Find the coordinates of the point that results from a $225^circ$ rotation counterclockwise around the origin on the unit circle.

To find the coordinates of a point on the unit circle after a $225^\circ$ rotation counterclockwise, we can use the trigonometric functions cosine and sine:

The general formula for finding the coordinates $(x, y)$ on the unit circle is:

$x = \cos(\theta)$

$y = \sin(\theta)$

For $\theta = 225^\circ$:

$x = \cos(225^\circ)$

$y = \sin(225^\circ)$

Since $225^\circ = 180^\circ + 45^\circ$, we can use reference angles:

$\cos(225^\circ) = \cos(180^\circ + 45^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}$

$\sin(225^\circ) = \sin(180^\circ + 45^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$

Therefore, the coordinates are:

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

We start by noting that a $225^circ$ rotation on the unit circle can be converted into radians:

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

On the unit circle, the coordinates $(x, y)$ after rotating $frac{5pi}{4}$ radians are found using:

$x = cos(frac{5pi}{4})$

$y = sin(frac{5pi}{4})$

For the angle $frac{5pi}{4}$, we know:

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

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

Thus, the coordinates are:

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

For a $225^circ$ rotation (or $frac{5pi}{4}$ radians) on the unit circle, the coordinates are:

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

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

Thus, the coordinates are:

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

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