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title: "Inverse Trigonometric Functions"
board: "CBSE"
curriculum: "CBSE"
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subject: "Mathematics"
book: "Mathematics Part - I"
chapter: "Inverse Trigonometric Functions"
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---

# Inverse Trigonometric Functions
In this chapter, we will study the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behavior through graphical representations. Besides, some elementary properties will also be discussed. The inverse trigonometric functions play an important role in calculus for they serve to define many integrals.

---

## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part - I |
| Chapter | Inverse Trigonometric Functions |
| Pages | 18-33 |

---

## Chapter Summary

### Short Summary
This chapter focuses on understanding inverse trigonometric functions, their domains, ranges, and properties, utilizing graphical representations to elucidate their behavior.

### Detailed Summary
We begin by recalling that if a function is one-one and onto, its inverse can be defined. We explore trigonometric functions, which are not one-one and onto over their natural domains and ranges, and thus do not have inverses unless their domains are restricted. Each inverse trigonometric function (such as $\sin^{-1}$, $\cos^{-1}$, etc.) is introduced with its specific domain and range, with emphasis on the principal value branches. Relationships and properties of these functions are examined, facilitating their application in calculus.

---

## Topic-Wise Explanation

### Introduction
Inverse trigonometric functions arise from the need to reverse the trigonometric functions, ensuring they are defined within specific ranges and domains.

### Basic Concepts
We define the trigonometric functions as follows:
- Sine: $\sin: \mathbb{R} 	o [-1, 1]$
- Cosine: $\cos: \mathbb{R} 	o [-1, 1]$
- Tangent: $	an: \mathbb{R} - \{x : x = (2n + 1)(rac{\pi}{2}), n \in \mathbb{Z}\} 	o \mathbb{R}$
- Cotangent: $\cot: \mathbb{R} - \{x : x = n\pi, n \in \mathbb{Z}\} 	o \mathbb{R}$
- Secant: $\sec: \mathbb{R} - \{x : x = (2n + 1)(rac{\pi}{2}), n \in \mathbb{Z}\} 	o \mathbb{R} - (-1, 1)$
- Cosecant: $\csc: \mathbb{R} - \{x : x = n\pi, n \in \mathbb{Z}\} 	o \mathbb{R} - (-1, 1)$

### Properties of Inverse Trigonometric Functions
Key properties include the reflection of graphs about the line $y = x$ and the relationship between inverse functions and their original functions.

### Inverse Function Formulas
Each inverse function has distinct formulations based on their restricted domains.

### Principal Value Branches
This section details the principal value branches for the inverse functions, defining their respective domains and ranges explicitly. The cases for different functions are summarized as follows:
- $\sin^{-1}: [-1, 1] 	o [-rac{\pi}{2}, rac{\pi}{2}]$
- $\cos^{-1}: [-1, 1] 	o [0, \pi]$
- $\csc^{-1}: \mathbb{R} - (-1, 1) 	o [-rac{\pi}{2}, rac{\pi}{2}] - \{0\}$
- $\sec^{-1}: \mathbb{R} - (-1, 1) 	o [0, \pi] - \{rac{\pi}{2}\}$
- $	an^{-1}: \mathbb{R} 	o [-rac{\pi}{2}, rac{\pi}{2}]$
- $\cot^{-1}: \mathbb{R} 	o (0, \pi)$

---

## Core Ideas

| Idea | Explanation |
| :--- | :--- |
| Existence of Inverses | A function has an inverse if it is one-one and onto. |
| Restrictions | The domains of trigonometric functions need to be restricted to define their inverses. |

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## Key Concepts

| Concept | Meaning |
| :--- | :--- |
| Inverse Function | A function that reverses the action of another function. |
| Principal Value Branch | The main interval where the inverse function is defined. |

---

## Important Points for Revision

* The sine function has to be restricted to $[-rac{\pi}{2}, rac{\pi}{2}]$ for its inverse to exist.
* The cosine function is bijective when restricted to $[0, \pi]$.
* The tangent function is defined as one-one when restricted to $(-rac{\pi}{2}, rac{\pi}{2})$.
* Each inverse trigonometric function has a principal value branch.
* $\sin^{-1}(x)$ and $(\sin x)^{-1}$ are not the same thing.
* Graphs of inverse functions can be derived from their respective trigonometric functions by reflection about the line $y = x$.
* The inverse trigonometric functions are integral in calculus.
* Mutual relationships exist among inverse trigonometric functions.

---

## Vocabulary and Glossary

| Word / Phrase | Meaning |
| :--- | :--- |
| Trigonometric Function | Functions related to angles and sides of triangles. |
| One-one Function | A function where each output is associated with exactly one input. |

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## Practice Questions

### Short Answer Questions
1. Define the domain of the sine function.
2. What is the principal value branch of $\cos^{-1}$?
3. How is $	an^{-1}(x)$ defined?
4. Identify the range for $\csc^{-1}(x)$?
5. Explain the term “principal value.”

### Long Answer Questions
1. Discuss the significance of restricting the domains of trigonometric functions when defining their inverses.
2. Provide the derivation of the relationships between $\sin^{-1}$ and $\sin$.
3. Explain the graphical representation of an inverse function with respect to its original function.

---

## Related Concepts

* Trigonometric Functions
* Calculus
* Graphical Representations

---

## Source Attribution

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| :--- | :--- |
| Source | Edzy |
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