Find the exact values of sine and cosine for the angle $frac{5pi}{4}$ using the unit circle.

To find the exact values of sine and cosine for the angle $\frac{5\pi}{4}$, we start by determining in which quadrant the angle lies.

The angle $\frac{5\pi}{4}$ is in the third quadrant because $\frac{5\pi}{4} > \pi$ and $\frac{5\pi}{4} < \frac{3\pi}{2}$.

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

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

We know that $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ and $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

Hence, for the third quadrant:

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

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

First, convert the angle $frac{5pi}{4}$ into degrees:

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

The angle 225 degrees is in the third quadrant, where both sine and cosine values are negative.

The reference angle is $225^circ – 180^circ = 45^circ$.

The sine and cosine of $45^circ$ are:

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

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

Since we are in the third quadrant, the sine and cosine values are:

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

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

Given angle $frac{5pi}{4}$ is in the third quadrant.

Referencing $frac{pi}{4}$:

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

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

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