Exercise No. 2.2 Chapter 2 - Inverse Trigonometric Functions NCERT Solutions for Class 12 Science Math

Chapter 2 - Inverse Trigonometric Functions

NCERT Solutions for Class 12 Science Math

Exercise No. 2.2

Question 1:

Prove

ANSWER:

To prove:

Let x = sinθ. Then,

We have,

R.H.S. =

= 3θ

= L.H.S.

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Question 2:

Prove

ANSWER:

To prove:

Let x = cosθ. Then, cos−1 x =θ.

We have,

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Question 3:

Prove

ANSWER:

To prove:

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Question 4:

Prove

ANSWER:

To prove:

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Question 5:

Write the function in the simplest form:

ANSWER:

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Question 6:

Write the function in the simplest form:

ANSWER:

Put x = cosec θ ⇒ θ = cosec−1 x

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Question 7:

Write the function in the simplest form:

ANSWER:

Question 8:

Write the function in the simplest form:

ANSWER:

$\tan^{- 1} (\frac{\cos x - \sin x}{\cos x + \sin x}) = \tan^{- 1} (\frac{1 - \frac{\sin x}{\cos x}}{1 + \frac{\sin x}{\cos x}}) = \tan^{- 1} (\frac{1 - \tan x}{1 + \tan x}) = \tan^{- 1} (1) - \tan^{- 1} (\tan x) (\tan^{- 1} \frac{x - y}{1 + x y} = \tan^{- 1} x - \tan^{- 1} y) = \frac{π}{4} - x$

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Question 9:

Write the function in the simplest form:

ANSWER:

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Question 10:

Write the function in the simplest form:

ANSWER:

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Question 11:

Find the value of

ANSWER:

Let

. Then,

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Question 12:

Find the value of

ANSWER:

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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.

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Question 14:

If

, then find the value of x.

ANSWER:

On squaring both sides, we get:

Hence, the value of x is

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Question 15:

If

, then find the value of x.

ANSWER:

Hence, the value of x is

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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:

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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:

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Question 18:

Find the values of

ANSWER:

Let

. Then,

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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.

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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.