Inverse Trigonometric Functions

Introduction

• Discussion on Chapter 2: Inverse Trigonometric Functions (ITF)
• Previous chapter: Relations and Functions
• Importance for CBSE board exams and competitive exams
• Key topics: Principal value branch, domain, co-domain, graphs, basics, and property-based questions

What is Inverse Trigonometric Function?

• Definition: An inverse function undoes the action of the function
• A function F from A to B is invertible if there exists a function G from B to A
  • G = inverse of F
• Necessary condition for invertibility: F must be bijective (one-one and onto)

Types of Functions

• Emphasized on types of functions discussed in the first chapter
• For inverses to exist, functions must be bijective

Trigonometric Functions Recap

• Trigonometric ratios: sin(x), cos(x), tan(x), cot(x), sec(x), cosec(x)
• Discussed domain and range for each trigonometric function
  • Example: sin(x) and cos(x) both have domain as Real numbers and range as [-1, 1]
• In-depth analysis of each function with specific intervals and undefined values

Inverse Trigonometric Functions (ITFs)

• Key Inverse Trigonometric Functions: sin^-1(x), cos^-1(x), tan^-1(x), cot^-1(x), sec^-1(x), cosec^-1(x)
• Domain and range interchange when dealing with inverses

Inverse Function Graphs

• Graphing techniques for sin^-1(x), cos^-1(x), tan^-1(x), cot^-1(x), sec^-1(x), and cosec^-1(x)
• Explained how to derive graphs from the original trigonometric functions
• Importance of being familiar with the trigonometric principles of graph transformations

Properties of Inverse Trigonometric Functions

Type 1: Inverse Property

• sin^-1(sin(x)) = x if x is in principal value branch [-π/2, π/2]
• Similar rules for cos^-1(x), tan^-1(x), cot^-1(x), sec^-1(x), and cosec^-1(x)

Type 2: Identity Property

• sin(sin^-1(x)) = x if x is in domain [-1, 1]
• Similar rules for cos^-1(x), tan^-1(x), cot^-1(x), sec^-1(x), and cosec^-1(x)

Type 3: Negation Property

• sin^-1(-x) = -sin^-1(x)
• Variants for cos^-1(x) and other functions

Type 4: Additive Property

• tan^-1(x) + tan^-1(y) = tan^-1((x + y) / (1 - xy))

Type 5: Multiplicative Property

• tan^-1(x) - tan^-1(y) = tan^-1((x - y) / (1 + xy))

Type 6: Angle Transformation Formulas

• 2sin^-1(x) = sin^-1(2x√(1-x²))
• 3sin^-1(x) = sin^-1(3x - 4x³)

Application Examples

Example 1: Finding the Domain

• Example problem on finding the domain of sin^-1(2x)

Example 2: Problem on Property Utilization

• sin(2sin^-1(0.6)) and use properties to solve

Conclusion

• Emphasized the importance of properties and problem-solving
• Encouraged practice for confidence and familiarity
• Invitation to next lecture and request to like and subscribe for more content