Answer
a. $\frac{\pi}{9}+\frac{2k\pi}{3}$, $\frac{5\pi}{9}+\frac{2k\pi}{3}$
b. $\frac{\pi}{9}$, $\frac{5\pi}{9}$, $\frac{7\pi}{9}$, $\frac{11\pi}{9}$ $\frac{13\pi}{9}$, $\frac{17\pi}{9}$
Work Step by Step
a. $2\cos 3\theta=1$
$\cos 3\theta=\frac{1}{2}$
$3\theta=\frac{\pi}{3}+2k\pi$, $\frac{5\pi}{3}+2k\pi$
$\theta=\frac{\pi}{9}+\frac{2k\pi}{3}$, $\frac{5\pi}{9}+\frac{2k\pi}{3}$
b. If $\theta=\frac{\pi}{9}+\frac{2k\pi}{3}$, the only solutions in $[0, 2\pi)$ are $\frac{\pi}{9}$, $\frac{7\pi}{9}$, $\frac{13\pi}{9}$. If $\theta=\frac{5\pi}{9}+\frac{2k\pi}{3}$, the only solutions in $[0, 2\pi)$ are $\frac{5\pi}{9}$, $\frac{11\pi}{9}$, $\frac{17\pi}{9}$.
