This is really a trig question. I have been using atn(1/6) for our standard tapered ball joint angles. All derivations and definitions of the function say it should give me the angle. It's close, but when I force a 1:6 ratio angle it results in an angle of 9.527283˚. atn(1/6) yields 9.462322˚ Can anyone explain why there is a difference? Using a calculator yields the same thing as SW. The difference is enough for our tapers not to seat correctly.

What you have drawn is not atn(1/6). It's 2*atn(1/12), which is pretty close, but (as you have discovered) is not the same. Atn only applies to a right triangle. Yours is isosceles.