I want to prove that sin(−θ)=−sin(θ)sin(−θ)=−sin(θ) and cos(−θ)=cos(θ)cos(−θ)=cos(θ) are true without using the summation identities for sin(x±y)sin(x±y) or cos(x±y)cos(x±y). This is easy enough to do with those formulas if I use 0−x0−x as my arguments there.
Without the formulas: Is it usually just by virtue of the unit circle? If I move −θ−θ around (i.e. clockwise) on the unit circle, I'll end up at some position and the sinsin will correspond to that yy value. If I had instead moved θθ around counter-clockwise, I'd be at the same point but just the vertical mirror (this is a result of us starting at (1,0)(1,0) on the xx-axis), so the yy position would be the negative flip of the first one I found. So that one makes sense as to why sin(−θ)=−sin(θ)sin(−θ)=−sin(θ).
For coscos, for the xx-coordinates, the vertical mirroring doesn't impact this in either case, so cos(θ)=cos(−θ)cos(θ)=cos(−θ).
These are a little informal even though it seems to be right, but is there any other way to derive these results without using the summation formula?
