2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2

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2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2

2nd PUC Maths Inverse Trigonometric Functions NCERT Text Book Questions and Answers Ex 2.2

Question 1.
\(3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right), \quad x \in\left[-\frac{1}{2}, \frac{1}{2}\right]\)
Answer:
Let sin^-1 x be θ ⇒ x = sin θ
In RHS putting the value of x
sin^-1 (3x – 4x³) ⇒ sin^-1 (3 sinθ – 4 sin³θ)
= sin^-1 (sin 3θ)
[∵ 3sinθ – 4sin³ θ = sin3θ]
= 3θ
putting the value of θ
3θ = 3 sin^-1 x = LHS
∴ 3sin^-1 x = sin^-1 (3x – 4x³)
Hence proved.

Question 2.
\(3 \cos ^{-1} x=\cos ^{-1}\left(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right]\)
Answer:
Let cos^-1 x be θ ⇒ x = cos θ
In RHS, putting the value of x
cos^-1 (4x³ – 3x) ⇒ cos¹ (4cos³ θ – 3 cosθ)
⇒ cos^-1 (cos 3θ)
[∵ 4cos³ θ – 3cosθ = cos 3θ]
= 30
putting the value of 0 = 30
⇒ 3 cos^-1 x = LHS
LHS = RHS
Hence proved

Question 3.
\(\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}=\tan ^{-1} \frac{1}{2}\)
Answer:

Question 4.
\(2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}\)
Answer:

Write the following function in the simplest form:

Question 5.
\(\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}, x \neq 0\)
Answer:

Question 6.
\(\tan ^{-1} \frac{1}{\sqrt{x^{2}-1}},|x|>1\)
Answer:
Let cosec^-1 x = 0
⇒ x = cosec 0

Question 7.
\(\tan ^{-1}(\sqrt{\frac{1-\cos x}{1+\cos x}}), 0<x<\pi\)
Answer:

Question 8.
\(\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right), \frac{-\pi}{4}<x<\frac{3 \pi}{4}\)
Answer:
Divide each term of numerator and denominator inside the brackets by cos x
\(\Rightarrow \tan ^{-1}\left[\frac{(\cos x-\sin x) / \cos x}{(\cos x+\sin x) / \cos x}\right] \Rightarrow \tan ^{-1}\left[\frac{1-\tan x}{1+\tan x}\right]\)

Question 9.
\(\tan ^{ -1 } \frac { { x } }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } ,|{ x }|<{ a }\)
Answer:

Question 10.
\(\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right), a>0 ; \frac{-a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}\)
Answer:

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:

Question 12.
cot (tan^-1 a + cot^-1 a)
Answer:

Question 13.
\(\begin{aligned}&\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right]\\&|x|<1, y>0 \text { and } x y<1\end{aligned}\)
Answer:

Question 14.
In \(\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1\)
Answer:

Question 15.
\(\text { If } \tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4}\)Then find the value of x
Answer:


Find the values of each of the expressions in Exercises 16 to 18.

Question 16.
\(\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)\)
Answer:

Question 17.
\( \tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)\)
Answer:

Question 18.
\(\tan \left(\sin ^{-1} \frac{3}{5}+\cot ^{-1}\left(\frac{3}{2}\right)\right)\)
Answer:

Choose the correct Answer:

Question 19.
\(\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right) \text { is equal to }\)
(A) \(\frac{7 \pi}{6}\)
(B) \(\frac{5 \pi}{6}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{\pi}{6}\)
Answer:

Question 20.
\(\sin \left(\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right)\)is equal to
(A) π
(B) \(-\frac{\pi}{2}\)
(C) 0
(D) \(2 \sqrt{3}\)
Answer:

Question 21.
\(\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})\) is equal to
(A) π
(B) \(-\frac{\pi}{2}\)
(C) 0
(D) \(2 \sqrt{3}\)
Answer: