Definition:Inverse Cotangent/Real
Definition
Let $x \in \R$ be a real number such that $-1 \le x \le 1$.
The inverse cotangent of $x$ is the multifunction defined as:
- $\map {\cot^{-1} } x := \set {y \in \R: \map \cot y = x}$
where $\map \cot y$ is the cotangent of $y$.
Arccotangent
From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$.
From the same source, we also have that:
- $\cot x \to + \infty$ as $x \to 0^+$
- $\cot x \to - \infty$ as $x \to \pi^-$
Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.
This function is called arccotangent of $x$ and is written $\arccot x$.
Thus:
- The domain of the arccotangent is $\R$
- The image of the arccotangent is $\openint 0 \pi$.
Terminology
There exists the popular but misleading notation $\cot^{-1} x$, which is supposed to denote the inverse cotangent function.
However, note that as $\cot x$ is not an injection, it does not have a well-defined inverse.
The $\arccot$ 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 $\cot^{-1} x$, which (as pointed out) can be confusing and misleading.
Sometimes, $\operatorname {Cot}^{-1}$ (with a capital $\text C$) is taken to mean the same as $\arccot$.
However this can also be confusing due to the visual similarity between that and the lowercase $\text c$.
Some sources hyphenate: arc-cotangent.