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Inverse Trigonometric Functions

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Inverse trigonometric functions are the inverse functions of the trigonometric functions with restricted domains and are used for computing the angle from any of its trigonometric ratios.
Inverse trigonometric functions are multivalued as they require branch cuts in the complex plane. Range of inverse functions are proper subsets of the domains of the original functions.

As none of the six trigonometric functions are one-to-one, they have inverse functions.

Given below is the table listing the inverse trigonometric functions:

 Definition Usual Notation Domain of x Range of Principal value in degrees x = sin y y = arcsin x -1 $\leq$ x $\leq$ 1 -90$^{\circ}\leq y\leq 90^{\circ}$ x = cos y y = arccos x -1 $\leq$ x $\leq$ 1 0$^{\circ}\leq y\leq 180^{\circ}$ x = tan y y = arctan x All real numbers -90$^{\circ}$ x = cot y y = arccot x All real numbers 0$^{\circ}$ x = sec y y = arcsec x x $\leq$  -1 0$^{\circ}\leq y<90^{\circ}$ x =csc y y = arccsc x x $\leq$  -1 -90$^{\circ}\leq y<0^{\circ}$

Inverse Sine function (arcsin)

The inverse sine function, denoted by sin$^{-1}$ is the function with domain [-1, 1] and range [$\frac{-\pi}{2}$,$\frac{\pi}{2}$] defined by x = $\sin^{-1}(y)$

$\Rightarrow$ y = sinx.
The inverse sine function is also called as arcsine. It is denoted by arcsin.

Derivative of inverse sine function is given below:
$\frac{d}{dx}$ $\sin^{-1}$x = $\frac{1}{\sqrt{1-x^{2}}}$; x $\neq$ -1,1

Inverse Cosine Function (arccos)

The inverse cosine function, denoted by cos$^{-1}$ is the function with domain [-1, 1] and range [0, $\pi$] defined by x = $\cos^{-1}(y)$

$\Rightarrow$ y = cosx.
The inverse cosine function is also called as arccosine. It is denoted by arccos.

Derivative of inverse cosine function is given below:
$\frac{d}{dx}$ $\cos^{-1}$x = $\frac{-1}{\sqrt{1-x^{2}}}$; x $\neq$ -1,1

Inverse Tangent Function (arctan)

The inverse tangent function, denoted $\tan^{-1}$ is the function with domain -$\infty < x <\infty$ and range ($\frac{-\pi}{2},\frac{\pi}{2}$) defined by x = tan$^{-1}$ y

$\Rightarrow$ y = tan x
The inverse tangent function is also called arctangent. It is denoted by arctan.

Derivative of inverse tangent function is given below:
$\frac{d}{dx}$ $\tan^{-1}$x = $\frac{1}{1+x^{2}}$; x $\neq$ -i, i

Inverse Cosecant Function (arccsc)

The inverse cosecant function, denoted by csc$^{-1}$ is the function with domain x $\leq$ -1 or x $\geq$ 1 and range $\frac{-\pi}{2}$ $\leq y < 0$ defined by x = $\csc^{-1}(y)$

$\Rightarrow$ y = cscx.
The inverse cosecant function is also called as arccosecant. It is denoted by arccsc.

Derivative of inverse cosecant function is given below:
$\frac{d}{dx}$ $\csc^{-1}$x = $\frac{-1}{|x|\sqrt{x^{2}-1}}$; $\left | x \right | > 1$

Inverse Secant Function (arcsec)

The inverse secant function, denoted by cot$^{-1}$ is the function with domain x $\leq$ -1 or x $\geq$ 1 and range $0 \leq y < $$\frac{\pi}{2} defined by x = \sec^{-1}(y) \Rightarrow y = sec x. The inverse secant function is also called arcsecant. It is denoted by arcsec. Derivative of inverse secant function is given below: \frac{d}{dx}$$\sec^{-1}$x = $\frac{1}{|x|\sqrt{x^{2}-1}}$; $\left | x \right |> 1$

Inverse Cotangent function

The inverse cotangent function, denoted by cot$^{-1}$ is the function with domain -$\infty < x <\infty$, and range 0 < y < $\pi$ defined by x = $\cot^{-1}(y)$

$\Rightarrow$ y = cot x.
The inverse cotangent function is also called as arccotangent. It is denoted by arccot.

Derivative of inverse cotangent function is given below:
$\frac{d}{dx}$$\cot^{-1}$x = $\frac{-1}{{1+x^{2}}}$; x $\neq$ -i, i

Properties of Inverse Trigonometric Functions

Given below are the important properties of inverse trigonometric functions:

$\sin^{-1}(-x)$ = - $\sin^{-1} x$

$\tan^{-1}(-x)$ = - $\tan^{-1} x$

$\cos^{-1}(-x)$ = $\pi-\cos^{-1} x$

$\sin^{-1}(\sin x)$ = $x$

$\sin (\sin^{-1} x)$ = $x$

$\tan^{-1}(\tan x)$ = $x$

$\tan(\tan^{-1}x)$ = $x$

$\csc^{-1} x$ = $sin^{-1}$ $(\frac{1}{x})$

$\cot^{-1} x$ = $\tan^{-1}$ $(\frac{1}{x})$

$\sec^{-1} x$ = $\cos^{-1}$ $(\frac{1}{x})$

$\sin^{-1}x+\cos^{-1} x$ = $\frac{\pi}{2}$

$\tan^{-1}x+\cot^{-1}x$ = $\frac{\pi}{2}$

cosec$^{-1}x+\sec^{-1} x$ =$\frac{\pi}{2}$

$2 \tan^{-1}x$ = $\left\{\begin{matrix} \sin^{-1}(\frac{2x}{1+x^{2}})\\ \cos^{-1}(\frac{1-x^{2}}{1+x^{2}})\\ \tan^{-1}(\frac{2x}{1-x^{2}}) \end{matrix}\right.$

$\tan^{-1}x+\tan^{-1} y$ = $\tan^{-1}$ $(\frac{x+y}{1-xy})$

$\tan^{-1}x-\tan^{-1} y$ = $\tan^{-1}$ $(\frac{x-y}{1+xy})$

$\sin^{-1}x\pm\sin^{-1}y$ = $\sin^{-1}[x\sqrt{1-y^{2}}\pm\ y\sqrt{1-x^{2}}]$