Answer
(a) $0.15+\frac{2k\pi}{3}$ and $0.89+\frac{2k\pi}{3}$
(b) $0.15,2.24,4.34,0.89,2.98,5.08$
Work Step by Step
(a) As $csc3\theta=\frac{1}{sin3\theta}$, the equation becomes $sin^23\theta=\frac{1}{5}$ which gives
$sin3\theta=\pm\frac{\sqrt 5}{5}$, $3\theta=0.46+2k\pi$ and $3\theta=2.68+2k\pi$. Thus the solutions are
$\theta=0.15+\frac{2k\pi}{3}$ and $\theta=0.89+\frac{2k\pi}{3}$ where $k$ is an integer.
(b) Based on the solutions from (a), within $[0,2\pi)$, $\theta=0.15,2.24,4.34,0.89,2.98,5.08$
