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Class XII                   Chapter 2 – Inverse Trigonometric Functions                       Maths

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:

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:

We know that the range of the principal value branch of tan⁻¹ is

Therefore, the principal value of

Question 5:

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:

We know that the range of the principal value branch of sec⁻¹ is

Therefore, the principal value of

Question 8:

We know that the range of the principal value branch of cot⁻¹ is (0,π) and

Therefore, the principal value of

Question 9:

We know that the range of the principal value branch of cos⁻¹ is [0,π] and

.

Therefore, the principal value of

Question 10:

We know that the range of the principal value branch of cosec⁻¹ is

Therefore, the principal value of

Question 11:

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

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:

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:

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:

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:

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:

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.

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:

Question 3:

Now, we have:

Question 4:

Now, we have:

Question 5:

Prove Answer

Now, we will prove that:

Question 6:

Now, we have:

Question 7:

Using (1) and (2), we have

Question 8:

Prove Answer

Question 9:

Question 10:

Question 11:

Prove                                                                     [Hint: putx = cos 2θ]

Answer

Question 12:

Prove  Answer

Question 13:

Question 14:

Solve Answer

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

(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:

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

Hence, the correct answer is C.