Definition:Inverse Secant/Real/Arcsecant
Definition
From Shape of Secant Function, we have that $\sec x$ is continuous and strictly increasing on the intervals $\hointr 0 {\dfrac \pi 2}$ and $\hointl {\dfrac \pi 2} \pi$.
From the same source, we also have that:
- $\sec x \to + \infty$ as $x \to \dfrac \pi 2^-$
- $\sec x \to - \infty$ as $x \to \dfrac \pi 2^+$
Let $g: \hointr 0 {\dfrac \pi 2} \to \hointr 1 \to$ be the restriction of $\sec x$ to $\hointr 0 {\dfrac \pi 2}$.
Let $h: \hointl {\dfrac \pi 2} \pi \to \hointl \gets {-1}$ be the restriction of $\sec x$ to $\hointl {\dfrac \pi 2} \pi$.
Let $f: \closedint 0 \pi \setminus \dfrac \pi 2 \to \R \setminus \openint {-1} 1$:
- $\map f x = \begin{cases} \map g x & : 0 \le x < \dfrac \pi 2 \\ \map h x & : \dfrac \pi 2 < x \le \pi \end{cases}$
From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\hointr 1 \to$.
From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly increasing on $\hointl \gets {-1}$.
As both the domain and range of $g$ and $h$ are disjoint, it follows that:
- $\inv f x = \begin {cases} \inv g x & : x \ge 1 \\ \inv h x & : x \le -1 \end {cases}$
This function $\inv f x$ is called the arcsecant of $x$.
Thus:
- The domain of the arcsecant is $\R \setminus \openint {-1} 1$
- The image of the arcsecant is $\closedint 0 \pi \setminus \dfrac \pi 2$.
Terminology
There exists the popular but misleading notation $\sec^{-1} x$, which is supposed to denote the inverse secant function.
However, note that as $\sec x$ is not an injection, it does not have a well-defined inverse.
The $\arcsec$ 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 $\sec^{-1} x$, which (as pointed out) can be confusing and misleading.
Sometimes, $\operatorname {Sec^{-1} }$ (with a capital $\text S$) is taken to mean the same as $\arcsec$.
However, this can also be confusing due to the visual similarity between that and the lowercase $\text{s}$.
Some sources hyphenate: arc-secant.
Symbol
The symbol used to denote the arcsecant function is variously seen as follows:
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arcsecant function is $\arcsec$.
A variant symbol used to denote the arcsecant function is $\operatorname {asec}$.
Also see
- Results about inverse secant can be found here.
Other inverse trigonometrical ratios
- Definition:Arcsine
- Definition:Arccosine
- Definition:Arctangent
- Definition:Arccotangent
- Definition:Arccosecant
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Principal Values for Inverse Trigonometrical Functions