# Definition:Inverse Tangent/Real/Arctangent

## Contents

## Definition

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 denoted as

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

## Caution

There exists the a 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 an 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 $\arctan$.

## Also see

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

### Other inverse trigonometrical ratios

- Definition:Arcsine
- Definition:Arccosine
- Definition:Arccotangent
- Definition:Arcsecant
- Definition:Arccosecant

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.10)$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: Principal Values for Inverse Trigonometrical Functions - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 16.5 \ (4)$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**arc cosine, arc sine, arc tangent,**etc.**${}$** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**inverse trigonometric function**