Step 1 :Since the cosecant is the reciprocal of the sine function, we need to find two angles with a sine of $-\frac{\sqrt{3}}{2}$. We know that sine is negative in the third and fourth quadrants.
Step 2 :First, let's find the reference angle, which is the angle with a sine of $\frac{\sqrt{3}}{2}$. The reference angle is approximately $60^\circ$.
Step 3 :Now, we need to find the angles in the third and fourth quadrants with this reference angle. The angles are $240^\circ$ and $300^\circ$.
Step 4 :\(\boxed{240^\circ, 300^\circ}\)