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 $\left[{0 \,.\,.\, \dfrac \pi 2}\right)$ and $\left({\dfrac \pi 2 \,.\,.\, \pi}\right]$.

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 $g: \left[{0 \,.\,.\, \dfrac \pi 2}\right) \to \left[{1 \,.\,.\, \infty}\right)$ be the restriction of $\sec x$ to $\left[{0 \,.\,.\, \dfrac \pi 2}\right)$.

Let $h: \left({\dfrac \pi 2 \,.\,.\, \pi}\right] \to \left({-\infty \,.\,.\, -1}\right]$ be the restriction of $\sec x$ to $\left({\dfrac \pi 2 \,.\,.\, \pi}\right]$.

Let $f: \left[{0 \,.\,.\, \pi}\right] \setminus \dfrac \pi 2 \to \R \setminus \left({-1 \,.\,.\, 1}\right)$:

$f\left({x}\right) = \begin{cases} g\left({x}\right) & : 0 \le x < \dfrac \pi 2 \\ h\left({x}\right) & : \dfrac \pi 2 < x \le \pi \end{cases}$


From Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly increasing on $\left[{1 \,.\,.\, \infty}\right)$.

From Inverse of Strictly Monotone Function, $h \left({x}\right)$ admits an inverse function, which will be continuous and strictly increasing on $\left({-\infty \,.\,.\, -1}\right]$.

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

$f^{-1}\left({x}\right) = \begin{cases} g^{-1}\left({x}\right) & : x \ge 1 \\ h^{-1}\left({x}\right) & : x \le -1 \end{cases}$


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

Thus:

  • The domain of $\operatorname{arcsec} x$ is $\R \setminus \left({-1 \,.\,.\, 1}\right)$
  • The image of $\operatorname{arcsec} x$ is $\left[{0 \,.\,.\, \pi}\right] \setminus \dfrac \pi 2$.


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 $\operatorname{arcsec}$ 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 $\operatorname{arcsec}$, 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