Arctan and Arccot — Cheat sheet

Definitions

$y = \arctan x \iff \tan y = x$ , with $D = (-\infty, \infty)$ and $R = \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ .

$y = \text{arccot}\,x \iff \cot y = x$ , with $D = (-\infty, \infty)$ and $R = (0, \pi)$ .

Both accept every real number. Both ranges are open: the ends are approached, never reached.

Negative inputs

Function	Rule	Like
$\arctan(-x) = -\arctan x$	odd - flips sign	arcsin
$\text{arccot}(-x) = \pi - \text{arccot}\,x$	reflects - never negative	arccos

$-45^\circ$ and $315^\circ$ are the same point, but arctan reports $-45^\circ$ - the reading inside its range.

Exact values

$x$	$-\sqrt{3}$	$-1$	$-\dfrac{1}{\sqrt{3}}$	$0$	$\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$
$\arctan x$	$-60^\circ$	$-45^\circ$	$-30^\circ$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$
$\text{arccot}\,x$	$150^\circ$	$135^\circ$	$120^\circ$	$90^\circ$	$60^\circ$	$45^\circ$	$30^\circ$

$\text{arccot}\,0 = 90^\circ$ , while $\arctan 0 = 0$ . That difference catches people out.

Asymptotes

Arctan flattens towards $\pm\dfrac{\pi}{2}$ ; arccot towards $0$ and $\pi$ . Every real input, and the output never leaves a fixed interval - the standard example when limits at infinity are taught.

Identities

Identity	Condition
$\tan(\arctan x) = x$	any real $x$
$\cot(\text{arccot}\,x) = x$	any real $x$
$\arctan x + \text{arccot}\,x = \dfrac{\pi}{2}$	any real $x$ - the conversion between them

All four inverses, side by side

Function	Range	Quadrants	Negative input
$\arcsin$	$\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$	first, fourth	negative angle
$\arctan$	$\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$	first, fourth	negative angle
$\arccos$	$[0, \pi]$	first, second	obtuse angle
$\text{arccot}$	$(0, \pi)$	first, second	obtuse angle

When there is no special angle

Name it $\alpha$ . For arctan, $\tan\alpha$ is opposite over adjacent; for arccot, $\cot\alpha$ is adjacent over opposite. Draw the triangle, let Pythagoras give the hypotenuse, read off whatever is asked. The angle itself is never needed.