# Definition:Inverse Tangent

## Definition

### Real Numbers

Let $x \in \R$ be a real number.

The inverse tangent of $x$ is the multifunction defined as:

$\tan^{-1} \left({x}\right) := \left\{{y \in \R: \tan \left({y}\right) = x}\right\}$

where $\tan \left({y}\right)$ is the tangent of $y$.

### Complex Plane

The inverse tangent is a multifunction defined on $S$ as:

$\forall z \in S: \tan^{-1} \left({z}\right) := \left\{{w \in \C: \tan \left({w}\right) = z}\right\}$

where $\tan \left({w}\right)$ is the tangent of $w$.

## Arctangent

From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From the same source, we also have that:

$\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$
$\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$

Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\R$.

This function is called arctangent of $x$ and is written $\arctan x$.

Thus:

The domain of $\arctan x$ is $\R$
The image of $\arctan x$ is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

## Also see

• Results about the inverse tangent function can be found here.