### Instruction: Compute $\cos 225^\circ$.### Response: To find the cosine of 225 degrees, I need to use some trigonometric identities or properties.
One property that seems useful is that $\cos (180^\circ + \theta) = -\cos \theta$ for any angle $\theta$.
This means that I can rewrite $\cos 225^\circ$ as $\cos (180^\circ + 45^\circ)$.
Using the property, this is equal to $-\cos 45^\circ$.
Now, I need to recall what the cosine of 45 degrees is.
One way to remember it is to use a right triangle with sides of length 1 and hypotenuse of length $\sqrt{2}$.
This triangle has two 45-degree angles, and the cosine of either one is the adjacent side over the hypotenuse, which is $1/\sqrt{2}$.
Alternatively, I can use the unit circle and see that the point on the circle that corresponds to 45 degrees has coordinates $(\cos 45^\circ, \sin 45^\circ) = (1/\sqrt{2}, 1/\sqrt{2})$.
Either way, I get that $\cos 45^\circ = 1/\sqrt{2}$.
Therefore, $\cos 225^\circ = -\cos 45^\circ = -1/\sqrt{2}$.
This is the exact value of the cosine of 225 degrees, but I could also write it as a decimal approximation if needed.