This gets into SOH CAH TOA.

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In the unit circle, the length of the hypotenuse is 1. Anything divided by 1 is itself, so we don't have to really think about the division: the sine is just the opposite leg, and the cosine is just the adjacent leg.

But even though we don't have to think about the division in the unit circle, technically it still happens. This is what keeps rhe sine and cosine from changing when the size of the circle does. All circles are geometrically similar, so the sizes of the triangle's legs change in proportion to one another. That means their ratios -the sine and cosine- don't change. The unit circle just makes the math a little easier.