Intuitive formulation of trigonometry?
hackerblues: My best trick for being able to derive trig identies from scratch is to use the complex analysis identity:exp{i \theta} = cos(\theta) + i sin{\theta}where i^2 = -1.As an example, lets derive the double angle formula:cos(2\theta) + i sin{2\theta} = exp{i 2\theta}= (exp{i \theta})^2= (cos(\theta) + i sin{\theta})^2= cos(\theta)^2  + i 2 sin(\theta) cos(\theta) + i^2 sin{\theta}^2= cos(\theta)^2  - sin{\theta}^2 + i 2 sin(\theta) cos(\theta)Since the real part must equal the real part and the imaginary part equal the imaginary part we conclude that:cos(2\theta) = cos(\theta)^2 - sin{\theta}^2andsin{2*\theta} = 2 sin(\theta) cos(\theta)~~~The general procedure is:cos(complicated) = ???1) cos(complicated) + i sin(complicated) = exp{i complicated}2) Break it up:  exp{i complicated} =  exp{i simple1 + ... + i simple4} =  exp{i simple1}...exp{i simple4}3) Translate back to sins and cos: (cos(simple1) + i sin(simple1))...(cos(simple4) + i sin(simple4))4) Then multiply them out and choose the real part to get cos(complicated) or the imaginary part to get sin(complicated)where the simple1,...,simple4 was an arbitrarily chosen number of simple bits.