Compute $\cos 180^\circ$.
To solve this problem, I will use the unit circle and the definition of cosine as the x-coordinate of a point on the unit circle. I know that the angle $180^\circ$ corresponds to the point $(-1,0)$ on the unit circle, because it is half a revolution counterclockwise from the positive x-axis. Therefore, $\cos 180^\circ$ is the same as the x-coordinate of $(-1,0)$, which is $-1$. 