# Inverse Trigonometric Functions Class 12 Maths

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Summary:

The ranges (principal value branches) of inverse trigonometric functions are given in the following table:

 Functions Range $\sin^{-1}x$ $\left[ \frac{-\pi}{2}, \frac{\pi}{2} \right]$ $\cosec^{-1}x$ $\left[ \frac{-\pi}{2}, \frac{\pi}{2} \right] - \{0\}$ $\tan^{-1}x$ $\left( \frac{-\pi}{2}, \frac{\pi}{2} \right)$ $\cos^{-1}x$ $[0, \pi]$ $\sec^{-1}x$ $[0, \pi] - \{ \frac{\pi}{2} \}$ $\cot^{-1}x$ $(0, \pi)$

$\sin^{-1}x$ should not be confused with $(\sin x)^{-1}$. In fact $(\sin x)^{-1} = \frac{1}{\sin x}$ and similarly for other trigonometric functions.

The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions.

For Suitable values of domain, we have

$\tan ^{-1} A+\tan ^{-1} B=\tan ^{-1} \left(\frac{A+B}{1-AB} \right)$

$\tan ^{-1} A-\tan ^{-1} B=\tan ^{-1} \left(\frac{A-B}{1+AB} \right)$

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

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

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

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

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

$cosec^{-1}(-\theta)= -cosec^{-1 }\theta$

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

$sec^{-1}(-\theta)= \pi - sec^{-1 }\theta$

$cot^{-1}(-\theta)= \pi - cot^{-1 }\theta$

$sin^{-1}\theta = cosec^{-1}{\frac{1}{\theta}}$

$cosec^{-1}\theta = sin^{-1}{\frac{1}{\theta}}$

$cos^{-1}\theta = sec^{-1}{\frac{1}{\theta}}$

$sec^{-1}\theta = cos^{-1}{\frac{1}{\theta}}$

$tan^{-1}\theta = cot^{-1}{\frac{1}{\theta}}$

$cot^{-1}\theta = tan^{-1}{\frac{1}{\theta}}$

$2\tan ^{-1} \theta =\sin ^{-1} \left(\frac{2\theta }{1+\theta ^{2} } \right)\, \, =\, \, \cos ^{-1} \left(\frac{1-\theta ^{2} }{1+\theta ^{2} } \right)\, \, \, =\, \, \tan ^{-1} \left(\frac{2\theta }{1-\theta ^{2} } \right)$

## Lecture - 1

In this lecture I am discussing derivations and explanation of ranges of inverse trigonometric functions with the help of open source tool GeoGebra.

## Lecture - 2

In this lecture, I am discussing representation of Inverse Trigonometric Functions, derivation of Identities and questions based on Principal Range of Inverse Trigonometric Functions.

## Lecture - 3

In this lecture, I am discussing questions from My booklet which is based on NCERT Exercise 2.1, Exercise 2.2, Miscellaneous Exercise, NCERT Examples, NCERT Exemplar Problems, CBSE Support Material, Board’s Question Bank, Books from Private Publishers like RD Sharma, RS Aggarwal etc.

## Lecture - 4

In this lecture, I am discussing questions from My booklet which is based on NCERT Exemplar Problems, CBSE Support Material, Board’s Question Bank, Books from Private Publishers like RD Sharma, RS Aggarwal etc.

## Lecture - 5

In this lecture, I am discussing questions from My booklet which is based on NCERT textbooks, CBSE Question Bank, Books from Private Publishers etc.

## Exercise - 2.2

Here you will find video explanation of all NCERT solutions for Exercise 2.2

## Miscellaneous Exercise

Here you will find video explanation of all NCERT solutions for Miscellaneous Exercise.