# Definition:Inverse Tangent

## Definition

### Real Numbers

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

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

- $\map {\tan^{-1} } x := \set {y \in \R: \map \tan y = x}$

where $\map \tan y$ 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 $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

## Terminology

There exists the popular but misleading notation $\tan^{-1} x$, which is supposed to denote the **inverse tangent function**.

However, note that as $\tan x$ is not an injection, it does not have a well-defined inverse.

The $\arctan$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\tan^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname {Tan}^{-1}$ (with a capital $\text T$) is taken to mean the same as $\arctan$.

In computer software packages, the notation $\operatorname {atan}$ or $\operatorname {atn}$ can sometimes be found.

Some sources hyphenate: **arc-tangent**.

## Also see

- Definition:Inverse Sine
- Definition:Inverse Cosine
- Definition:Inverse Cotangent
- Definition:Inverse Secant
- Definition:Inverse Cosecant

- Results about
**the inverse tangent function**can be found**here**.