Category:Arcsecant Function

This category contains results about Arcsecant Function.

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:

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

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 $\closedint 0 \pi \setminus \dfrac \pi 2$.

Also see

• Category:Arcsine Function
• Category:Arccosine Function
• Category:Arctangent Function
• Category:Arccotangent Function
• Category:Arccosecant Function