Answer
$240^{o},\quad 300^{o}$
Work Step by Step
Reference Angle $\theta^{\prime}$ for $\theta$ in $(0^{\mathrm{o}},\ 360^{\mathrm{o}})$
$\left[\begin{array}{lllll}
Quad.: & I & II & III & IV\\
\theta' & \theta & 180^{o}-\theta & \theta-180^{o} & 360^{o}-\theta
\end{array}\right]$
Cosecant is negative in quadrants III and IV.
Browsing through: Function Values of Special Angles,
we find that $\displaystyle \csc 60^{o}=\frac{2\sqrt{3}}{3}$, so the reference angle is $60^{o}$
In quadrant III, $\theta^{\prime}=\theta-180^{o}$ so
$\theta=180^{o}+\theta^{\prime}=180^{o}+60^{o}=240^{o}$
In quadrant IV, $\theta^{\prime}=360^{o}-\theta$ so
$\theta=360^{o}-\theta^{\prime}=360^{o}-60^{o}=300^{o}$
