Inverse Trigonometric Functions Class 12th Notes - Free NCERT Class 12 Maths Chapter 2 Notes

Q: Which is the best book for CBSE class 12 maths ?

As most of the questions in the CBSE class 12 board exam are directly asked from the NCERT textbook, students are advised to do rigorous practice of the questions given in the NCER textbook.

Q: How does the NCERT Notes are helpful in the board exam ?

NCERT notes are very important for getting conceptual clarity. These notes are provided in a very simple language, so they can be understood very easily and can be used to revise important concepts.

Q: What is the weightage of the chapter Inverse Trigonometric Functions for CBSE board exam ?

The total weightage of Inverse Trigonometric Functions is 4 marks in the final board exam.

Q: Does CBSE provides the revision notes for NCERT class 12?

No, CBSE doesn't provide any short notes or revision notes for any class.

Q: What is official website of NCERT ?

NCERT is the official website of the NCERT where students can find the important resources from class 6 to 12.

Inverse Trigonometric Functions Class 12th Notes - Free NCERT Class 12 Maths Chapter 2 Notes - Download PDF

Edited By Ravindra Pindel | Updated on Apr 23, 2022 01:59 PM IST

NCERT Class 12 Maths Chapter 2 Inverse Trigonometric Functions Notes:- Inverse Trigonometric Functions is one of the most important chapter for the students in the board exam as well as in competitive exams such as JEE Main. In this article, you will get the NCERT Class 12 Maths Chapter 2 Notes. These chapter notes are beneficial for students to revise the important concepts before the exams.

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Inverse Trigonometric Functions:-
Important Trigonometric Functions Formula:-
NCERT Books and Syllabus

In the previous classes, you have already learnt about trigonometric functions and their applications in fields like geometry, navigation, science, and engineering. In this chapter, you will learn about Inverse Trigonometric functions. If you have a good command of trigonometric functions, it won't take much effort to understand this chapter. Class 12 Maths chapter 2 notes are designed in a detailed manner. First, you need to practice all the NCERT problems on your own. Inverse Trigonometric Functions Class 12 notes will give a short description of the topics covered in the NCERT Book.

Also, see,

Inverse Trigonometric Functions:-

The inverse of any functions exists if the function 'f' is one-one and onto. The inverse of function 'f' is denoted by f ^-1. The trigonometric functions are neither one-one and onto over their domain and natural ranges. So the domains and ranges of trigonometric functions are restricted to ensure the existence of their inverse. The inverse trigonometric functions, denoted by sin ^-1x or (arc sin x), cos ^-1x etc. denote the angles whose sine, cosine etc, is equal to x.

Note- Do not confuse inverse function ( $f^-^1$ ) with $\frac{1}{f}$ . The inverse trigonometric functions like $\sin^-^1x$ is not the same as $\frac{1}{\sin x}$ .

Domain & Range of Inverse Trigonometric Functions:-

$\begin{array}{|l|l|l|l|} \hline \text { S.No } & \boldsymbol{f}(\mathbf{x}) & \text { Domain } & \text { Range } \\ \hline \text { (1) } & \sin ^{-1} \mathbf{x} & |\mathrm{x}| \leq 1 & {\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]} \\ \text { (2) } & \cos ^{-1} \mathrm{x} & |\mathrm{x}| \leq 1 & {[0, \pi]} \\ \text { (3) } & \tan ^{-1} \mathrm{x} & \mathrm{x} \in \mathrm{R} & \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \\ \text { (4) } & \sec ^{-1} \mathrm{x} & |\mathrm{x}| \geq 1 & {[0, \pi]-\left\{\frac{\pi}{2}\right\} \text { or }\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]} \\ \text { (5) } & \operatorname{cosec}^{-1} \mathrm{x} & |\mathrm{x}| \geq 1 & {\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}} \\ \text { (6) } & \cot ^{-1} \mathrm{x} & \mathrm{x} \in \mathrm{R} & (0, \pi) \\ \hline \end{array}$

Graphs Of Inverse Trigonometric Functions :-

(i) Graphs of sin x and sin^-1 x-

$\begin{aligned} &\mathrm{f}:\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \rightarrow[-1,1]\\ &\mathrm{f}(\mathrm{x})=\sin \mathrm{x} \end{aligned}$ fireshot-capture-212-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

$\begin{aligned} &\mathrm{f}^{-1}:[-1,1] \rightarrow[-\pi / 2, \pi / 2] \\ &\mathrm{f}^{-1}(\mathrm{x})=\sin ^{-1}(\mathrm{x}) \end{aligned}$ fireshot-capture-213-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

The domain of the inverse sine function is [-1,1] and the range is $[-\pi / 2, \pi / 2]$ .

(ii) Graphs of cas x and cas^-1 x

$\begin{aligned} &f:[0, \pi] \rightarrow[-1,1] \\ &f(x)=\cos x \end{aligned}$ fireshot-capture-214-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

$\begin{aligned} &\mathrm{f}^{-1}:[-1,1] \rightarrow[0, \pi] \\ &\mathrm{f}^{-1}(\mathrm{x})=\cos ^{-1} \mathrm{x} \end{aligned}$ fireshot-capture-215-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

The domain of the inverse cosine function is [-1,1] and the range is $[0, \pi ]$ .

(iii) Graphs of cosec x and cosec^-1 x

$\begin{aligned} &f:[-\pi / 2,0) \cup(0, \pi / 2] \rightarrow(-\infty,-1] \cup[1, \infty) \\ &f(x)=\operatorname{cosec} x \end{aligned}$ fireshot-capture-216-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

$\begin{aligned} &\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty) \rightarrow[-\pi / 2,0) \cup(0, \pi / 2] \\ &\mathrm{f}^{-1}(\mathrm{x})=\operatorname{cosec}^{-1} \mathrm{x} \end{aligned}$ fireshot-capture-217-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

The domain of the inverse cosec function is $(-\infty,-1] \cup[1, \infty)$ and the range is $\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]$ .

(iv) Graphs of sec x and sec^-1 x

$\begin{aligned} &\mathrm{f}:[0, \pi / 2) \cup(\pi / 2, \pi] \rightarrow(-\infty,-1] \cup[1, \infty) \\ &\mathrm{f}(\mathrm{x})=\sec \mathrm{x} \end{aligned}$ fireshot-capture-218-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

$\begin{aligned} &\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty) \rightarrow[0, \pi / 2) \cup(\pi / 2, \pi] \\ &\mathrm{f}^{-1}(\mathrm{x})=\sec ^{-1} \mathrm{x} \end{aligned}$ fireshot-capture-219-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

The domain of the inverse cosec function is $(-\infty,-1] \cup[1, \infty)$ and the range is $\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]$ .

(v) Graphs of tan x and tan^-1 x

$\begin{aligned} &\mathrm{f}:(-\pi / 2, \pi / 2) \rightarrow \mathrm{R} \\ &\mathrm{f}(\mathrm{x})=\tan \mathrm{x} \end{aligned}$ fireshot-capture-220-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

$\begin{aligned} &\mathrm{f}^{-1}: \mathrm{R} \rightarrow(-\pi / 2, \pi / 2) \\ &\mathrm{f}^{-1}(\mathrm{x})=\tan ^{-1} \mathrm{x} \end{aligned}$ fireshot-capture-221-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

The domain of the inverse cosec function is $(-\infty, \infty)$ and the range is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

(vi) Graphs of cot x and cot^-1 x

$\begin{aligned} &\mathrm{f}:(0, \pi) \rightarrow \mathrm{R} \\ &\mathrm{f}(\mathrm{x})=\cot \mathrm{x} \end{aligned}$ fireshot-capture-222-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

$\begin{aligned} &\mathrm{f}^{-1}: \mathrm{R} \rightarrow(0, \pi) \\ &\mathrm{f}^{-1}(\mathrm{x})=\cot ^{-1} \mathrm{x} \end{aligned}$ fireshot-capture-223-12-maths-ncert-chapter-2pdf-google-drive-drivegooglecom

The domain of the inverse cosec function is $(-\infty, \infty)$ and the range is $(0, \pi)$ .

Note:-

All the inverse trigonometric functions represent an angle
If $x \geq 0$ , then all six inverse trigonometric functions viz sin^-¹ x, cos^-¹ x, tan^-¹ x, sec^-¹x, cosec^-¹x, cot^-¹x represent an acute angle
If x < 0, then sin^-¹x, tan^-¹x & cosec^-¹x represent an angle from $\frac{-\pi}{2}$ to 0
If x < 0, then cos^-¹ x, cot^-¹x & sec^-¹x represent an obtuse angle
IIIrd quadrant is never used in inverse trigonometric function

Important Trigonometric Functions Formula:-

There are some important formulas in NCERT Class 12 Maths chapter 2 Inverse trigonometric functions which will be useful while solving NECRT problems. These formulas are conditional, that can be used for a certain range of values of x.

(i) $\sin ^{-1} \frac{1}{x}=\operatorname{cosec}^{-1} x, x \geq 1 \text { or } x \leq-1$

$\cos ^{-1} \frac{1}{x}=\sec ^{-1} x, x \geq 1 \text { or } x \leq-1$

$\tan ^{-1} \frac{1}{x}=\cot ^{-1} x, x>0$

(ii) $\sin ^{-1}(-x)=-\sin ^{-1} x, x \in[-1,1]$

$\tan ^{-1}(-x)=-\tan ^{-1} x, x \in \mathbf{R}$

$\operatorname{cosec}^{-1}(-x)=-\operatorname{cosec}^{-1} x,|x| \geq 1$

$\cos ^{-1}(-x)=\pi-\cos ^{-1} x, x \in[-1,1]$

$\sec ^{-1}(-x)=\pi-\sec ^{-1} x,|x| \geq 1$

$\cot ^{-1}(-x)=\pi-\cot ^{-1} x, x \in \mathbf{R}$

(iii)

$\begin{aligned} &\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, x \in[-1,1] \\ &\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, x \in \mathbf{R} \\ &\operatorname{cosec}^{-1} x+\sec ^{-1} x=\frac{\pi}{2},|x| \geq 1 \end{aligned}$

(iv)

$\begin{aligned} &\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}, x y<1 \\ &\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}, x y>-1 \\ &2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}},|x|<1 \end{aligned}$

(v)

$\begin{aligned} &2 \tan ^{-1} x=\sin ^{-1} \frac{2 x}{1+x^{2}},|x| \leq 1 \\ &2 \tan ^{-1} x=\cos ^{-1} \frac{1-x^{2}}{1+x^{2}}, x \geq 0 \\ &2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}},-1<x<1 \end{aligned}$

NCERT Class 12 Notes Chapter Wise.

NCERT Class 12 Maths Chapter 1 Notes

NCERT Class 12 Maths Chapter 2 Notes

NCERT Class 12 MathsChapter 3 Notes

NCERT Class 12 Maths Chapter 4 Notes

NCERT Class 12 Maths Chapter 5 Notes

NCERT Class 12 Maths Chapter 6 Notes

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NCERT Class 12 Maths Chapter 9 Notes

NCERT Class 12 Maths Chapter 10 Notes

NCERT Class 12 Maths Chapter 11 Notes

NCERT Class 12 Maths Chapter 12 Notes

NCERT Class 12 Maths Chapter 13 Notes

Subject Wise NCERT Exemplar Solutions

Subject Wise NCERT Solutions

NCERT Books and Syllabus

NCERT Book for Class 12

NCERT Syllabus for Class 12

Important Things to Remember:-

If you are finding difficulties in remembering inverse trigonometric functions formulas, try to relate these formulas with trigonometric functions formulas, so it will be easy for you to memorize these formulae.
NCERT Class 12 Maths Chapter 2 Notes will give you conceptual clarity about the chapter.
You can also download CBSE Class 12 Maths chapter 2 notes in pdf format.
If you have solved all NCERT problems, try to solve CBSE 12 Board Previous Year's Papers, so you will get familiar with the pattern of the board exam question paper.
Use these Inverse Trigonometric Functions Class 12 notes for quick revision before the exams.

Happy learning!!!

Frequently Asked Question (FAQs)

1. Which is the best book for CBSE class 12 maths ?

As most of the questions in the CBSE class 12 board exam are directly asked from the NCERT textbook, students are advised to do rigorous practice of the questions given in the NCER textbook.

2. How does the NCERT Notes are helpful in the board exam ?

NCERT notes are very important for getting conceptual clarity. These notes are provided in a very simple language, so they can be understood very easily and can be used to revise important concepts.

3. What is the weightage of the chapter Inverse Trigonometric Functions for CBSE board exam ?

The total weightage of Inverse Trigonometric Functions is 4 marks in the final board exam.

4. Does CBSE provides the revision notes for NCERT class 12?

No, CBSE doesn't provide any short notes or revision notes for any class.

5. What is official website of NCERT ?

NCERT is the official website of the NCERT where students can find the important resources from class 6 to 12.

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NCERT Books for Class 12 - Download All Subjects PDFs Here

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NCERT Class 12 Physics Chapter 7 Notes Alternating Current - Download PDF

NCERT Class 12 Physics Chapter 6 Notes Electromagnetic Induction - Download PDF

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NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals

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NCERT Exemplar Class 12 Physics Solutions Chapter 8 Electromagnetic Waves

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