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.