### Instruction: Compute $\cos 225^\circ$.### Response: I know that $\cos 225^\circ$ is the same as $\cos (180^\circ + 45^\circ)$, because adding 180 degrees to an angle does not change its cosine value.
I also know that $\cos (A + B) = \cos A \cos B - \sin A \sin B$, by the cosine sum formula.
So I can apply this formula to $\cos (180^\circ + 45^\circ)$.
I need to know the values of $\cos 180^\circ$, $\sin 180^\circ$, $\cos 45^\circ$, and $\sin 45^\circ$.
I remember that $\cos 180^\circ = -1$ and $\sin 180^\circ = 0$, because 180 degrees is a straight angle and the cosine and sine are the x- and y-coordinates of a point on the unit circle.
I also remember that $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$, because 45 degrees is half of a right angle and the cosine and sine are the lengths of the legs of an isosceles right triangle with hypotenuse 1.
So I can plug these values into the cosine sum formula and simplify.
$\cos (180^\circ + 45^\circ) = \cos 180^\circ \cos 45^\circ - \sin 180^\circ \sin 45^\circ$
$= (-1)(\frac{\sqrt{2}}{2}) - (0)(\frac{\sqrt{2}}{2})$