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.