Here is an amusing method for finding the sine of an angle that makes intuitive sense:

suppose we  want to find the sine of 1 radian, about 57 degrees.

First we choose a power of 2 that is large relative to the angle. I'll choose 32.

1/32 is fairly small and so sin(1/32)  ≈ 1/32, and cos(1/32)  ≈ 1.

Now that we know sin(1/32) and cos(1/32) we can use 5 applications of the double angle identities: sin(2x) = 2sin(x)cos(x) and cos(2x)=cos^2(x) - sin^2(x)  to get the final sine and cosine!
I worte a simple program to do it for me. Have a look:

sin(1/32): 0.03125 cos: 1
sin(2/32): 0.0625 cos: 0.999023
sin(4/32): 0.124878 cos: 0.994142
sin(8/32): 0.248293 cos: 0.972723
sin(16/32): 0.48304 cos: 0.884541
final (32/32): sin: 0.854537 cos: 0.549085

results by windows calculator:
sin: 0.8414709848078965066525023216303, cos: 0.54030230586813971740093660744298

Hey! Thats pretty close! Of course! you can make it more accurate by choosing a larger power of 2. Choosing 2^10 for instance yields: sin: 0.841881 cos: 0.540566

Its hardly the fastest way to do it, but its easy to understand why it works and I think that makes it fun!
Originally I envisioned this as a series of line reflections. But I eventually realized I was essentially using the double angle formulas. Maybe I'll post a diagram later.

Last edited by mikau (2009-12-08 19:23:22)