NCERT Class 12 Mathematics Mathematics Part-I: Chapter 2 — MATHEMATICS
This chapter, NCERT Class 12 Mathematics Part-I, Chapter 2, introduces Inverse Trigonometric Functions. It builds upon the concept of inverse functions from Chapter 1, explaining that inverses exist only for one-one and onto functions. Trigonometric functions, in their natural domains, are not one-one, necessitating restrictions on their domains and ranges to define their inverses. The chapter details the basic definitions of trigonometric functions and their domains/ranges. It then focuses on defining the inverse sine function (sin⁻¹) by restricting the sine function's domain to [–π/2, π/2], making it one-one and onto with a range of [–1, 1]. The principal value branch of sin⁻¹ is established with domain [–1, 1] and range [–π/2, π/2]. Key identities like sin(sin⁻¹x) = x and sin⁻¹(sin x) = x are introduced, along with the relationship y = sin⁻¹x implies sin y = x. The chapter also touches upon obtaining the graph of inverse functions by interchanging axes. This foundational understanding is crucial for calculus and various scientific applications.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-I |
| Chapter | Chapter 2 — MATHEMATICS |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 5 minutes |
| Word count | 836 |
Learning outcomes
- Understand the necessity of restricting domains for trigonometric functions to define their inverses.
- Define and understand the principal value branch of inverse trigonometric functions, starting with sin⁻¹.
- Recall the domains and ranges of basic trigonometric functions.
- Understand the relationship between a function and its inverse, specifically for trigonometric functions.
- Recognize the identity sin(sin⁻¹x) = x and sin⁻¹(sin x) = x under specific conditions.
Vocabulary
| Word | Meaning |
|---|---|
| Inverse function | A function that reverses the action of another function. |
| One-one function | A function where each element of the range corresponds to exactly one element of the domain. |
| Onto function | A function where every element in the codomain is mapped to by at least one element in the domain. |
| Domain | The set of all possible input values for a function. |
| Range | The set of all possible output values for a function. |
| Trigonometric functions | Functions that relate an angle of a right-angled triangle to the ratios of its sides (e.g., sine, cosine, tangent). |
| Inverse trigonometric functions | Functions that are the inverses of trigonometric functions (e.g., arcsine, arccosine). |
| Principal value branch | The specific branch of an inverse trigonometric function that is chosen to define the function uniquely. |
| Arc sine function | Another name for the inverse sine function (sin⁻¹). |
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Practice questions
- Why are trigonometric functions not one-one over their natural domains? Answer: Trigonometric functions are periodic, meaning they repeat their values over intervals, thus failing the one-one criterion where each output must correspond to a unique input.
- What is the domain and range of the principal value branch of the sine function? Answer: The domain is [–1, 1] and the range is [–π/2, π/2].
- If y = sin⁻¹x, what is the corresponding equation in terms of the sine function? Answer: sin y = x.
- How can the graph of an inverse function be obtained from the graph of the original function? Answer: By interchanging the x and y axes.
Practice MCQs
Q1. Which of the following is the natural domain of the sine function?
Explanation: The sine function is defined for all real numbers as input.
Q2. For the inverse sine function (sin⁻¹), what is the range of its principal value branch?
Explanation: The principal value branch of sin⁻¹ is defined to have a range of [–π/2, π/2] to ensure it is a function.
Q3. If sin(x) = 0.5, and x is in the principal value range of sin⁻¹, what is sin⁻¹(0.5)?
Explanation: sin(π/6) = 0.5, and π/6 lies within the principal value range [–π/2, π/2].
Q4. The identity sin(sin⁻¹x) = x is valid for which values of x?
Explanation: The identity sin(sin⁻¹x) = x holds true for all x in the domain of sin⁻¹, which is [–1, 1].
Q5. Which of the following trigonometric functions is NOT one-one over its natural domain?
Explanation: Sine, cosine, and tangent functions are all periodic and thus not one-one over their entire natural domains.
Frequently asked questions
What are inverse trigonometric functions?
Inverse trigonometric functions are the inverse functions of the basic trigonometric functions (like sine, cosine, tangent). They are used to find the angle corresponding to a given trigonometric ratio.
Why do we need to restrict the domain of trigonometric functions to define their inverses?
Trigonometric functions are periodic, meaning they repeat their values. To make them one-one and onto (necessary for an inverse to exist), their domains must be restricted to specific intervals.
What is the principal value branch of the inverse sine function (sin⁻¹)?
The principal value branch of sin⁻¹ is defined with a domain of [–1, 1] and a range of [–π/2, π/2].
What is the relationship between y = sin⁻¹x and sin y = x?
If y = sin⁻¹x, it means that sin y = x, where x is in the domain [–1, 1] and y is in the range [–π/2, π/2].
How is the graph of sin⁻¹x related to the graph of sin x?
The graph of sin⁻¹x can be obtained by reflecting the graph of sin x (within its principal domain) across the line y = x, which is equivalent to interchanging the x and y axes.
Are all trigonometric functions invertible over their natural domains?
No, trigonometric functions like sine, cosine, and tangent are not one-one over their natural domains, so their inverses do not exist without domain restrictions.
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NCERT Class 12 Mathematics — Mathematics Part-I — Chapter 2 — MATHEMATICS. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.