \sin x=\sin \frac{\pi }{6}$

Then,

$x=7\frac{\pi }{6}$

Again,

$\sin x=\frac{-1}{2}$

$\sin x=-5\sin \frac{\pi }{6}$

$\sin x=\sin (2\pi -\frac{\pi }{6})\dots \dots .\sin ({360}^o-\theta )=\sin \theta$

$\sin x=\sin 11\frac{\pi }{6}$

Then,

$x=11\frac{\pi }{6}$

Hence, this is the principle value of

$\sin x=-\frac{1}{2}$