Sorry for chiming in after all this time, but I can't let this pass. Scott, where on earth did you pick up your trig books ?
The mathematical functions sine and cosine are defined everywhere. There is absolutely 0 identity involving them which doesn't apply all over the real, or the complex plane. It's also true for other trigonometric functions, like tangent, with the obvious caveat that tangent goes to infinity when x-> pi/2 (or any congruent number, periodically). The infinite series for sine and cosine even converge all over the complex plane, since n! >> x^n for a given x, with n big enough (okay, the actual mathematical argument is a bit more complex, but that's the idea, n! goes to infinity a heck of a lot faster than x^n). I'm thinking you're confusing that stuff with either of two things: - since the trig functions are periodic, the reverse functions are obviously ambiguous, and you need some external input to solve the ambiguity. This makes for arbitrary definitions, and lots of fun in glueing the complex plane back together, and there's no way to avoid that, since it's the whole basis for the very useful theory of holomorphic functions and complex integration. And the math library usually has an atan2 function to take care of the ambiguity. - most software implementation of trig functions use approximation polynomial, usually a variation on Tchebichev polynomials, which converge much faster than the complete series, but MUST be restricted to a very small range, since they don't even converge to the right value outside this range. Now, the fact is that floating point arithmetic can be real tricky, and it's often necessary to (gasp) rework the equations and think to get some correct digits out of ill-applied trigonometric functions. But I haven't seen it that often in text books outside of specialized applied maths...
