Here we are going through the ncert solutions for class 12 maths chapter 2 – inverse trigonometric functions so before going through the ncert solutions make sure to go through the textbook that helps you to understand the solutions more easily
EXERCISE 2.1
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:
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:
tan−1(cosx−sinxcosx+sinx)=tan−1(1−sinxcosx1+sinxcosx)=tan−1(1−tanx1+tanx)=tan−1(1)−tan−1(tanx) (tan−1x−y1+xy=tan−1x−tan−1y)=π4−xtan-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
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−1(cos7π6) = cos−1[cos(π+π6)]cos−1(cos7π6) = cos−1[− cosπ6] [as, cos(π+θ) = − cos θ]cos−1(cos7π6) = cos−1[− cos(π−5π6)]cos−1(cos7π6) = cos−1[−{− cos (5π6)}] [as, cos(π−θ) = − cos θ]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.
Miscellaneous Exercise on Chapter 2
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