Find the point on the unit circle where the angle is $frac{pi}{3}$ and show all steps to verify the trigonometric coordinates.

To find the point on the unit circle where the angle is $\frac{\pi}{3}$, we start by noting that the unit circle has a radius of 1. The coordinates of any point on the unit circle can be found using the trigonometric functions cosine (cos) and sine (sin).

We know that for an angle $\theta$:

$ x = \cos(\theta) $

$ y = \sin(\theta) $

For $\theta = \frac{\pi}{3}$:

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

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

Therefore, the coordinates of the point on the unit circle where the angle is $\frac{\pi}{3}$ are:

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

To determine the point on the unit circle at angle $frac{pi}{3}$, we utilize the definitions of the trigonometric functions cosine and sine for this specific angle. The point’s coordinates can be given by:

$ (cos( heta), sin( heta)) $

Where $ heta = frac{pi}{3}$:

$ cosleft(frac{pi}{3}
ight) = frac{1}{2} $

$ sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2} $

Thus, the coordinates of the desired point are:

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

The point on the unit circle for $ heta = frac{pi}{3}$ is found by evaluating:

$ cosleft(frac{pi}{3}
ight) = frac{1}{2}, sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2} $

Hence, the coordinates are:

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

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