How do you calculate inverse sin, cos, and tan without a calculator?

My math book teaches me how to divide the opposite by the adjacent, opposite by the hypotenuse, adjacent by the hypotenuse etc..... this can be done without a calculator. but then it says to figure out the angle you press the inverse sin,cos,tan key on the calculator and it will display the angle. i want to do this without a calculator. i find it fascinating and do take a genuine interest in triangles and how useful they are, but if someday i'm out on a jobsite with no calculator then all this knowledge dont do much for me. i need to be able to do EVERYTHING without a calculator to be happy with myself.THANKS