# Simplification of ArcTan

Mathematica provides the following function to compute the arctangent of a number, preserving quadrant information:

ArcTan[x,y]


for real $x$ and $y$, when $y = 0$ and $x \ne 0$, $\arctan(y/x) = 0$. However, Mathematica doesn't perform this simplification:

FullSimplify[ArcTan[x,0], Assumptions->{x \[Element] Reals}]
(* ArcTan[x, 0] *)


Is this an error on my part or is there an underlying reason why this simplification isn't performed?

It is because M does not know the actual numerical value of x (or rather its sign).

Let say x was 1

ArcTan[1,0]


But what if x was -1 ?

ArcTan[-1,0]


• Ah, a mathematical error on my end then. I suppose I was overthinking it. Thanks! – Billy Kalfus Nov 21 '17 at 22:43
Assuming[0 < x < π, Simplify@ArcTan[x, 0]]


ArcTan simplifications are somehow limited:

Simplify[ArcTan[x, y], {x == y, 0 < y, 0 < x}]


ArcTan[y, y]