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

To find the coordinates of the point on the unit circle at an angle $\theta$:

1. Use the parametric equations for the unit circle:

$x = \cos(\theta)$

$y = \sin(\theta)$

2. Substitute the given angle $\theta = \frac{2\pi}{3}$ into the equations:

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

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

Thus, the coordinates of the point are:

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

To determine the coordinates on the unit circle at an angle $ heta$:

1. Use the formulas for the x and y coordinates:

$x = cos( heta)$

$y = sin( heta)$

2. For $ heta = frac{5pi}{4}$, substitute into the equations:

$x = cosleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$

$y = sinleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$

The coordinates are:

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

To find coordinates for $ heta = frac{7pi}{6}$:

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

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

Coordinates:

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

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