calculating trig. ratios w/ out a calculator

Hello,

So i know how to construct a table for 0, 30, 45, 60, and 90 degree angles (and the respective radians) that shows the trig. ratios for these degree/radian values. If i am asked to determine the trig. ratio for 30 degrees without a calculator i'm cool w/ that. What I cannot always do is determine trig ratios for csc 120 (or another degree value other than the values i construct the table for). I figured I could just take the inverse of 2 times the sin of 60 degrees, but this didn't result in the correct answer as determined w/ the use of a calculator. I would think this would always work (that is, adding up the trig ratios for an angle to determine a greater angle that is a multiple of the smaller angles). Sometimes it works, but sometimes it doesn't.

Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).