Determine the coordinates of the point on the unit circle corresponding to an angle of $120^{circ}$.

To find the coordinates of the point on the unit circle corresponding to an angle of $120^{\circ}$, we first convert the angle to radians. The conversion formula is:

$ \theta (\text{radians}) = \theta (\text{degrees}) \times \frac{\pi}{180} $

So,

$ 120^{\circ} \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3} $

Using the unit circle, the coordinates for an angle of $\frac{2\pi}{3}$ are given by:

$ (x, y) = (\cos(\frac{2\pi}{3}), \sin(\frac{2\pi}{3})) $

From trigonometric values:

$ \cos(\frac{2\pi}{3}) = -\frac{1}{2} $

$ \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} $

So, the coordinates are:

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

To determine the coordinates on the unit circle for an angle of $120^{circ}$, we must convert the angle from degrees to radians:

$ 120^{circ} = 120 imes frac{pi}{180} = frac{2pi}{3} $

From the unit circle properties, we know:

$ cos(frac{2pi}{3}) = -frac{1}{2} quad ext{and}quad sin(frac{2pi}{3}) = frac{sqrt{3}}{2} $

Thus, the coordinates are:

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

To find the point on the unit circle for $120^{circ}$:

$ 120^{circ} = frac{2pi}{3} $

Coordinates are:

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

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