Quote:
Originally Posted by

**miceblue**
I should re-learn Fourier series. The teacher who taught it to us didn't do a very good job I thought. Do you self-teach yourself a lot of things? My friend who is a math wizard (he's planning to get a bioengineering and electrical engineering degree with a math minor) makes proofs for fun and he showed me how to calculate the sine/cosine of any angle using some clever angles in a circle.

I self teach myself stuff if I can't be arsed to go to a lecture, and it's pretty doable. Though it's really more efficient to study math from lectures. In both cases a lot of practice is the key to success.

You can't explicitly calculate the sine/cosine of any angle, not even for rational products of pi. You can get pretty far, but you won't get there exactly.

For example it's possible to find the cos(x/2) if cos(x) is known. cos(x/3) may be possible, but in any case cos(x/5) or cos(x/7) do not have an explicit identity because they require you to find the roots of a respectively fifth and seventh degree polynomial, for which a formula in terms of elementary functions does not exist. And even if it does exist in the case of those formulas (although I'm pretty sure it doesn't), then it sure won't exist for an arbitrarily large prime number like cos(x/739). In order to calculate the cosine of any rational angle, you have to be able to get from the cosine of an angle you can derive geometrically to your angle of choice trigonometrically. This would require half/third/nth angle formulas of the prime factorization of the denominator of the ratio of pi of the angle you're trying to cosine from, which is not always possible as I just illustrated.

Edited by Tilpo - 4/23/13 at 2:16pm