pi / 180 is what you'd use if you were making the conversion from degrees. i don't think he is doing that, his input is just in radians, actually the input is the multiplier, like 1/4 * PI or 2 * PI, etc.
I didn't read the OP's prompts, so that's my bad.
@OP: To illustrate the problem of approximations that jephthah was explaining look at this graph of sin x versus it's series approximation http://en.wikipedia.org/wiki/File:Taylorsine.svg. It only holds over a limited interval. I think you can in fact improve this a bit by specifying a point about which you are approximating the function (see http://en.wikipedia.org/wiki/Taylor_series#Definition where they have the (x-a) instead of x) but there's only so much you can do.