Answer
a. $\frac{\pi}{3}+k\pi$, $\frac{2\pi}{3}+k\pi$
b. $\frac{\pi}{3}$, $\frac{2\pi}{3}$, $\frac{4\pi}{3}$, $\frac{5\pi}{3}$
Work Step by Step
a. $2\cos2\theta+1=0$
$2\cos2\theta=-1$
$\cos2\theta=-\frac{1}{2}$
$2\theta=\frac{2\pi}{3}+2k\pi$, $\frac{4\pi}{3}+2k\pi$
$\theta=\frac{\pi}{3}+k\pi$, $\frac{2\pi}{3}+k\pi$
b. If $\theta=\frac{\pi}{3}+k\pi$, the only solutions in $[0, 2\pi)$ are $\frac{\pi}{3}$ and $\frac{4\pi}{3}$. If $\theta=\frac{2\pi}{3}+k\pi$, the only solutions in $[0, 2\pi)$ are $\frac{2\pi}{3}$ and $\frac{5\pi}{3}$.
