Definition:Arctangent
Definition
From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.
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 $g: \left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right) \to \R$ be the restriction of $\tan x$ to $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.
Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ 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 $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.
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 $\arctan$.
Sources
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 16.5 \ (4)$
