# Definition:Inverse Cosecant

## Definition

### Real Numbers

Let $x \in \R$ be a real number such that $x \le -1$ or $x \ge 1$.

The inverse cosecant of $x$ is the multifunction defined as:

$\map {\csc^{-1} } x := \set {y \in \R: \map \csc y = x}$

where $\map \csc y$ is the cosecant of $y$.

### Complex Plane

Let $z \in \C_{\ne 0}$ be a non-zero complex number.

The inverse cosecant of $z$ is the multifunction defined as:

$\csc^{-1} \left({z}\right) := \left\{{w \in \C: \csc \left({w}\right) = z}\right\}$

where $\csc \left({w}\right)$ is the cosecant of $w$.

## Arccosecant

Arccosecant Function

From Shape of Cosecant Function, we have that $\csc x$ is continuous and strictly decreasing on the intervals $\hointr {-\dfrac \pi 2} 0$ and $\hointl 0 {\dfrac \pi 2}$.

From the same source, we also have that:

$\csc x \to + \infty$ as $x \to 0^+$
$\csc x \to - \infty$ as $x \to 0^-$

Let $g: \hointr {-\dfrac \pi 2} 0 \to \hointl {-\infty} {-1}$ be the restriction of $\csc x$ to $\hointr {-\dfrac \pi 2} 0$.

Let $h: \hointl 0 {\dfrac \pi 2} \to \hointr 1 \infty$ be the restriction of $\csc x$ to $\hointl 0 {\dfrac \pi 2}$.

Let $f: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0 \to \R \setminus \openint {-1} 1$:

$\map f x = \begin{cases} \map g x & : -\dfrac \pi 2 \le x < 0 \\ \map h x & : 0 < x \le \dfrac \pi 2 \end{cases}$

From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointl {-\infty} {-1}$.

From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointr 1 \infty$.

As both the domain and range of $g$ and $h$ are disjoint, it follows that:

$\map {f^{-1} } x = \begin{cases} \map {g^{-1} } x & : x \le -1 \\ \map {h^{-1} } x & : x \ge 1 \end{cases}$

This function $\map {f^{-1} } x$ is called arccosecant of $x$ and is written $\arccsc x$.

Thus:

The domain of $\arccsc x$ is $\R \setminus \openint {-1} 1$
The image of $\arccsc x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0$.

## Terminology

There exists the popular but misleading notation $\csc^{-1} x$, which is supposed to denote the inverse cosecant function.

However, note that as $\csc x$ is not an injection, it does not have a well-defined inverse.

The $\arccsc$ 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 $\csc^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname {Csc}^{-1}$ (with a capital $\text {C}$) is taken to mean the same as $\arccsc$.

However this can also be confusing due to the visual similarity between that and the lowercase $\text {c}$.

Some sources hyphenate: arc-cosecant.