# 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

### Real Numbers

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

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 $g: \left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right) \to \R$ be the restriction of $\tan x$ to $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ 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 $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

### Complex Plane

The principal branch of the complex inverse tangent function is defined as:

$\map \arctan z := \dfrac 1 {2 i} \, \map \Ln {\dfrac {i - z} {i + z} }$

where $\Ln$ denotes the principal branch of the complex natural logarithm.

## Also see

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