Answer
The solution set is $$\{\frac{\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3},\frac{5\pi}{3}\}$$
Work Step by Step
$$\cos2x=-\frac{1}{2}$$ over interval $[0,2\pi)$
1) Interval $[0,2\pi)$ can be written as
$$0\le x\lt2\pi$$
As a result, for $2x$, the interval would be
$$0\le2x\lt4\pi$$
or $$2x\in[0,4\pi)$$
2) Now consider back the equation $$\cos2x=-\frac{1}{2}$$
Over the interval $[0,4\pi)$, there are 4 values with $\cos$ equaling $-\frac{1}{2}$, which are $\frac{2\pi}{3},\frac{4\pi}{3},\frac{8\pi}{3},\frac{10\pi}{3}$, meaning that
$$2x=\{\frac{2\pi}{3},\frac{4\pi}{3},\frac{8\pi}{3},\frac{10\pi}{3}\}$$
So $$x=\{\frac{\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3},\frac{5\pi}{3}\}$$
