Topics and Sub Topics in Class 11 Maths Chapter 2 Inverse Trigonometric Functions:
Section Name | Topic Name |
2 | Inverse Trigonometric Functions |
2.1 | Introduction |
2.2 | Basic Concepts |
2.3 | Properties of Inverse Trigonometric Functions |
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NCERT Solutions for Class 12 Math Chapter 2
Page No 41:
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
Page No 42:
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:

Page No 47:
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:
tan-1cosx-sinxcosx+sinx=tan-11-sinxcosx1+sinxcosx=tan-11-tanx1+tanx=tan-11-tan-1tanx tan-1x-y1+xy=tan-1x-tan-1y=π4-x
Page No 48:
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.
Page No 51:
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:


Question 3:
Prove
Answer:

Now, we have:

Question 4:
Prove
Answer:

Now, we have:

Question 5:
Prove
Answer:

Now, we will prove that:

Question 6:
Prove
Answer:

Now, we have:

Question 7:
Prove
Answer:

Using (1) and (2), we have

Question 8:
Prove
Answer:

Page No 52:
Question 9:
Prove
Answer:

Question 10:
Prove
Answer:

Question 11:
Prove [Hint: putx = cos 2θ]
Answer:

Question 12:
Prove
Answer:

Question 13:
Solve
Answer:


Question 14:
Solve
Answer:

Question 15:
Solveis equal to
(A) (B)
(C)
(D)
Answer:
Let tan−1 x = y. Then,

The correct answer is D.
Question 16:
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.
Question 17:
Solveis equal to
(A) (B).
(C)
(D)
Answer:

Hence, the correct answer is C.
NCERT Solutions for Class 12 Maths
- Chapter 1 – Relations and Functions
- Chapter 2 – Inverse Trigonometric Functions
- Chapter 3 – Matrices
- Chapter 4 – Determinants
- Chapter 5 – Continuity and Differentiability
- Chapter 6 – Application of Derivatives
- Chapter 7 – Integrals
- Chapter 8 – Applications of Integrals
- Chapter 9 – Differential Equations
- Chapter 10 – Vector Algebra
- Chapter 11 – Three dimensional Geometry
- Chapter 12 – Linear Programming
- Chapter 13 – Probability
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