# Definition:Inverse Tangent/Arctangent

## Definition

### Real Numbers

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 $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

### 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.

## 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**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**arc cosine, arc sine, arc tangent,**etc.**${}$**