RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 Textbook Exercise Questions and Answers.

Rajasthan Board RBSE Solutions for Class 12 Maths in Hindi Medium & English Medium are part of RBSE Solutions for Class 12. Students can also read RBSE Class 12 Maths Important Questions for exam preparation. Students can also go through RBSE Class 12 Maths Notes to understand and remember the concepts easily.

RBSE Class 12 Maths Solutions Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 1.
3 sin-1 x = sin-1 (3x - 4x3), x ∈ \(\left[-\frac{1}{2}, \frac{1}{2}\right]\)
Answer:
Let sin-1 x = θ ⇒ sin θ = x
∵ sin 3θ = 3 sin θ - 4 sin3θ
⇒ sin 3θ = 3x - 4x3
⇒ 3θ = sin-1(3x - 4x3)
Thus, 3 sin-1 x = sin-1 (3x - 4x3)
Hence Proved.

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 2.
3 cos-1x = cos-1 (4x3 - 3x), x ∈ \(\left[\frac{1}{2}, 1\right]\)
Answer:
Let cos-1 x = θ ⇒ cos θ = x
⇒ cos 3θ = 4 cos 3θ - 3 cos θ
⇒ cos 3θ = 4x3 - 3x
⇒ 3θ = cos-1 (4x3 - 3x)
Thus, 3 cos-1 x = cos-1 (4x3 - 3x)
Hence Proved.

Question 3.
tan-1\(\frac{2}{11}\) + tan-1\(\frac{7}{24}\) = tan-1\(\frac{1}{2}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 1

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 4.
2 tan-1\(\frac{1}{2}\) + tan-1\(\frac{1}{7}\) = tan-1\(\frac{31}{17}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 2

write the following functions in the simplest form:

Question 5.
tan-1\(\frac{\sqrt{1+x^{2}}-1}{x}\), x ≠ 0
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 3

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 6.
tan-1\(\frac{1}{\sqrt{x^{2}-1}}\), |x| > 1
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 4

Question 7.
tan-1\(\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)\), 0 < x < π
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 5

Question 8.
tan-1\(\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right), \frac{-\pi}{4}\) < x < \(\frac{3 \pi}{4}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 6

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 9.
tan-1 \(\frac{x}{\sqrt{a^{2}-x^{2}}}\), |x| < a
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 7

Question 10.
tan-1\(\left\{\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right\}\), a > 0, - \(\frac{a}{\sqrt{3}}\) < x < \(\frac{a}{\sqrt{3}}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 8

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Find the values of each of the following:

Question 11.
tan-1\(\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]\).
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 9

Question 12.
cot (tan-1 a + cot-1 a)
Answer:
We have, cot (tan-1 a + cot-1 a)
= cot \(\frac{\pi}{2}\) = 0 (∵ tan-1 a + cot-1 a = \(\frac{\pi}{2}\))
Hence, cot(tan-1 a + cot-1 a) = 0

Question 13.
\(\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right]\), |x| < 1, y > 0 and xy > 1
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 10

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 14.
If sin(sin-1 \(\left(\frac{1}{5}\right)\) + cos-1 x) = 1, then find the value of x.
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 11

Question 15.
If tan-1\(\left(\frac{x-1}{x-2}\right) \)+ tan-1\(\left(\frac{x+1}{x+2}\right)\) = \(\frac{\pi}{4}\), then find value of x.
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 12

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Find the values of each of the expression in questions 16 to 18:

Question 16.
sin-1 (sin\(\frac{2 \pi}{3}\)).
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 13

Question 17.
tan-1 \(\left(\tan \frac{3 \pi}{4}\right)\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 14

Question 18.
tan \(\left(\sin ^{-1} \frac{3}{5}+\cot ^{-1} \frac{3}{2}\right)\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 15

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 19.
cos-1 \(\left(\cos \frac{7 \pi}{6}\right)\) is equal to:
(A) \(\frac{7 \pi}{6}\)
(B) \(\frac{5 \pi}{6}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{\pi}{6}\)
Answer:
The principal value branch of cos-1 is [0, π].
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 16
Thus, option (B) is correct.

Question 20.
sin [\(\frac{\pi}{3}\)-sin-1 \(\left(-\frac{1}{2}\right)\)] is equal to:
(A) \(\frac{1}{2}\)
(B) \(\frac{1}{3}\)
(C) \(\frac{1}{4}\)
(D) 1
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 17
Thus, option(D) is correct.

RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 21.
tan-1 √3 - cot-1 (- √3) is equal to:
(A) π
(B) - \(\frac{\pi}{2}\)
(C) 0
(D) 2√3
Answer:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 18

Bhagya
Last Updated on Nov. 1, 2023, 5:17 p.m.
Published Oct. 31, 2023