Answer
a. $\frac{5\pi}{18}+\frac{k\pi}{3}$
b. $\frac{5\pi}{18}$, $\frac{11\pi}{18}$, $\frac{17\pi}{18}$, $\frac{23\pi}{18}$, $\frac{29\pi}{18}$, $\frac{35\pi}{18}$
Work Step by Step
a. $\sqrt{3}\tan3\theta+1=0$
$\sqrt{3}\tan3\theta=-1$
$\tan3\theta=-\frac{1}{\sqrt{3}}$
$\tan3\theta=-\frac{\sqrt{3}}{3}$
$3\theta=\frac{5\pi}{6}+k\pi$
$\theta=\frac{5\pi}{18}+\frac{k\pi}{3}$
b. To get all solutions in $[0, 2\pi)$, we add multiples of $\frac{\pi}{3}$ to $\frac{5\pi}{18}$:
$\frac{5\pi}{18}$
$\frac{5\pi}{18}+\frac{\pi}{3}=\frac{11\pi}{18}$
$\frac{5\pi}{18}+\frac{2\pi}{3}=\frac{17\pi}{18}$
$\frac{5\pi}{18}+\frac{3\pi}{3}=\frac{23\pi}{18}$
$\frac{5\pi}{18}+\frac{4\pi}{3}=\frac{29\pi}{18}$
$\frac{5\pi}{18}+\frac{5\pi}{3}=\frac{35\pi}{18}$
All other solutions lie outside of the interval $[0, 2\pi)$.
