### Instruction: Compute $\cos 315^\circ$.### Response: I know that $\cos 315^\circ$ is the same as $\cos (-45^\circ)$, since they are coterminal angles.
I also know that $\cos -\theta = \cos \theta$, since the cosine function is even.
So, I can simplify the problem to $\cos 45^\circ$.
Now, I recall that 45 degrees is a special angle, and it is part of a 45-45-90 right triangle.
In such a triangle, the hypotenuse is $\sqrt{2}$ times the length of either leg, and the legs are equal.
Therefore, the cosine of 45 degrees is the ratio of the adjacent leg to the hypotenuse, which is $\frac{1}{\sqrt{2}}$.
I could also rationalize the denominator by multiplying by $\frac{\sqrt{2}}{\sqrt{2}}$, and get $\frac{\sqrt{2}}{2}$.
Either way, this is the final answer.