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.
Of course, you could always just define sin and cos for acute angles only and then extend the definition with trig identities, but that seems rather unmotivated too.