# NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Functions

Free NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions solved by Expert Teachers as per NCERT (CBSE) Book guidelines and brought to you by Toppers Bulletin. These Inverse Trigonometric Functions Exercise Questions with Solutions for Class 12 Maths covers all questions of Chapter Inverse Trigonometric Functions Class 12 and help you to revise complete Syllabus and Score More marks as per CBSE Board guidelines from the latest NCERT book for class 12 maths. You can read and download NCERT Book Solution to get a better understanding of all topics and concepts

2.1 Introduction

2.2 Basic Concepts

2.3 Properties of Inverse Trigonometric Functions.

## Inverse Trigonometric Functions NCERT Solutions – Class 12 Maths

**Q1 : Find the principal value of **

**Answer :**

Let =y. Then sin y=

We know that the range of the principal value branch of sin^{-1} is

and sin

Therefore, the principal value of

**Q2 : ****Find the principal value of**

** Answer :**

We know that the range of the principal value branch of cos ^{-1}is

Therefore, the principal value of.

**Q3 : ****Find the principal value of cosec ^{-1}(2)**

**Answer :**

Let cosec

^{ -1(}2) = y. Then,

We know that the range of the principal value branch of cosec

^{-1}is

Therefore, the principal value of

**Q4 : ****Find the principal value of**

** Answer :**

We know that the range of the principal value branch of tan^{ -1}is

Therefore, the principal value of

**Q5 : ****Find the principal value of**

** Answer :**

We know that the range of the principal value branch of cos^{ -1}is

Therefore, the principal value of

**Q6 : Find the principal value of tan ^{-1}(-1)**

**Answer :**

Let tan

^{-1}(-1) = y. Then,

We know that the range of the principal value branch of tan

^{-1}is

Therefore, the principal value of

**Q7 : Find the principal value of**

** Answer :**

We know that the range of the principal value branch of sec^{-1} is

Therefore, the principal value of

**Q8 :Find the principal value of**

** Answer :**

We know that the range of the principal value branch of cot^{-1} is

(0,π) and

Therefore, the principal value of

**Q9 : Find the principal value of**

** Answer :**

We know that the range of the principal value branch of cos^{-1} is [0,π] and

Therefore, the principal value of

**Q10 : Find the principal value of**

** Answer :**

We know that the range of the principal value branch of cosec^{-1} is

Therefore, the principal value of

**Q11 :****Find the value of**

** Answer : **

**Q12 :Find the value ofAnswer :**

**Q13 :Find the value of if sin – 1 x = y, then**

** (A) (B)**

**(C) (D)**

**Answer :**

It is given that sin^{-1} x = y.

We know that the range of the principal value branch of sin^{-1} is

Therefore,.

**Q14 :Find the value of is equal to **

**(A) π (B) (C) (D)**

**Answer**

**Exercise 2.2 : Solutions of Questions on Page Number : 47**

**Q1 :Prove **

** Answer :**

To prove:

Let x = sinθ. Then,

We have,

R.H.S. =

= 3θ

= L.H.S.

**Q2 :Prove**

** Answer :**

To prove:

Let x = cosθ. Then, cos^{-1} x =θ.

We have,

**Q3 :Prove **

** Answer :**

To prove:

**Q4 :Prove **

** Answer :**

To prove:

**Q6 :Write the function in the simplest form: **

**Answer :**

Put x = cosec θ ⇒ θ = cosec

^{-1}x

**Q7 :Write the function in the simplest form:**

** Answer :**

**Q8 :Write the function in the simplest form: **

**Answer :**

**Q9 :Write the function in the simplest form: **

**Answer :**

**Q10 :Write the function in the simplest form: **

**Answer :**

**Q11 :Find the value of**

** Answer :**

Let. Then,

**Q12 :Find the value of **

** Answer : **

**Q13 :Find the value of **

**Answer :**

Let x = tan θ. Then, θ = tan

^{-1}x.

Let y = tan Φ. Then, Φ = tan

^{-1}y.

**Q14 :If, then find the value of x.**

** Answer : **

On squaring both sides, we get:

Hence, the value of x is

**Q15 :If, then find the value of x.**

** Answer : **

Hence, the value of x is

**Q16 :Find the values of **

** Answer : **

We know that sin

^{-1}(sin x) = x if, which is the principal value branch of sin

^{-1}x.

Here,

Now, can be written as:

**Q17 :Find the values of**

** Answer :**

We know that tan^{-1} (tan x) = x if, which is the principal value branch of tan^{-1}x.

Here,

Now, can be written as:

**Q18 :****Find the values of **

**Answer :**

Let. Then,

**Q19 :Find the values of is equal to**

** (A) (B) (C) (D)**

** Answer :**

We know that cos^{-1} (cos x) = x if, which is the principal value branch of cos^{-1} x.

Here,

Now, can be written as:

The correct answer is B.

**Q20 :Find the values of is equal to **

**Answer :**

Let . Then

We know that the range of the principle value branch of

The correct answer is D.

**Q21 :Find the values of is equal to**

** (A) π (B) (C) 0 (D)**

** Answer :**

Let. Then,

We know that the range of the principal value branch of Let.

The range of the principal value branch of The correct answer is B.

**Exercise Miscellaneous : Solutions of Questions on Page Number : 51**

**Q1 :Find the value of**

** Answer :**

We know that cos^{-1} (cos x) = x if, which is the principal value branch of cos^{-1} x.

Here,

Now, can be written as:

**Q2 :****Find the value of **

**Answer :**

We know that tan

^{-1}(tan x) = x if, which is the principal value branch of tan

^{-1}x.

Here,

Now, can be written as:

**Q3 :Prove**

** Answer : **

Now, we have:

**Q4 :Prove **

**Answer :**

Now, we have:

**Q5 :Prove**

** Answer : **

Now, we will prove that:

**Q6 :Prove**

** Answer : **

Now, we have:

**Q7 :Prove**

** Answer : **

Using (1) and (2), we have

**Q8 :Prove **

**Answer :**

**Q9 :Prove**

** Answer : **

**Q10 :Prove**

** Answer : **

**Q11 :Prove [Hint: putx = cos 2θ]**

** Answer : **

**Q12 :Prove**

** Answer : **

**Q13 :Solve**

** Answer : **

**Q14: Solve **

**Answer:**

**Q15 :Solveis equal to**

** (A) (B) (C) (D)**

** Answer :**

Let tan – 1 x = y. Then,

The correct answer is D.

**Q16 :Solve, then x is equal to**

** (A) (B) (C) 0 (D)**

** Answer : **

Therefore, from equation (1), we have

Put x = sin y. Then, we have:

But, when, it can be observed that:

is not the solution of the given equation.

Thus, x = 0.

Hence, the correct answer is C.

**Q17 :Solve is equal to**

** (A) (B) (C) (D)**

** Answer :**

Hence, the correct answer is C.