I said I would have something more constructive later, and I don't think I ever got around to it. So in email yesterday, someone asked me about calculating sine by hand, and because the network went down while i was composing the email, I decided to give everybody the benefit of my experience and put up the techniques here.
Unit conversion is done.
Make sure the angle is between -Pi and Pi. Obviously, you have to stop at some point. You can't keep calculating forever. This is an incredibly fast way to calculate sine, and you can calculate to any level of precision desired. This is how computers and calculators do it, more or less.