Back in the day it was done via the unit circle. So the process goes something like this:

1. Define sin/cos as ratios in a triangle 2. Draw a unit circle with an appropriate triangle in the first quarter and note that for the vertex on the circle, x = cos alpha and y = sin alpha. 3. Assuming it holds for values of alpha > 90 deg, extend sin/cos definitions to match.

Don't quite remember how one got derivatives from that, because trig and calc were in separate modules. Possibly via the sum formula.