$Calculate the coordinates and angles on the unit circle$

To find the coordinates of the point on the unit circle corresponding to an angle of $\frac{5\pi}{4}$ radians, we use the trigonometric functions sine and cosine. For any angle $\theta$, the coordinates are given by:

$ (\cos \theta, \sin \theta) $

For $\theta = \frac{5\pi}{4}$, we have:

$ \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $

$ \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $

Thus, the coordinates of the point are:

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

To determine the coordinates at an angle of $frac{7pi}{6}$ on the unit circle, we use the cosine and sine functions. The coordinates for any angle $ heta$ are:

$ (cos heta, sin heta) $

For $ heta = frac{7pi}{6}$, the calculations are:

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

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

The resulting coordinates are:

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

To find the coordinates at an angle of $frac{2pi}{3}$ on the unit circle, use:

$ (cos heta, sin heta) $

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

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

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

So, the coordinates are:

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

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