Find the coordinates of a point on the unit circle, and determine the quadrant $ ext{Find the coordinates of a point on the unit circle, and determine the quadrant}$

Let the point on the unit circle have coordinates $(x, y)$, and let the angle it makes with the positive x-axis be $\theta = \frac{5\pi}{4}$ radians.

To find the coordinates:

$x = \cos \frac{5\pi}{4}$

$y = \sin \frac{5\pi}{4}$

Using the unit circle properties:

$x = -\frac{\sqrt{2}}{2}$

$y = -\frac{\sqrt{2}}{2}$

Since both coordinates are negative, the point lies in the third quadrant.

Given the angle $ heta = 300^{circ}$, convert it to radians: $ heta = frac{5pi}{3} $ radians.

To find the coordinates of the point on the unit circle:

$x = cos frac{5pi}{3}$

$y = sin frac{5pi}{3}$

Using the unit circle properties:

$x = frac{1}{2}$

$y = -frac{sqrt{3}}{2}$

Since x is positive and y is negative, the point lies in the fourth quadrant.

Given the angle $135^{circ}$ or $ heta = frac{3pi}{4}$ radians, find the coordinates of the point on the unit circle:

$x = cos frac{3pi}{4}$

$y = sin frac{3pi}{4}$

$x = -frac{sqrt{2}}{2}$

$y = frac{sqrt{2}}{2}$

The point lies in the second quadrant.

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