Question

$\sin x+\sqrt{3}\cos x=\sqrt{2}$ Find the value of $x$.

Answer 1

Dividing both sides by $\sqrt{1+3}=2$, we get
$\frac{\sqrt{3}}{2}\cos x+\frac{1}{2}\sin x=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{\left(2\right)}}$
$\cos x\cos \frac{\pi }{6}+\sin x\sin \frac{\pi }{6}=\cos \frac{\pi }{4}$
$\cos (x-\frac{\pi }{6})=\cos \frac{\pi }{4}$
$\therefore x=\frac{\pi }{6}=2n\pi \pm \frac{\pi }{4}$ or $x=2n\pi +\frac{\pi }{6}\pm \frac{\pi }{4}$.