”Given

First, recognize that $\sin(\theta) = -\frac{1}{2}$ means that $\theta$ corresponds to either $210^\circ$ or $330^\circ$ in degrees, or $\frac{7\pi}{6}$ or $\frac{11\pi}{6}$ in radians. Since $\cos(\theta) = -\frac{\sqrt{3}}{2}$, we identify that $\theta$ must be in the third quadrant.

Therefore, the coordinates are:

$ (x, y) = \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) $

The angle $\phi$ in the range $[0, 2\pi]$ that satisfies the same conditions is $\frac{7\pi}{6}$.

First, we identify that the conditions $sin( heta) = -frac{1}{2}$ and $cos( heta) = -frac{sqrt{3}}{2}$ correspond to specific standard angles. The angle $ heta$ must be $frac{7pi}{6}$ since it falls in the third quadrant. We then confirm that the point on the unit circle is:

$ (x, y) = left( -frac{sqrt{3}}{2}, -frac{1}{2}
ight) $

Next, we need to find another angle $phi$ in the range $[0, 2pi]$ with the same sine and cosine. The angle $phi$ that meets this requirement is $frac{7pi}{6}$ since it is the principal angle.

To find the coordinates, note that $sin( heta) = -frac{1}{2}$ and $cos( heta) = -frac{sqrt{3}}{2}$ implies:

$ (x, y) = left( -frac{sqrt{3}}{2}, -frac{1}{2}
ight) $

The corresponding angle $phi$ in the range $[0, 2pi]$ is $frac{7pi}{6}$.

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