To determine the angles that satisfy the equation $\sin(x) = -0.5$, we can utilize the properties of the unit circle and the sine function.

First, it’s important to remember that the sine function is periodic, with a period of $360^\circ$ (or $2\pi$ radians). This periodicity implies that the sine of an angle will repeat every $360^\circ$. Additionally, the sine function exhibits symmetry about $180^\circ$ (or $\pi$ radians), which aids in identifying all possible solutions.

On the unit circle, the sine of an angle equals $-0.5$ at specific locations. The primary angle where $\sin(x) = -0.5$ is at $210^\circ$ (or $\frac{7\pi}{6}$ radians) in the third quadrant. Another angle where $\sin(x) = -0.5$ occurs at $330^\circ$ (or $\frac{11\pi}{6}$ radians) in the fourth quadrant. These two angles represent the principal solutions within one complete cycle of $360^\circ$.

To find all possible solutions, we can leverage the periodic nature of the sine function. For any integer $k$, the general solutions can be expressed as:

or, equivalently,

In these equations, $k$ represents any integer, which accounts for the infinite number of cycles the sine function can undergo. By adding multiples of $360^\circ$ (or $2\pi$ radians), we can identify all angles that satisfy the equation $\sin(x) = -0.5$.