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.${}$