Does anyone know anything about the trig functions:

C(a) = b s.t. cos(a+b) = b

S(a) = b s.t. sin(a+b) = b

I can't find any mention of these anywhere. One way to get the values is to iterate the cos and sin functions. For example, C(0) = cos(cos(cos(...))) ~= 0.73908513321516064165531208767387. (As good as Windows calculator can do.) S(0) = 0. Also, these functions are periodic with period 2*pi.

Hmm... Thinking about it just now, it seems like you could work out a Taylor series for these functions using the series for cos, cos(cos), etc... but IIRC function composition covolutes the coefficients, right? Can an infinite convolution be worked out?