Find the coordinates of a point on the unit circle at a given angle.

To find the coordinates of a point on the unit circle at an angle of 120 degrees, we first convert the angle to radians, as the unit circle typically uses radians.

$ \theta = 120^{\circ} = \frac{2\pi}{3} \text{ radians} $

The coordinates on the unit circle are given by:

$ (x, y) = (\cos \theta, \sin \theta) $

Substituting our angle:

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

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

Thus, the coordinates of the point on the unit circle at 120 degrees are:

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

To determine the coordinates of a point on the unit circle at 135 degrees, we first convert the angle to radians:

$ heta = 135^{circ} = frac{3pi}{4} ext{ radians} $

The coordinates on the unit circle are:

$ (x, y) = (cos heta, sin heta) $

Substituting our angle:

$ x = cos left(frac{3pi}{4}
ight) = -frac{sqrt{2}}{2} $

$ y = sin left(frac{3pi}{4}
ight) = frac{sqrt{2}}{2} $

Thus, the coordinates of the point on the unit circle at 135 degrees are:

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

Find the coordinates of a point on the unit circle at 210 degrees:

$ heta = 210^{circ} = frac{7pi}{6} ext{ radians} $

Coordinates are given by:

$ (x, y) = (cos heta, sin heta) $

$ x = cos left(frac{7pi}{6}
ight) = -frac{sqrt{3}}{2} $

$ y = sin left(frac{7pi}{6}
ight) = -frac{1}{2} $

Thus, the coordinates are:

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

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