Convert the angle $225^circ$ to radians, find its coordinates on the unit circle, and determine the sine, cosine, and tangent values.

To convert 225 degrees to radians, we use the conversion factor $\frac{\pi}{180}$:

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

Next, identify the coordinates on the unit circle at $\frac{5\pi}{4}$ radians:

The angle $\frac{5\pi}{4}$ is in the third quadrant where both sine and cosine are negative. The reference angle is $\frac{\pi}{4}$ with coordinates $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$.

Thus, the Cartesian coordinates are:

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

Finally, calculate the trigonometric values:

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

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

$\tan(\frac{5\pi}{4}) = \frac{\sin(\frac{5\pi}{4})}{\cos(\frac{5\pi}{4})} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$

Starting with the conversion of 225 degrees to radians:

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

Now, locate $frac{5pi}{4}$ on the unit circle:

The angle $frac{5pi}{4}$ lies in the third quadrant. The coordinates for any $frac{pi}{4}$ related angle in the third quadrant are $(-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2})$.

Thus:

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

The trigonometric values are:

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

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

$ an(frac{5pi}{4}) = 1$

Convert the angle:

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

Find the coordinates on the unit circle:

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

Calculate trigonometric values:

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

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

$ an(frac{5pi}{4}) = 1$

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