Definition:Inverse Trigonometric Function
Definition
As there are $6$ basic trigonometric functions, it follows that there are $6$ inverse trigonometric functions.
As follows:
Arcsine
From Shape of Sine Function, we have that $\sin x$ is continuous and strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
From Sine of Half-Integer Multiple of Pi:
- $\map \sin {-\dfrac {\pi} 2} = -1$
and:
- $\sin \dfrac {\pi} 2 = 1$
Therefore, let $g: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \to \closedint {-1} 1$ be the restriction of $\sin x$ to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Thus from Inverse of Strictly Monotone Function, $g \paren x$ admits an inverse function, which will be continuous and strictly increasing on $\closedint {-1} 1$.
This function is called the arcsine of $x$.
Thus:
- The domain of arcsine is $\closedint {-1} 1$
- The image of arcsine is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Arccosine
From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$.
From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.
Therefore, let $g: \closedint 0 \pi \to \closedint {-1} 1$ be the restriction of $\cos x$ to $\closedint 0 \pi$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$.
This function is called the arccosine of $x$.
Thus:
Arctangent
From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
From the same source, we also have that:
- $\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$
- $\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$
Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\R$.
This function is called the arctangent of $x$ and is written $\arctan x$.
Thus:
- The domain of the arctangent is $\R$
- The image of the arctangent is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Arccotangent
From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$.
From the same source, we also have that:
- $\cot x \to + \infty$ as $x \to 0^+$
- $\cot x \to - \infty$ as $x \to \pi^-$
Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.
This function is called the arccotangent of $x$ and is written $\arccot x$.
Thus:
- The domain of the arccotangent is $\R$
- The image of the arccotangent is $\openint 0 \pi$.
Arcsecant
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 $\closedint 0 \pi \setminus \dfrac \pi 2$.
Arccosecant
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:
- $\inv f x = \begin {cases} \inv g x & : x \le -1 \\ \inv h x & : x \ge 1 \end {cases}$
This function $\map {f^{-1} } x$ is called the arccosecant of $x$.
Thus:
- The domain of the arccosecant is $\R \setminus \openint {-1} 1$
- The image of the arccosecant is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0$.
Also known as
An inverse trigonometric function can also be seen as:
Also see
- Results about inverse trigonometric functions can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse trigonometric functions (antitrigonometric function)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse trigonometric functions (antitrigonometric function)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inverse trigonometric function