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Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse functions of trigonometric ratios. All trigonometric functions are periodic which means that they repeat their values after a certain period. Inverse trigonometric functions are obtained by restricting the domains of respective trigonometric functions.
The inverse of trigonometric functions sin, cos, tan, csc, sec, cot are represented by $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$, $\csc^{-1}$, $\sec^{-1}$, $\cot^{-1}$ respectively. These are also written as arcsin, arccos, arctan, arccsc, arcsec, arccot respectively.

Domain and range of inverse trigonometric functions are given in the table below:

Function
Domain Range
$y=\sin^{-1}x$ [-1, 1] $[-$$\frac{\pi }{2}$,$\frac{\pi }{2}$$]$
$y=\cos^{-1}x$
[-1, 1]
$[0,\pi ]$
$y=\tan^{-1}x$
R $(-$$\frac{\pi }{2}$,$\frac{\pi }{2}$$)$
$y=\csc^{-1}x$
$(-\infty ,-1]\bigcup[1,\infty )$
$[-$$\frac{\pi }{2}$,$\frac{\pi }{2}$$]$ except 0
$y=\sec^{-1}x$
$(-\infty ,-1]\bigcup[1,\infty )$ $[0,\pi ]$ except $\frac{\pi }{2}$
$y=\cot^{-1}x$
R $(0,\pi )$


Few important identities related to Inverse trigonometric functions are described below:

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

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

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

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

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

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

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

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

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

Related Calculators
Six Trigonometric Functions Calculator Calculate Inverse Function
 

More topics in  Inverse Trigonometric Functions
Arctan Inverse Sine Functions
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