Find the exact values of $sin$, $cos$, and $ an$ for the angle $225^{circ}$ using the unit circle.

To find the exact values of $\sin$, $\cos$, and $\tan$ for the angle $225^{\circ}$ using the unit circle, we first note that $225^{\circ}$ is in the third quadrant.

In the third quadrant, both sine and cosine values are negative, and tangent value is positive since tangent is the ratio of sine to cosine.

The reference angle for $225^{\circ}$ is $225^{\circ} – 180^{\circ} = 45^{\circ}$.

The values for $45^{\circ}$ are:

$ \sin 45^{\circ} = \frac{\sqrt{2}}{2} $

$ \cos 45^{\circ} = \frac{\sqrt{2}}{2} $

Therefore, the values in the third quadrant (for $225^{\circ}$) are:

$ \sin 225^{\circ} = – \frac{\sqrt{2}}{2} $

$ \cos 225^{\circ} = – \frac{\sqrt{2}}{2} $

$ \tan 225^{\circ} = \frac{\sin 225^{\circ}}{\cos 225^{\circ}} = \frac{- \frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = 1 $

Hence, the values are:

$ \sin 225^{\circ} = – \frac{\sqrt{2}}{2} $

$ \cos 225^{\circ} = – \frac{\sqrt{2}}{2} $

$ \tan 225^{\circ} = 1 $

To determine the exact trigonometric values for $225^{circ}$ on the unit circle, begin by recognizing that $225^{circ}$ resides in the third quadrant.

Here, sine and cosine are negative, while tangent remains positive.

The reference angle is calculated as:

$225^{circ} – 180^{circ} = 45^{circ}$

Reference angle values:

$ sin 45^{circ} = frac{sqrt{2}}{2} $

$ cos 45^{circ} = frac{sqrt{2}}{2} $

Converting to third quadrant values:

$ sin 225^{circ} = – frac{sqrt{2}}{2} $

$ cos 225^{circ} = – frac{sqrt{2}}{2} $

The tangent is:

$ an 225^{circ} = 1 $

For $225^{circ}$:

$ sin 225^{circ} = – frac{sqrt{2}}{2} $

$ cos 225^{circ} = – frac{sqrt{2}}{2} $

$ an 225^{circ} = 1 $

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