i stumbled upon this exact trigonometric values, very good!
Exact trigonometric constants - Wikipedia, the free encyclopedia
[edit] 3°: 60-sided polygon
[edit] 6°: 30-sided polygon
[edit] 9°: 20-sided polygon
[edit] 12°: 15-sided polygon
[edit] 15°: dodecagon
[edit] 18°: decagon
[edit] 21°: sum 9° + 12°
[edit] 22.5°: octagon
[edit] 24°: sum 12° + 12°
[edit] 27°: sum 12° + 15°
[edit] 30°: hexagon
[edit] 33°: sum 15° + 18°
[edit] 36°: pentagon
[edit] 39°: sum 18° + 21°
[edit] 42°: sum 21° + 21°
[edit] 45°: square
[edit] 60°: triangle