### Instruction: Compute $\cos 135^\circ$.### Response: I know that cosine is related to the x-coordinate of a point on the unit circle, so I need to find a point that has an angle of 135 degrees with the positive x-axis.
I also know that 135 degrees is the same as 90 degrees plus 45 degrees, so I can use the angle sum formula for cosine: $\cos (90^\circ + 45^\circ) = \cos 90^\circ \cos 45^\circ - \sin 90^\circ \sin 45^\circ$.
I remember that $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$, so I can simplify the formula to $\cos (90^\circ + 45^\circ) = -\sin 45^\circ$.
I also remember that $\sin 45^\circ = \frac{\sqrt{2}}{2}$, which is the same as $\cos 45^\circ$, so I can write $\cos (90^\circ + 45^\circ) = -\frac{\sqrt{2}}{2}$.
This is the x-coordinate of the point on the unit circle that has an angle of 135 degrees with the positive x-axis, so this is the answer.