Ask HN: Intuitive formulation of trigonometry?

Hi again HN,

In keeping with my theme of math questions, I ask you, is there an intuitive formulation/explanation of the rules of trigonometry?

I understand the basics (def of sin/cos, sin^2+cos^2=1, etc.) in terms of the unit circle, but as soon as you get to things like double angle, half angle, the other identities, estimating the functions, and so on, you've lost me totally, because the proofs I've seen of those are just random things based on triangles and then you just memorize the formula and are done with it. Unfortunately, I hate memorizing things without being able to at least grasp the derivation from first principles. Any references?