Answer
a. $\frac{7\pi}{18}+\frac{2k\pi}{3}$, $\frac{11\pi}{18}+\frac{2k\pi}{3}$
b. $\frac{7\pi}{18}$, $\frac{11\pi}{18}$, $\frac{19\pi}{18}$, $\frac{23\pi}{18}$, $\frac{31\pi}{18}$, $\frac{35\pi}{18}$
Work Step by Step
a. $2\sin3\theta+1=0$
$2\sin3\theta=-1$
$\sin3\theta=-\frac{1}{2}$
$3\theta=\frac{7\pi}{6}+2k\pi$, $\frac{11\pi}{6}+2k\pi$
$\theta=\frac{7\pi}{18}+\frac{2k\pi}{3}$, $\frac{11\pi}{18}+\frac{2k\pi}{3}$
b. If $\theta=\frac{7\pi}{18}+\frac{2k\pi}{3}$, the only solutions in $[0, 2\pi)$ are $\frac{7\pi}{18}$, $\frac{19\pi}{18}$ and $\frac{31\pi}{18}$. If $\theta=\frac{11\pi}{18}+\frac{2k\pi}{3}$, the only solutions in $[0, 2\pi)$ are $\frac{11\pi}{18}$, $\frac{23\pi}{18}$ and $\frac{35\pi}{18}$.
