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![Page 1: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/1.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 1 of 38
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
![Page 2: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/2.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 2 of 38
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
![Page 3: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/3.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 3 of 38
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
![Page 4: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/4.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 4 of 38
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
![Page 5: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/5.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 5 of 38
Question 11:
Find the value of
Answer
Question 12:
Find the value of
Answer
![Page 6: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/6.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 6 of 38
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 7: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/7.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 7 of 38
![Page 8: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/8.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 8 of 38
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,
![Page 9: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/9.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 9 of 38
Question 3:
Prove
Answer
To prove:
Question 4:
Prove
Answer
To prove:
![Page 10: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/10.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 10 of 38
Question 5:
Write the function in the simplest form:
Answer
![Page 11: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/11.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 11 of 38
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:
![Page 12: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/12.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 12 of 38
Answer
Question 8:
Write the function in the simplest form:
Answer
![Page 13: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/13.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 13 of 38
Question 9:
Write the function in the simplest form:
Answer
Question 10:
Write the function in the simplest form:
Answer
![Page 14: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/14.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 14 of 38
Question 11:
Find the value of
Answer
Let . Then,
Question 12:
![Page 15: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/15.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 15 of 38
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:
![Page 16: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/16.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 16 of 38
If , then find the value of x.
Answer
On squaring both sides, we get:
![Page 17: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/17.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 17 of 38
Hence, the value of x is
Question 15:
If , then find the value of x.
Answer
![Page 18: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/18.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 18 of 38
Hence, the value of x is
Question 16:
Find the values of
Answer
![Page 19: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/19.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 19 of 38
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:
![Page 20: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/20.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 20 of 38
Question 18:
Find the values of
Answer
Let . Then,
Question 19:
![Page 21: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/21.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 21 of 38
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:
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 .
![Page 22: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/22.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 22 of 38
∴
The correct answer is D.
![Page 23: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/23.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 23 of 38
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:
![Page 24: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/24.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 24 of 38
Question 3:
Prove
Answer
Now, we have:
![Page 25: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/25.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 25 of 38
Question 4:
Prove
Answer
Now, we have:
![Page 26: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/26.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 26 of 38
Question 5:
Prove
Answer
![Page 27: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/27.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Now, we will prove that:
![Page 28: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/28.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 28 of 38
Question 6:
Prove
Answer
Now, we have:
![Page 29: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/29.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 29 of 38
Question 7:
Prove
Answer
Using (1) and (2), we have
![Page 30: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/30.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 30 of 38
Question 8:
Prove
Answer
![Page 31: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/31.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 31 of 38
Question 9:
Prove
Answer
![Page 32: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/32.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 32 of 38
Question 10:
Prove
Answer
Question 11:
Prove [Hint: putx = cos 2θ]
Answer
![Page 33: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/33.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 33 of 38
Question 12:
Prove
Answer
![Page 34: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/34.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 34 of 38
Question 13:
Solve
Answer
Question 14:
Solve
Answer
![Page 35: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/35.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 35 of 38
Question 15:
Solve is 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
![Page 36: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/36.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 36 of 38
(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:
![Page 37: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/37.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 37 of 38
is not the solution of the given equation.
Thus, x = 0.
Hence, the correct answer is C.
Question 17:
Solve is equal to
(A) (B). (C) (D)
Answer
![Page 38: Chapter 2 Inverse Trigonometric Functions - Amazon S3 · PDF fileClass XII Chapter 2 – Inverse Trigonometric Functions Maths Page 2 of 38 Let cosec−1 (2) = y. Then, We know that](https://reader034.fdocuments.in/reader034/viewer/2022052515/5a727e0e7f8b9a98538da151/html5/thumbnails/38.jpg)
Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Page 38 of 38
Hence, the correct answer is C.