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

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.

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 -1 x 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).


  • 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-1 x & sec-1 x 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}


\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}


\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}


\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.

Subject Wise NCERT Exemplar Solutions

Subject Wise NCERT Solutions

NCERT Books and Syllabus

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.


Get answers from students and experts

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg


An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)


Option 2)

\; K\;

Option 3)


Option 4)


In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)


Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)


Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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