Find the coordinates of the point on the unit circle corresponding to an angle of $ frac{5pi}{4} $ radians.

To find the coordinates of the point on the unit circle corresponding to the angle $ \frac{5\pi}{4} $ radians, we need to use the unit circle properties.

The angle $ \frac{5\pi}{4} $ radians is in the third quadrant.

The reference angle for $ \frac{5\pi}{4} $ is $ \pi/4 $ radians.

In the third quadrant, both sine and cosine values are negative.

From the unit circle, we know:

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

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

Therefore, the coordinates for $ \frac{5\pi}{4} $ are:

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

To determine the coordinates of the point on the unit circle at an angle of $ frac{5pi}{4} $, we should remember that the unit circle’s radius is 1.

The angle $ frac{5pi}{4} $ places the terminal side in the third quadrant, with a reference angle of $ pi/4 $.

In the third quadrant, both the x and y coordinates are negative.

For the reference angle $ pi/4 $, the sine and cosine are:

$ sin left( frac{pi}{4}
ight) = frac{sqrt{2}}{2} $

$ cos left( frac{pi}{4}
ight) = frac{sqrt{2}}{2} $

Thus, the coordinates are:

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

To find the point on the unit circle at $ frac{5pi}{4} $, note it is in the third quadrant.

Reference angle is $ pi/4 $. Sine and cosine for this reference angle are $ frac{sqrt{2}}{2} $.

Therefore, coordinates are:

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

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