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

Imagine you have shifted to a new town, and you need to go to the nearest bank urgently. You know the distance, but you don't know the angles where you should turn to reach there quickly. This is where inverse trigonometric functions are very helpful. In Maths chapter 2 class 12, students can find this chapter. Students already know about trigonometric functions from previous year lessons, and if they understood those concepts thoroughly, it won't take much effort to understand this chapter. This chapter will also be important in JEE Main, and students should initially attempt the exercises to find solutions on their own.

Like trigonometric functions, these are also used in geometry, navigation, science, and engineering. Students should study the inverse trigonometric functions class 12 questions and answers from the textbook to strengthen all the basic concepts. It is also recommended that NCERT Exemplar Class 12 Maths Solutions Chapter 2 Inverse Trigonometric Functions be checked for a better understanding of the chapter. After completing all these, students need a handy study material for quick revision purposes. That is when inverse trigonometric functions class 12 NCERT notes come into play. These notes are prepared by experienced Careers360 experts keeping in mind the importance of the students' learning process. Inverse Trigonometric Functions Class 12 notes will give a short description of the topics covered in the NCERT Book.

Inverse Trigonometric Functions:-

The inverse of any function 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 nor 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 ${\sin }^{-1}x$ is not the same as $\frac{1}{\sin x}$.

Domain & Range of Inverse Trigonometric Functions:-

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

Graphs Of Inverse Trigonometric Functions:-

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

$\begin{array}{rl}& \mathrm{f}:[\frac{-\pi }{2},\frac{\pi }{2}]\to [-1,1]\\ & \mathrm{f}(\mathrm{x})=\sin \mathrm{x}\end{array}$

$\begin{array}{rl}& {\mathrm{f}}^{-1}:[-1,1]\to [-\frac{\pi }{2},\frac{\pi }{2}]\\ & {\mathrm{f}}^{-1}(\mathrm{x})={\sin }^{-1}(\mathrm{x})\end{array}$

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

(ii) Graphs of cos x and cos^-1 x:

$\begin{array}{rl}& f:[0,\pi ]\to [-1,1]\\ & f(x)=\cos x\end{array}$

$\begin{array}{rl}& {\mathrm{f}}^{-1}:[-1,1]\to [0,\pi ]\\ & {\mathrm{f}}^{-1}(\mathrm{x})={\cos }^{-1}\mathrm{x}\end{array}$

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

$\begin{array}{rl}& f:[-\frac{\pi }{2},0)\cup (0,\frac{\pi }{2}]\to (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })\\ & f(x)=\mathrm{cosec}x\end{array}$

$\begin{array}{rl}& {\mathrm{f}}^{-1}:(-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })\to [-\frac{\pi }{2},0)\cup (0,\frac{\pi }{2}]\\ & {\mathrm{f}}^{-1}(\mathrm{x})={\mathrm{cosec}}^{-1}\mathrm{x}\end{array}$

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

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

$\begin{array}{rl}& \mathrm{f}:[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]\to (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })\\ & \mathrm{f}(\mathrm{x})=\sec \mathrm{x}\end{array}$

$\begin{array}{rl}& {\mathrm{f}}^{-1}:(-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })\to [0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]\\ & {\mathrm{f}}^{-1}(\mathrm{x})={\sec }^{-1}\mathrm{x}\end{array}$

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

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

$\begin{array}{rl}& \mathrm{f}:(-\frac{\pi }{2},\frac{\pi }{2})\to \mathrm{R}\\ & \mathrm{f}(\mathrm{x})=\tan \mathrm{x}\end{array}$

$\begin{array}{rl}& {\mathrm{f}}^{-1}:\mathrm{R}\to (-\frac{\pi }{2},\frac{\pi }{2})\\ & {\mathrm{f}}^{-1}(\mathrm{x})={\tan }^{-1}\mathrm{x}\end{array}$

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

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

$\begin{array}{rl}& \mathrm{f}:(0,\pi )\to \mathrm{R}\\ & \mathrm{f}(\mathrm{x})=\cot \mathrm{x}\end{array}$

$\begin{array}{rl}& {\mathrm{f}}^{-1}:\mathrm{R}\to (0,\pi )\\ & {\mathrm{f}}^{-1}(\mathrm{x})={\cot }^{-1}\mathrm{x}\end{array}$

The domain of the inverse cotangent function is $(-\mathrm{\infty },\mathrm{\infty })$ and the range is $(0,\pi )$.

Note:-

• All the inverse trigonometric functions represent an angle.
• If $x\ge 0$, then all six inverse trigonometric functions viz sin^-1 x, cos^-1 x, tan^-1 x, sec^-1x, cosec^-1x, and 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.
• (III)rd quadrant is never used in inverse trigonometric functions.

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.

NCERT Class 12 Notes Chapter Wise

Subject Wise NCERT Exemplar Solutions

These links will help students to find the step-by-step NCERT exemplar solutions.

• NCERT Exemplar Class 12 Solutions
• NCERT Exemplar Class 12 Maths
• NCERT Exemplar Class 12 Physics
• NCERT Exemplar Class 12 Chemistry
• NCERT Exemplar Class 12 Biology

Subject Wise NCERT Solutions

Students can also explore these well-made subject-wise solutions.

• NCERT Solutions for Class 12 Mathematics
• NCERT Solutions for Class 12 Chemistry
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for Class 12 Biology

NCERT Books and Syllabus

The following links can lead students to the latest CBSE syllabus and some important reference books for better conceptual clarity.

Important Things to Remember:-

• If you are having difficulty in remembering inverse trigonometric functions formulas, try relating these formulas to trigonometric functions formulas, so it will be easy for you to memorize these formulas.

• 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.