# NCERT Solutions for Exercise 2.1 Class 12 Maths Chapter 2 - Inverse Trigonometric Functions

NCERT Solutions for Class 12 Maths chapter 2 exercise 2.1 discusses the inverse of various trigonometric functions. From the Exercise 2.1 Class 12 Maths it can be observed that most of the questions are related to finding the value of inverse of various trigonometric functions like sine, cos, tan etc. From NCERT Solutions for Class 12 Maths chapter 2 exercise 2.1, direct questions related to finding the inverse are asked many times in board exams.Hence it is highly recommended to practice this exercise before CBSE class 12 board exam. Sometimes questions are asked in competitive exams also like JEE main etc. The NCERT chapter Inverse Trigonometric Functions has a lot of applications in subsequent chapters of maths as well as Physics also. Hence Practice of the questions is mandatory for this exercise.

Inverse Trigonometric Functions Exercise 2.2

Inverse Trigonometric Functions Miscellaneous Exercise

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions: Exercise 2.1

## Question:1 Find the principal values of the following : $\sin^{-1}\left ( \frac{-1}{2} \right )$

Let $x = \sin^{-1}\left ( \frac{-1}{2} \right )$

$\implies \sin x = \frac{-1}{2}= -\sin(\frac{\pi}{6}) = \sin(-\frac{\pi}{6})$
We know, principle value range of $\dpi{100} sin^{-1}$ is $\dpi{100} [-\frac{\pi}{2}, \frac{\pi}{2}]$

$\dpi{100} \therefore$ The principal value of $\dpi{100} \sin^{-1}\left ( \frac{-1}{2} \right )$ is $\dpi{100} -\frac{\pi}{6},$

Question:2 Find the principal values of the following: $\cos^{-1}\left(\frac{\sqrt3}{2} \right )$

So, let us assume that $\cos^{-1}\left(\frac{\sqrt3}{2} \right ) = x$ then,

Taking inverse both sides we get;

$cos\ x = (\frac{\sqrt{3}}{2})$ , or $cos (\frac{\pi}{6}) = (\frac{\sqrt{3}}{2})$

and as we know that the principal values of $cos^{-1}$ is from [0, $\pi$ ],

Hence $cos\ x = (\frac{\sqrt{3}}{2})$ when x = $\frac{\pi}{6}$ .

Therefore, the principal value for $\cos^{-1}\left(\frac{\sqrt3}{2} \right )$ is $\frac{\pi}{6}$ .

Question:3 Find the principal values of the following: $\textup{cosec}^{-1}(2)$

Let us assume that $\textup{cosec}^{-1}(2) = x$ , then we have;

$Cosec\ x = 2$ , or

$Cosec( \frac{\pi}{6}) = 2$ .

And we know the range of principal values is $[\frac{-\pi}{2},\frac{\pi}{2}] - \left \{ 0 \right \}.$

Therefore the principal value of $\textup{cosec}^{-1}(2)$ is $\frac{\pi}{6}$ .

Question:4 Find the principal values of the following: $\tan^{-1}(-\sqrt3)$

Let us assume that $\tan^{-1}(-\sqrt3) = x$ , then we have;

$\tan x = (-\sqrt 3)$ or

$-\tan (\frac{\pi}{3}) = \tan \left ( \frac{-\pi}{3} \right ).$

and as we know that the principal value of $\tan^{-1}$ is $\left ( \frac{-\pi}{2}, \frac{\pi}{2} \right )$ .

Hence the only principal value of $\tan^{-1}(-\sqrt3)$ when $x = \frac{-\pi}{3}$ .

Question:5 Find the principal values of the following: $\cos^{-1}\left(-\frac{1}{2} \right )$

Let us assume that $\cos^{-1}\left(-\frac{1}{2} \right ) =y$ then,

Easily we have; $\cos y = \left ( \frac{-1}{2} \right )$ or we can write it as:

$-\cos \left ( \frac{\pi}{3} \right ) = \cos \left ( \pi - \frac{\pi}{3} \right ) = \cos \left ( \frac{2\pi}{3} \right ).$

as we know that the range of the principal values of $\cos^{-1}$ is $\left [ 0,\pi \right ]$ .

Hence $\frac{2\pi}{3}$ lies in the range it is a principal solution.

Question:6 Find the principal values of the following : $\tan^{-1}(-1)$

Given $\tan^{-1}(-1)$ so we can assume it to be equal to 'z';

$\tan^{-1}(-1) =z$ ,

$\tan z = -1$

or

$-\tan (\frac{\pi}{4}) = \tan(\frac{-\pi}{4})= -1$

And as we know the range of principal values of $\tan^{-1}$ from $\left ( \frac{-\pi}{2}, \frac{\pi}{2} \right )$ .

As only one value z = $-\frac{\pi}{4}$ lies hence we have only one principal value that is $-\frac{\pi}{4}$ .

Question:7 Find the principal values of the following : $\sec^{-1}\left (\frac{2}{\sqrt3}\right)$

Let us assume that $\sec^{-1}\left (\frac{2}{\sqrt3}\right) = z$ then,

we can also write it as; $\sec z = \left (\frac{2}{\sqrt3}\right)$ .

Or $\sec (\frac{\pi}{6}) = \left (\frac{2}{\sqrt3}\right)$ and the principal values lies between $\left [ 0, \pi \right ] - \left \{ \frac{\pi}{2} \right \}$ .

Hence we get only one principal value of $\sec^{-1}\left (\frac{2}{\sqrt3}\right)$ i.e., $\frac{\pi}{6}$ .

Question:8 Find the principal values of the following: $\cot^{-1}(\sqrt3)$

Let us assume that $\cot^{-1}(\sqrt3) = x$ , then we can write in other way,

$\cot x = (\sqrt3)$ or

$\cot (\frac{\pi}{6}) = (\sqrt3)$ .

Hence when $x=\frac{\pi}{6}$ we have $\cot (\frac{\pi}{6}) = (\sqrt3)$ .

and the range of principal values of $\cot^{-1}$ lies in $\left ( 0, \pi \right )$ .

Then the principal value of $\cot^{-1}(\sqrt3)$ is $\frac{\pi}{6}$

Question:9 Find the principal values of the following: $\cos^{-1}\left(-\frac{1}{\sqrt2} \right )$

Let us assume $\cos^{-1}\left(-\frac{1}{\sqrt2} \right ) = x$ ;

Then we have $\cos x = \left ( \frac{-1}{\sqrt 2} \right )$

or

$-\cos (\frac{\pi}{4}) = \left ( \frac{-1}{\sqrt 2} \right )$ ,

$\cos (\pi - \frac{\pi}{4}) = \cos (\frac{3\pi}{4})$ .

And we know the range of principal values of $\cos^{-1}$ is $[0,\pi]$ .

So, the only principal value which satisfies $\cos^{-1}\left(-\frac{1}{\sqrt2} \right ) = x$ is $\frac{3\pi}{4}$ .

Question:10 Find the principal values of the following: $\textup{cosec}^{-1}(-\sqrt2)$

Let us assume the value of $\textup{cosec}^{-1}(-\sqrt2) = y$ , then

we have $cosec\ y = (-\sqrt 2)$ or

$-cosec\ (\frac{\pi}{4}) = (-\sqrt 2) = cosec\ (\frac{-\pi}{4})$ .

and the range of the principal values of $\textup{cosec}^{-1}$ lies between $\left [ \frac{-\pi}{2},\frac{\pi}{2} \right ] - \left \{ 0 \right \}$ .

hence the principal value of $\textup{cosec}^{-1}(-\sqrt2)$ is $\frac{-\pi}{4}$ .

Question:11 Find the values of the following: $\tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2} \right ) + \sin^{-1}\left(-\frac{1}{2} \right )$

To find the values first we declare each term to some constant ;

$tan^{-1}(1) = x$ , So we have $\tan x = 1$ ;

or $\tan (\frac{\pi}{4}) = 1$

Therefore, $x = \frac{\pi}{4}$

$cos^{-1}(\frac{-1}{2}) = y$

So, we have

$\cos y = \left ( \frac{-1}{2} \right ) = -\cos \left ( \frac{\pi}{3} \right ) = \cos(\pi - \frac{\pi}{3}) = \cos \left ( \frac{2\pi}{3} \right )$ .

Therefore $y = \frac{2\pi}{3}$ ,

$\sin^{-1}(\frac{-1}{2}) = z$ ,

So we have;

$\sin z = \frac{-1}{2}$ or $-\sin (\frac{\pi}{6}) =\sin (\frac{-\pi}{6}) = \frac{-1}{2}$

Therefore $z = -\frac{\pi}{6}$

Hence we can calculate the sum:

$= \frac{\pi}{4}+\frac{2\pi}{3}-\frac{\pi}{6}$

$=\frac{3\pi + 8\pi -2\pi}{12} = \frac{9\pi}{12}=\frac{3\pi}{4}$ .

Question:12 Find the values of the following: $\cos^{-1}\left(\frac{1}{2} \right ) + 2\sin^{-1}\left(\frac{1}{2} \right )$

Here we have $\cos^{-1}\left(\frac{1}{2} \right ) + 2\sin^{-1}\left(\frac{1}{2} \right )$

let us assume that the value of

$\cos^{-1}\left ( \frac{1}{2} \right ) = x, \:and\:\sin^{-1}\left(\frac{1}{2} \right ) = y$ ;

then we have to find out the value of x +2y.

Calculation of x :

$\Rightarrow \cos^{-1}\left ( \frac{1}{2} \right ) = x$

$\Rightarrow \cos x = \frac{1}{2}$

$\Rightarrow \cos \frac{\pi}{3} = \frac{1}{2}$ ,

Hence $x = \frac{\pi}{3}$ .

Calculation of y :

$\Rightarrow \sin^{-1}\left(\frac{1}{2} \right ) = y$

$\Rightarrow \sin y = \frac{1}{2}$

$\Rightarrow \sin \frac{\pi}{6} = \frac{1}{2}$ .

Hence $y = \frac{\pi}{6}$ .

The required sum will be = $\frac{\pi}{3}+2(\frac{\pi}{6}) = \frac{2\pi}{3}$ .

Question:13 If $\sin^{-1}x = y$ then

(A) $0\leq y \leq \pi$

(B) $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

(C) $0 < y < \pi$

(D) $-\frac{\pi}{2} < y < \frac{\pi}{2}$

Given if $\sin^{-1}x = y$ then,

As we know that the $\sin^{-1}$ can take values between $\left [ \frac{-\pi}{2}, \frac{\pi}{2} \right ].$

Therefore, $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$ .

Hence answer choice (B) is correct.

Question:14 $\tan^{-1}(\sqrt3)-\sec^{-1}(-2)$ is equal to

(A) $\pi$

(B) $-\frac{\pi}{3}$

(C) $\frac{\pi}{3}$

(D) $\frac{2\pi}{3}$

Let us assume the values of $\tan^{-1}(\sqrt3)$ be 'x' and $\sec^{-1}(-2)$ be 'y'.

Then we have;

$\tan^{-1}(\sqrt3) = x$ or $\tan x = \sqrt 3$ or $\tan \frac{\pi}{3} = \sqrt 3$ or

$x = \frac{\pi}{3}$ .

and $\sec^{-1}(-2) = y$ or $\sec y = -2$

or $-\sec (\frac{\pi}{3}) =\sec ({\pi - \frac{\pi}{3}}) = \sec{\frac{2\pi}{3}}$

$y = \frac{2\pi}{3}$

also, the ranges of the principal values of $\tan^{-1}$ and $\sec^{-1}$ are $(\frac{-\pi}{2},\frac{\pi}{2})$ . and

$[0,\pi] - \left \{ \frac{\pi}{2} \right \}$ respectively.

$\therefore$ we have then;

$\tan^{-1}(\sqrt3)-\sec^{-1}(-2)$

$= \frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3}$

More About NCERT Solutions for Class 12 Maths Chapter 2 Exercise 2.1

The NCERT book for Class 12 Maths chapter Inverse Trigonometric Functions has a total of 3 exercises including miscellaneous exercise. Exercise 2.1 Class 12 Maths covers solutions to a total of 21 questions mostly based on finding the inverse value of various trigonometric functions. NCERT Solutions for Class 12 Maths chapter 2 exercise 2.1 is an authentic source to learn concepts related to finding the inverse of a trigonometric function.

Also Read| Inverse Trigonometric Functions NCERT Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 2 Exercise 2.1

• The NCERT syllabus Class 12th Maths chapter 2 exercise provided here is in detail which is solved by subject matter experts .

• Students are recommended to practice Exercise 2.1 Class 12 Maths to prepare for exams, direct questions are asked in Board exams as well as competitive exams.

• These Class 12 maths chapter 2 exercise 2.1 solutions can be referred by students to revise just before the exam for revision.

• NCERT Solutions for Class 12 Maths chapter 2 exercise 2.1 can be used to understand physics topics on similar concepts.

### Also see-

• NCERT exemplar solutions class 12 maths chapter 2

• NCERT solutions for class 12 maths chapter 2

NCERT Solutions Subject Wise

• NCERT solutions class 12 chemistry

• NCERT solutions for class 12 physics

• NCERT solutions for class 12 biology

• NCERT solutions for class 12 mathematics

## Subject wise NCERT Exemplar solutions

• NCERT Exemplar Class 12th Maths

• NCERT Exemplar Class 12th Physics

• NCERT Exemplar Class 12th Chemistry

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