[Borodog]

Well, here's how to do sines.

For small angles, less than 35 degrees, you can just divide the angle in degrees by 60. For example sin(15) = 0.25, to about 3%.

For angles greater than 30 degrees but less than 90, the trick is a bit harder, but gives great results. Take 90 minus the angle, divide by 60, square the result, and subtract half that from 1.

So, sin(75) = 1 - 0.5*[(90-75)/60}^2 = 1 - 0.5*[15/60]^2 = 1 - 0.5*[1/4]^2 = 31/32, to within 0.3%.

sin(45) = 23/32, to within 2%.

Since cos(x) = sin(90-x), you can use these same tricks to do cosines.

For small angles, tan(x) ~ sin(x), so you can use the same divide by 60 trick, but it's less accurate and becomes really inaccurate fast. For even 10 degrees, this method is off by almost 5.5%

Instead, you can use tan(x)=sin(x)/cos(x)=sin(x)/sin(90-x) and the above tricks. So, tan(15)= sin(15)/sin(75) = 0.25/(31/32) = 8/31, to less than 4%.

The most useful of these is obviously the first, for small angle sines.