NCERT Solutions For Class 12 Maths Chapter 2

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NCERT Solutions Of Class 12 Maths Chapter 2 – Inverse Trigonometric Functions

Name OF The Section	Topic Name
2	Inverse Trigonometric Functions
2.1	Introduction
2.2	Basic Concepts
2.3	Properties of Inverse Trigonometric Functions

Class 12 Maths Chapter 2 NCERT Solutions – 2.1 Introduction

Question 1:
Find the principal value of
Answer:
Let sin^{-1 Then sin y =}

We know that the range of the principal value branch of sin⁻¹ is
and sin
Therefore, the principal value of

Question 2:
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

Question 3:

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

Question 4:
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

Question 5:
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

Question 6:

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

Question 7:
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

Question 8:
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

Question 9:
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

Question 10:
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

Question 11:
Find the value of
Answer:

Question 12:
Find the value of
Answer:

Question 13:

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,

Question 14:
Find the value of is equal to
(A) π (B) (C) (D)
Answer:

NCERT Solutions For Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Exercise 2.2

Question 1:
Prove
Answer:
To prove,
Let x = sinθ. Then,

We have,
R.H.S. =

= 3θ

= L.H.S.

Question 2:
Prove
Answer:
To prove:

Let x = cosθ. Then, cos⁻¹ x =θ.

We have,

Question 3:
Prove
Answer:
To prove:

Question 4:
Prove
Answer:
To prove:

Question 5:

Write the function in the simplest form:

Answer:

Question 6:

Write the function in the simplest form:
Answer:

Put x = cosec θ ⇒ θ = cosec⁻¹ x

Question 7:

Write the function in the simplest form:

Answer:

Question 8:

Write the function in the simplest form:

Answer:

Dividing numerator and denominator by cos x

Question 9:

Write the function in the simplest form:

Answer:

Question 10:

Write the function in the simplest form:

Answer:

Question 11:
Find the value of
Answer:
Let Then,

Question 12:
Find the value of
Answer:

Question 13:
Find the value of
Answer:

Let x = tan θ. Then, θ = tan⁻¹ x.

Let y = tan Φ. Then, Φ = tan⁻¹ y.

Question 14:
If , then find the value of x
Answer:

On squaring both sides, we get:

Hence, the value of x is

Question 15:
If then find the value of x.
Answer:

Hence, the value of x is

Question 16:
Find the values of
Answer:
We know that sin⁻¹ (sin x) = x if which is the principal value branch of sin⁻¹x.
Here,
Now, can be written as:

Question 17:
Find the values of
Answer:
We know that tan⁻¹ (tan x) = x if which is the principal value branch of tan⁻¹x.
Here
Now, can be written as:

Question 18:
Find the values of
Answer:
Let Then,

Question 19:
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 ⁻¹x.
Here,
Now, can be written as:
cos-1cos7π6 = cos-1cosπ+π6cos-1cos7π6 = cos-1- cosπ6 as, cosπ+θ = – cos θcos-1cos7π6 = cos-1- cosπ-5π6cos-1cos7π6 = cos-1– cos 5π6 as, cosπ-θ = – cos θ

The correct answer is B.

Question 20:
Find the values of is equal to
(A) (B) (c) (D) 1
Answer:
Let Then,
We know that the range of the principal value branch of

The correct answer is D.

Question 21:
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.

NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Miscellaneous Solutions

Question 1:
Find the value of
Answer:
We know that cos⁻¹ (cos x) = x if , which is the principal value branch of cos ⁻¹x.
Here,
Now, can be written as:

Question 2:
Find the value of
Answer:
We know that tan⁻¹ (tan x) = x if , which is the principal value branch of tan ⁻¹x.
Here,
Now,
can be written as: