For trigonmetric identities you might want to look at Eulers Identity In Complex Numbers where e^(iax) = cos(ax) + isin(ax) = [e^(ix)]^a. This is used to show a lot of identities.
In terms of functions, just note that sin(x+2pi) = sin(x) and cos(x+2pi) = cos(x) with tan(x+pi) = tan(x) and also use the fact that e^(ix) = cos(x) + isin(x).
Right angled triangles can be used to find formulas for things like arcsin(cos(x)) and arccos(sin(x)) and things involving compositions of inverse trig with trig functions.
The right angle is just a special case of the circle geometrically.
Those should give you somewhere to start.