Elementary trigonometric equations with sine and cosine

Problem

Solve, giving all the infinitely many solutions: $a)\ \sin x=\frac{\sqrt2}{2}\,;\qquad b)\ \cos(3x)=\frac12.$

Solution

a) On the unit circle $\sin x=\dfrac{\sqrt2}{2}$ at the angles $\dfrac{\pi}{4}$ and $\dfrac{3\pi}{4}$: $x=\frac{\pi}{4}+2k\pi\quad\lor\quad x=\frac{3\pi}{4}+2k\pi,\qquad k\in\mathbb{Z}.$ b) $\cos(3x)=\dfrac12$ gives $3x=\pm\dfrac{\pi}{3}+2k\pi$, hence $x=\pm\frac{\pi}{9}+\frac{2k\pi}{3},\qquad k\in\mathbb{Z}.$ $\boxed{\,a)\ x=\tfrac{\pi}{4}+2k\pi\ \lor\ \tfrac{3\pi}{4}+2k\pi\,;\quad b)\ x=\pm\tfrac{\pi}{9}+\tfrac{2k\pi}{3}\,}$