Find the coordinates of points where the angle is $ frac{2pi}{3} $ on the unit circle

To find the coordinates of the points where the angle is $ \frac{2\pi}{3} $ on the unit circle, we use the unit circle definition where any point can be given by $(\cos(\theta), \sin(\theta))$.

Here, $ \theta = \frac{2\pi}{3} $.

Therefore, the coordinates are:

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

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

Thus, the coordinates are:

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

Using the unit circle, we need to find the coordinates for the angle $ frac{2pi}{3} $. The coordinates on the unit circle are given by $ (cos( heta), sin( heta)) $.

Given:

$ heta = frac{2pi}{3} $

We can find:

$ cos left( frac{2pi}{3}
ight) = cos left( pi – frac{pi}{3}
ight) = -cos left( frac{pi}{3}
ight) = -frac{1}{2} $

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

Hence, the coordinates are:

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

The unit circle coordinates for $ heta = frac{2pi}{3} $ are found using $ (cos( heta), sin( heta)) $.

Thus:

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

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

Coordinates:

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

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