$ ext{Find the coordinates of point P on the unit circle}$

Given a point P on the unit circle at an angle $\theta = \frac{\pi}{3}$ radians, we need to find its coordinates.

The coordinates of a point on the unit circle are given by $ (\cos \theta, \sin \theta) $.

So, we will use the values of cosine and sine for $\theta = \frac{\pi}{3}$.

$ \cos \frac{\pi}{3} = \frac{1}{2} $

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

Therefore, the coordinates of point P are:

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

To find the coordinates of a point P on the unit circle at an angle $ heta = frac{pi}{4}$ radians, we use the unit circle properties.

The coordinates are $ (cos heta, sin heta) $.

For $ heta = frac{pi}{4}$, we calculate:

$ cos frac{pi}{4} = frac{sqrt{2}}{2} $

$ sin frac{pi}{4} = frac{sqrt{2}}{2} $

Hence, the coordinates of point P are:

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

For a point P on the unit circle at $ heta = frac{pi}{6}$ radians:

The coordinates are:

$ cos frac{pi}{6} = frac{sqrt{3}}{2} $

$ sin frac{pi}{6} = frac{1}{2} $

So, the coordinates of point P are:

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

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