# Definition:Inverse Trigonometric Function

## Definition

As there are six basic trigonometric functions, so each of these has its inverse 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 **arcsine of $x$** and is written $\arcsin x$.

Thus:

- The domain of $\arcsin x$ is $\closedint {-1} 1$
- The image of $\arcsin x$ 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 **arccosine of $x$** and is written $\arccos x$.

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 **arctangent** of $x$ and is written $\arctan x$.

Thus:

- The domain of $\arctan x$ is $\R$
- The image of $\arctan x$ 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 **arccotangent** of $x$ and is written $\arccot x$.

Thus:

### 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:

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

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

### 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:

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

This function $\map {f^{-1} } x$ is called **arccosecant** of $x$ and is written $\arccsc x$.

Thus:

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

## Also known as

An **inverse trigonometric function** can also be seen as:

**antitrigonometric function****cyclometric function****arcus function**

## Also see

- Results about
**inverse trigonometric functions**can be found here.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**antitrigonometric function** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**inverse trigonometric functions (antitrigonometric function)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**antitrigonometric function** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**inverse trigonometric functions (antitrigonometric function)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**inverse trigonometric function**