Unless I'm missing something the right half of that picture is simply wrong, not just another quadrant.
tan(-1/3 pi) = -sqrt(3), so arctan(sqrt(3)/-1) = arctan(-sqrt(3)) = -1/3 pi != 1/3 pi
Why do they use sqrt(3)/-1 instead of simply -sqrt(3)? I have the impression something else is going on. Are we seeing the last line of a multi-line expression? Why do we see the bottom curls of the parentheses but not the top curls?