Simplification of ArcTan

Question

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?

Nasser · Accepted Answer

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]

David G. Stork · Answer

Assuming[0 < x < π, Simplify@ArcTan[x, 0]]

Robert Nowak · Answer

ArcTan simplifications are somehow limited:
Simplify[ArcTan[x, y], {x == y, 0 < y, 0 < x}]

Sadly only gives:
ArcTan[y, y]

3 Answers