Question

Find the principal solution or solution of $\sin x=\frac{1}{\sqrt{2}}$.

Answer 1

We know that
$\sin x=\frac{1}{\sqrt{2}}=\sin \frac{\pi }{4}$ and by allied angles, we also know that
$\sin (\pi -x)=\sin x$
$\therefore \sin \frac{\pi }{4}=\sin (\pi -\frac{\pi }{4})=\sin \frac{3\pi }{4}$
$\therefore \sin \frac{\pi }{4}=\sin 3\pi 4=\frac{1}{\sqrt{2}}$
such that $0<\frac{\pi }{4}<2\pi$ and $0<\frac{3\pi }{4}<2\pi$
$\therefore$ the required principal solution are $x=\frac{\pi }{4}$ and $x=\frac{3\pi }{4}$