NCERT Solutions for Class 12 Maths Chapter 2 Exercise 2.2 (Inverse Trigonometric Function)

Class 12 Maths Chapter 2 Exercise 2.2 Inverse Trigonometric Function:- Inverse Trigonometric Functions are the inverses of the basic trigonometric functions like sine, cosine, and tangent, allowing us to determine angles from given trigonometric ratios. These functions are essential in solving equations involving trigonometric expressions. Exercise 2.2 in Chapter 2 of Class 12 Maths involves evaluating these functions, proving related identities, and solving equations. For instance, 

Sin − 1(12)sin −1 ( 21​ ) evaluates to 𝜋66π​ , as sin⁡30∘=12sin30 ∘ = 21. The mastery of this chapter is crucial for advanced applications in calculus and other mathematical analyses. Get the detailed NCERT Solutions for Class 12 Maths Chapter 2 Exercise 2.2 Inverse Trigonometric Function below.

NCERT Solutions for Class 12 Maths Chapter 2 Exercise 2.2 Inverse Trigonometric Function

Read the Class 12 Maths Chapter 2 Exercise 2.2 Inverse Trigonometric Function NCERT Solutions below. 

Question1.

Solution :

Proved.

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Question2.

Solution :

Proved.

Question3.

Solution :

Proved.

Question4.

Solution :

Proved.

NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2

Write the following functions in the simplest form:

Question5.

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NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.3

Question6.

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NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4

Question7.

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Question8.

Solution :

Question9.

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Question10.

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Find the values of each of the following:

Question11.

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Question12.

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Question13.

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Question14. If then find the value of x

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Question15. Ifthen find the value of x

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

Question16.

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Question17.

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Question18.

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Question19.

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Question20. is equal to:

(A) 1/2

(B) 1/3

(C) 1/4

(D) 1

Solution :

Therefore, option (D) is correct.

Question21.is equal to:

(A) π

(B) -π/2

(C) 0

(D) 2√3

Solution :

Therefore, option (B) is correct

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Class 12 Maths Chapter 2 Exercise 2.2 Inverse Trigonometric Function Summary

1. Inverse Trigonometric Functions

Inverse trigonometric functions, or arctrigonometric functions, are introduced to find the angle whose trigonometric ratio (sine, cosine, or tangent) is a given number.

2. Common Inverse Trigonometric Functions

The three most commonly encountered inverse trigonometric functions are:

• Sine Inverse: denoted by sin⁻¹(x) or arcsin(x)

• Cosine Inverse: denoted by cos⁻¹(x) or arccos(x)

• Tangent Inverse: denoted by tan⁻¹(x) or arctan(x)

3. Important Points

• Inverse trigonometric functions are defined only for specific input value ranges. This is because the original trigonometric functions (sine, cosine, tangent) have periodic repetitions. For instance, sine function repeats its values every 2π. Therefore, to ensure a unique output angle for a given sine value, we restrict the input range for the sine inverse function.

• The range of sin(x) is [-1, 1], and correspondingly, the range of arcsin(x) is also [-1, 1]. Similar restrictions apply to other inverse trigonometric functions.

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Class 12 Maths Chapter 2 Exercise 2.2 Inverse Trigonometric Function FAQs

Q1. What are Inverse Trigonometric Functions?

Ans. Inverse Trigonometric Functions are the inverses of trigonometric functions like sine, cosine, and tangent, used to find angles from given trigonometric ratios.

Q2. What is the principal value branch?

Ans. The principal value branch is the range within which the inverse trigonometric functions are defined to be single-valued. For example, 

sin−1𝑥sin −1 x is defined for [−𝜋2,𝜋2][− 2π​ , 2π​ ].

Q3. How do you prove identities involving inverse trigonometric functions?

Ans. Identities can be proved using known trigonometric identities, properties of inverse functions, and algebraic manipulation.

Q4. What is the domain and range of tan −1𝑥tan −1 x?

Ans. The domain of tan −1𝑥tan −1 x is all real numbers (−∞,∞)(−∞,∞) and the range is (−𝜋2,𝜋2)(− 2π​ , 2π​ ).

Q5. How are inverse trigonometric functions used in calculus?

Ans. They are used in integration and differentiation, for example, ∫11−𝑥2𝑑𝑥=sin⁡−1𝑥+𝐶∫ 1−x 2 ​ 1​ dx=sin −1 x+C.