Determine the coordinates of the point where the terminal side of an angle $ heta = frac{5pi}{4}$ radians intersects the unit circle.

To find the coordinates of the point where the terminal side of an angle $\theta = \frac{5\pi}{4}$ intersects the unit circle, we start by expressing the angle in degrees:

$\theta = \frac{5\pi}{4} \cdot \frac{180}{\pi} = 225^{\circ}$

This angle is in the third quadrant where both sine and cosine are negative. For the unit circle, we can use the reference angle:

$ 225^{\circ} – 180^{\circ} = 45^{\circ} $

The coordinates corresponding to $45^{\circ}$ are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Since $225^{\circ}$ is in the third quadrant:

$ \cos(225^{\circ}) = -\frac{\sqrt{2}}{2}, \sin(225^{\circ}) = -\frac{\sqrt{2}}{2} $

Therefore, the coordinates are:

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

Let’s determine the coordinates of the point where the terminal side of $ heta = frac{5pi}{4}$ intersects the unit circle.

First, recognize that $frac{5pi}{4}$ radians is equal to:

$frac{5pi}{4} cdot frac{180}{pi} = 225^{circ}$

This angle lies in the third quadrant. The reference angle is calculated as:

$225^{circ} – 180^{circ} = 45^{circ}$

For a $45^{circ}$ angle, the coordinates on the unit circle are:

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

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

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

For $ heta = frac{5pi}{4}$ radians, convert to degrees:

$ heta = 225^{circ}$

This is in the third quadrant:

Reference angle $= 45^{circ}$:

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

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