I. Prove sin θ= −sin(− θ), cos θ(− θ)

Let X′OX, Y′OY be rectangular axes, and let OP, OQ, starting from the position OX, describe angles θ, –θ. Then OP, OQ are in every case symmetrically placed with respect to OX. (Illustrate by taking positive and negative values of θ). Hence xP = xQ, yP = –yQ, and the results follow from the definitions of sine and cosine.