Arctan and Arccot — Essentials

Definitions

$y = \arctan x \iff \tan y = x$ , with $R = \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ . $y = \text{arccot}\,x \iff \cot y = x$ , with $R = (0, \pi)$ . Both accept every real number; both ranges are open.

Traps

Trap	Rule
Arctan is odd	$\arctan(-x) = -\arctan x$
Arccot is not	$\text{arccot}(-x) = \pi - \text{arccot}\,x$ - reflects, like arccos
At zero	$\text{arccot}\,0 = \dfrac{\pi}{2}$ , while $\arctan 0 = 0$

Values

$x$	$0$	$\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$
$\arctan x$	$0$	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$
$\text{arccot}\,x$	$\dfrac{\pi}{2}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{6}$

Negative inputs: arctan flips sign, arccot reflects.

Behaviour at infinity

$\arctan x \to \pm\dfrac{\pi}{2}$ ; $\text{arccot}\,x \to 0$ or $\pi$ . Horizontal asymptotes, never reached.

The conversion

$\arctan x + \text{arccot}\,x = \dfrac{\pi}{2}$ for every real $x$ - the fastest and safest way to turn one into the other.