Definition:Inverse Secant/Real/Arcsecant

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Definition

Arcsecant Function

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:

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


This function $f^{-1} \left({x}\right)$ is called arcsecant of $x$ and is written $\arcsec x$.

Thus:

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


Caution

There exists the a 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 an 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 $\arcsec$, although this can also be confusing due to the visual similarity between that and the lowercase $\text{s}$.


Also see

  • Results about inverse secant can be found here.


Other inverse trigonometrical ratios


Sources