Find the sine and cosine of $45^circ$ and $135^circ$ using the unit circle.

Using the unit circle, we know that:

$45^\circ = \frac{\pi}{4}$

and

$135^\circ = \frac{3\pi}{4}$

For $45^\circ$:

$\sin 45^\circ = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

$\cos 45^\circ = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

For $135^\circ$:

$\sin 135^\circ = \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}$

$\cos 135^\circ = \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$

From the properties of the unit circle, we know:

$45^circ$ is in the first quadrant, and $135^circ$ is in the second quadrant.

For $45^circ$:

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

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

For $135^circ$:

$sin 135^circ = frac{sqrt{2}}{2}$

$cos 135^circ = -frac{sqrt{2}}{2}$

The sine and cosine for:

$45^circ$ are $frac{sqrt{2}}{2}$ and $frac{sqrt{2}}{2}$ respectively.

$135^circ$ are $frac{sqrt{2}}{2}$ and $-frac{sqrt{2}}{2}$ respectively.

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