Class 12th Mathematics: Inverse Trigonometric Functions

Introduction

• Reviewing Inverse Trigonometric Functions
• Importance in science and engineering
• Prerequisites: Basic trigonometric functions and their properties, domain, and range

Key Concepts

Inverse Functions

• Functions need to be one-to-one (bijective) for their inverses to exist
• Trigonometric functions (sine, cosine, etc.) are not inherently one-to-one
• Restriction of domain to make them one-to-one

Restricting Domain for Inverse Functions

• Example: Sine function
  • Restricted domain: (-\frac{\pi}{2}) to (\frac{\pi}{2})
  • Corresponding range: -1 to 1
  • Inverse defined within this restricted interval
• Example: Cosine function
  • Restricted domain: 0 to (\pi)
  • Corresponding range: -1 to 1
  • Inverse is defined within this interval
• General note: Each trigonometric function has a principal value range for their inverses

Principal Value Range

• Definition and significance
• Example process for fixing domain and range for sine and cosine functions

Graph of Inverse Trigonometric Functions

• Inverse functions' graphs are reflections of original functions along the line y=x
• Examples shown for sine, cosine, secant, cosecant, tangent, and cotangent
• Important tip: Distinction between (\sin^{-1}(x)) and (\sin(x)^{-1})

Sample Problems

1. Compute principle value for (\sin^{-1} (-\frac{\sqrt{3}}{2}))
   • Convert to principal value: ( \sin^{-1} (-\frac{\sqrt{3}}{2}) = -\frac{\pi}{3})
2. Simplify complex expressions using known inverse properties
   • Example: Solve ( \sin (\sin^{-1} (\frac{1}{2}) + \cos^{-1} (x)) = 1)
3. Expand expressions: converting to simplest trigonometric form

Properties of Inverse Functions

• Properties for combinations: ( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} )
• Tan and Cot properties
• Working through complex problems with these properties

Conclusion

• Importance of mastering the principal value range and properties
• Relevance for future courses in integration and higher mathematics
• Practice with graph plotting and solving inverse problems