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

**NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions: ** Two chapters 'relation and function' and 'inverse trigonometry' of NCERT Class 12 has 10 % weightage in the board examination. In this article, you will find NCERT solutions for class 12 maths chapter 2 inverse trigonometric functions which will help you to understand the concepts in a detailed manner. In class 11 maths you have already learnt about trigonometric functions. It won't take much effort to command on inverse trigonometric functions if you have good knowledge of trigonometric functions. You just need to practice NCERT questions including examples and miscellaneous exercise. You may find some difficulties in solving the problems, so you can take the help of these solutions of NCERT for class 12 maths chapter 2 inverse trigonometric functions. These solutions of NCERT for class 12 maths chapter 2 inverse trigonometric functions are very important for the board exam and as well as for competitive exams like JEE Main, BITSAT, VITEEE, etc. In this chapter, there are 2 exercises with 35 questions. All these questions are prepared and explained in a step-by-step method in the NCERT solutions for class 12 maths chapter 2 inverse trigonometric functions article. So, it will be very easy for you to understand the concept. Check all ** NCERT solutions ** from class 6 to class 12 to get a better understanding of the concepts. Here you will get NCERT solutions for two exercises & a miscellaneous exercise of this chapter. Here you will get NCERT solutions for class 12 also.

There are important applications of ITF in geometry, navigation, science, and engineering. Also, inverse trigonometric functions play an important role in the calculus part of mathematics to define many integrals. Many students have a misconception in ** ** class 12 maths chapter inverse trigonometric functions ** ** like . But the inverse function ( ) is not the same as for example .

For inverse to exist, the function must be one-one and onto but trigonometric functions are neither one-one and onto over their domain and natural ranges. So, to ensure the existence of their inverse we restrict domains and ranges of trigonometric functions. And this range is known as principal value. In the following table the principal value branches of inverse trigonometric functions(ITF) are given:

**Topics of NCERT Grade 12 Maths Chapter-2 Inverse Trigonometric Functions **

2.1 Introduction

2.2 Basic Concepts

2.3 Properties of Inverse Trigonometric Functions

## **NCERT Solutions for class 12 maths chapter 2 Inverse Trigonometric Functions: Exercise 2.1 **

** Question:1 ** Find the principal values of the following :

** Answer: **

Let

We know, principle value range of is

The principal value of is

** Question:2 ** Find the principal values of the following:

** Answer: **

So, let us assume that ** then, **

** Taking inverse both sides we get; **

** , or **

and as we know that the principal values of is from [0, ],

Hence when x = . ** **

Therefore, the principal value for is .

** Question:3 ** Find the principal values of the following: ** **

** Answer: **

Let us assume that , then we have;

, or

.

And we know the range of principal values is

Therefore the principal value of is .

** Question:4 ** Find the principal values of the following:

** Answer: **

Let us assume that , then we have;

or

and as we know that the principal value of is .

** Hence the only principal value of when . **

** Question:5 ** Find the principal values of the following:

** Answer: **

Let us assume that ** then, **

Easily we have; or we can write it as:

as we know that the range of the principal values of is .

** Hence lies in the range it is a principal solution. **

** Question:6 ** Find the principal values of the following :

** Answer: **

Given so we can assume it to be equal to 'z';

** , **

** or **

And as we know the range of principal values of from .

** As only one value z = lies hence we have only one principal value that is ** ** . **

** Question:7 ** Find the principal values of the following :

** Answer: **

Let us assume that then,

we can also write it as; .

Or and the principal values lies between .

** Hence we get only one principal value of i.e., . **

** Question:8 ** Find the principal values of the following:

** Answer: **

Let us assume that , then we can write in other way,

or

.

Hence when we have .

and the range of principal values of lies in .

** Then the principal value of is **

** Question:9 ** Find the principal values of the following:

** Answer: **

Let us assume ;

Then we have

** or **

,

.

And we know the range of principal values of is .

** So, the only principal value which satisfies is . **

** Question:10 ** Find the principal values of the following:

** Answer: **

Let us assume the value of , then

we have ** or **

.

and the range of the principal values of lies between .

** hence the principal value of is . **

** Question:11 ** Find the values of the following:

** Answer: **

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

, So we have ;

or ** **

** Therefore, **

So, we have

.

** Therefore , **

,

So we have;

or

** Therefore **

Hence we can calculate the sum:

.

** Question:12 ** Find the values of the following:

** Answer: **

Here we have

let us assume that the value of

;

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

** Calculation of x : **

,

Hence .

** Calculation of y : **

.

Hence .

** The required sum will be = . **

** Question:13 ** If then

** Answer: **

Given if then,

As we know that the can take values between

Therefore, .

** Hence answer choice (B) is correct. **

** Question:14 ** is equal to

** Answer: **

Let us assume the values of be ** 'x' ** and be ** 'y'. **

** Then we have; **

or or or

.

and ** or **

or ** **

also, ** the ranges of the principal values of and are . and **

** respectively. **

we have then;

## ** NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions: Exercise 2.2 **

** Question:1 ** Prove the following: ** **

** Answer: **

Given to prove:

where, .

Take or

Take R.H.S value

=

=

=

= = L.H.S

** Question:2 ** Prove the following:

** Answer: **

Given to prove .

** Take or ; **

Then we have;

R.H.S.

=

=

=

= = L.H.S

** Hence Proved. **

** Question:4 ** Prove the following:

** Answer: **

Given to prove

** Then taking L.H.S. **

** We have **

** **

** **

** = R.H.S. **

** Hence proved. **

** Question:5 ** Write the following functions in the simplest form:

** Answer: **

** We have **

** Take **

** **

** is the simplified form. **

** Question:7 ** Write the following functions in the simplest form:

** Answer: **

Given that

We have in inside the root the term :

Put and ,

Then we have,

** Hence the simplest form is **

** Question:8 ** Write the following functions in the simplest form:

** Answer: **

Given where

So,

Taking common from numerator and denominator.

We get:

= as,

** = is the simplest form. **

** Question:9 ** Write the following functions in the simplest form:

** Answer: **

Given that

Take or

and putting it in the equation above;

** is the simplest form. **

** Question:10 ** Write the following functions in the simplest form:

** Answer: **

Given

Here we can take

So,

** will become; **

** and as ** ** ; **

** hence the simplest form is . **

** Question:11 ** Find the values of each of the following: ** **

** Answer: **

Given equation:

So, solving the inner bracket first, we take the value of

Then we have,

Therefore, we can write .

.

** Question:12 ** Find the values of each of the following:

** Answer: **

We have to find the value of

As we know so,

** Equation reduces to . **

** Question:13 ** Find the values of each of the following: and

** Answer: **

Taking the value ** or ** and ** or ** then we have,

** = , **

** = **

** Then, **

** ** ** **

** Ans. **

** Question:14 ** ** ** If , then find the value of .

** Answer: **

As we know the identity;

. it will just hit you by practice to apply this.

So, or ,

we can then write ,

putting in above equation we get;

=

** Ans. **

** Question:15 ** ** ** If , then find the value of .

** Answer: **

Using the identity ,

We can find the value of x;

So,

on applying,

=

= or ,

Hence, the possible values of x are .

** Question:16 ** Find the values of each of the expressions in Exercises 16 to 18.

** Answer: **

Given ;

We know that

If the value of x belongs to then we get the principal values of .

Here,

We can write is as:

=

= where

** Question:17 ** Find the values of each of the expressions in Exercises 16 to 18.

** Answer: **

As we know

If which is the principal value range of .

So, as in ;

Hence we can write as :

=

Where

and

** Question:18 ** Find the values of each of the expressions in Exercises 16 to 18.

** Answer: **

Given that

** we can take , **

** then **

** or **

** We have similarly; **

Therefore we can write

** from **

** Question:19 ** ** ** is equal to

** Answer: **

As we know that if and is principal value range of .

In this case ,

hence we have then,

** Hence the correct answer is (B). **

** Question:20 ** is equal to

** Answer: **

Solving the inner bracket of ;

or

Take then,

and we know the range of principal value of

Therefore we have .

Hence,

** Hence the correct answer is D. **

** Question:21 ** ** ** is equal to

** Answer: **

We have ;

finding the value of :

Assume then,

and the range of the principal value of is .

Hence, principal value is

Therefore

and

so, we have now,

or,

** Hence the answer is option (B). **

**NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions: Miscellaneous Exercise **

** Question:1 ** Find the value of the following:

** Answer: **

If then , which is principal value of .

So, we have

** Therefore we have, **

** . **

** Question:2 ** Find the value of the following:

** Answer: **

We have given ;

so, as we know

So, here we have .

Therefore we can write as:

** . **

** Question:3 ** Prove that ** **

** Answer: **

To prove: ;

Assume that

then we have .

or

** Therefore we have **

** Now, **

** We can write L.H.S as **

** **

as we know

** L.H.S = R.H.S **

** Question:4 ** Prove that

** Answer **

Taking

then,

Therefore we have-

** .............(1). **

,

Then,

** .............(2). **

** So, we have now, **

** L.H.S. **

using equations (1) and (2) we get,

** = R.H.S. **

** Question:5 ** Prove that

** Answer: **

Take and and

then we have,

Then we can write it as:

or

** ...............(1) **

Now,

So, ** ...................(2) **

Also we have similarly;

Then,

** ...........................(3) **

Now, we have

** L.H.S **

so, using ** (1) ** and ** (2) ** we get,

or we can write it as;

** = R.H.S. **

** Hence proved. **

** Question:6 ** Prove that

** Answer: **

Converting all terms in ** tan ** form;

Let ** , ** and .

now, converting all the terms:

** or **

** We can write it in tan form as: **

** . **

** or ................(1) **

** or **

** We can write it in tan form as: **

** **

** or ......................(2) **

Similarly, for ** ; **

we have ** .............(3) **

** Using (1) and (2) we have L.H.S **

** On applying **

** We have, **

** ...........[Using (3)] **

** =R.H.S. **

** Hence proved. **

** Question:7 ** Prove that

** Answer: **

** Taking R.H.S; **

We have

** Converting sin and cos terms in tan forms: **

Let and

now, we have or

** ............(1) **

Now,

** ................(2) **

** Now, Using (1) and (2) we get, **

R.H.S.

as we know

so,

** equal to L.H.S **

** Hence proved. **

** Question:8 ** Prove that ** **

** Answer: **

Applying the formlua:

on two parts.

we will have,

** Hence it s equal to R.H.S **

** Proved. **

** Question:9 ** Prove that ** **

** Answer: **

By observing the square root we will first put

.

Then,

we have

or, ** R.H.S. **

.

** L.H.S. **

hence ** L.H.S. = R.H.S proved. **

** Question:10 ** Prove that

** Answer: **

Given that

By observing we can rationalize the fraction

We get then,

Therefore we can write it as;

** As L.H.S. = R.H.S. **

** Hence proved. **

** Question:11 ** Prove that ** **

** Answer: **

By using the Hint we will put ;

we get then,

dividing numerator and denominator by ,

we get,

using the formula

** **

** As L.H.S = R.H.S **

** Hence proved **

** Question:12 ** Prove that

** Answer: **

We have to solve the given equation:

** Take as common in L.H.S, **

or from

Now, assume,

Then,

Therefore we have now,

So we have L.H.S then

** That is equal to R.H.S. **

** Hence proved. **

** Question:13 ** Solve the following equations: ** **

** Answer: **

Given equation ;

Using the formula:

We can write

So, we can equate;

that implies that .

or or

** Hence we have solution . **

** Question:14 ** Solve the following equations:

** Answer: **

Given equation is

:

L.H.S can be written as;

Using the formula

So, we have

** Hence the value of . **

** Question:16 ** then is equal to

** Answer: **

Given the equation:

we can migrate the term to the ** R.H.S. **

** then we have; **

or ** ............................(1) **

from

Take or .

So, we conclude that;

Therefore we can put the value of in equation ** (1) we get, **

Putting ** x= sin y ** , in the above equation; we have then,

So, we have the solution;

Therefore we have .

When we have , we can see that :

** So, it is not equal to the R.H.S. **

** Thus we have only one solution which is x = 0 **

** Hence the correct answer is (C). **

** NCERT solutions for class 12 maths - Chapter wise **

** NCERT solutions for class 12 subject wise **

** NCERT Solutions class wise **

**NCERT solutions for class 12****NCERT solutions for class 11****NCERT solutions for class 10****NCERT solutions for class 9**

Students who wish to perform well in the CBSE 12 board examination, solutions of NCERT for class 12 maths chapter 2 inverse trigonometric functions are very helpful but here are some tips to make command on Inverse Trigonometric Functions.

- The inverse trigonometric function is inverse of the trigonometric function, so if you have a command on the trigonometric function then it will be easy for you to understand inverse trigonometric functions.
- Try to relate trigonometric functions formulas with inverse trigonometric functions formulas, so that memorizing the formulae becomes easier.
- Before starting to solve exercise, first solve the examples that are given in the NCERT class 12 maths textbook.
- Also, try to solve every exercise including miscellaneous exercise, NCERT chapter examples, miscellaneous examples on your own, if you are finding difficulties, you can take the help of CBSE NCERT solutions for class 12 maths chapter 2 inverse trigonometric functions.
- If you have solved all NCERT then you can solve previous years paper CBSE board to get familiar with the pattern of board exam question paper

** Happy learning!!! **

## Frequently Asked Question (FAQs) - NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

**Question: **What are the important topics in chapter inverse trigonometric functions ?

**Answer: **

Some basic concepts of inverse trigonometry and properties of inverse trigonometric functions are important topics of this chapter.

**Question: **Does CBSE provides the solutions of NCERT class 12 maths ?

**Answer: **

No, CBSE doesn’t provided NCERT solutions for any class or subject.

**Question: **Where can I find the complete solutions of NCERT class 12 maths ?

**Answer: **

A Here you will get the detailed NCERT solutions for class 12 maths by clicking on the link.

**Question: **What is the weightage of the chapter inverse trigonometry for CBSE board exam ?

**Answer: **

Relation and function and inverse trigonometry combined has 10 % weightage in the CBSE class 12 board examination.

**Question: **Which are the most difficult chapters of NCERT Class 12 Maths syllabus?

**Answer: **

Students consider Integration and it's application as most difficult unit in CBSE class 12 maths. With the regular practice you will get conceptual clarity and will be able to have a strong grip on Integration.

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

**Answer: **

NCERT textbook is the best book for CBSE class 12 maths. Most of the questions in CBSE class 12 board exam are directly asked from NCERT textbook. All you need to do is rigorous practice of the questions given in the NCER textbook.

## Latest Articles

##### OFSS Bihar Admission 2020 Started on ofssbihar.in - Apply Now

OFSS Bihar admission 2020 Online - Bihar School Examination Bo...

##### MP Board 12th Time Table 2020 Released for Supplementary Exams...

MP Board 12th Time Table 2020 -MPBSE has released MP board cla...

##### Bihar Board Compartmental Result 2020 Declared!- Check BSEB 10...

Bihar Board Compartmental Result 2020 has released in on Aug 7...

##### NCERT Books for Class 11 English 2020-21 - Download Pdf here

NCERT Books for Class 11 English 2020-21 - NCERT books of Engl...

##### Mahadbt Scholarship 2020 -Dates, Eligibility, Application Form

Mahadbt Scholarship 2020 - Students can check eligibility, ben...

##### UP Board 12th Result 2020 Declared! - Application for Compartm...

UP Board 12th Result 2020 - Uttar Pradesh Madhyamik Shiksha Pa...

##### Odisha CHSE Result 2020 Date- Check Odisha Board 12th Result @...

Odisha CHSE Result 2020 - CHSE, Odisha will declare the Odisha...

##### NIOS Subjects for 10th & 12th Class 2020-21- Check NIOS Subjec...

NIOS Subjects for 10th & 12th Class 2020-21 - Students appeari...

##### NIOS Exam Dates 2020 for 10th & 12th Cancelled

NIOS Exam Dates 2020 (10th & 12th Class) - NIOS has cancelled ...

##### NIOS Result 2020 for 10th & 12th Declared - Check NIOS Board R...

NIOS Result 2020 - NIOS has declared class 10th and 12th NIOS ...