A Moving Circle's Relationship to Trigomentry

We were moving a concrete slap by rolling it on logs, and it reminded me of how a circle maps out a sin or cos.  It made me wonder if it is possible to derive the trigonometric functions from following a point on a circle as it moves at a constant velocity.

It is sort of like what is happening on the left, but imagine the entire circle moving, and that following a point on that circle maps out the trig function on the right-hand side.