Question

How does the unit circle allow the trigonometric functions to be defined for all real numbers instead of just for acute angles?

Answer 1

Every point in the unit circle is identified either by its coordinates $(x,y)$ or by the angle it forms with the x-axis, $\alpha$.

The trigonometric functions associate with every angle $\alpha$ and the correspondant $(x,y)$ coordinates the two values

$\cos(\alpha)=x,\quad\sin(\alpha)=y$

This procedure can be done for every angle $\alpha$, so you don't have to work with acute angles only.