> BUT, in actually computing > the real and imaginary parts of 2 + 11i, I am left evaluating > cos(arctan 11/2) (or sin of same). Is there a way to compute > this analytically (series?) so as to get exactly 2 or 1, rather > than a numerical approximation? [I suspect so, as my calculator > gives the exact answers.]
Any (hand-held) calculator will hardly be able to give the exact answer (even if the display seems to say so), nor will be the usual floating-based programming languages or spreadsheets (due to the inherent limitations of floating point arithmetic with the usually implemented constant "word-length"), but using a computer algebra system usually will do.