Inverse Trigonometric Functions

Introduction

• Comprehensive one-shot lecture planned.
• Chapter involves detailed concepts, best understood in sequence.
• Prepare to focus for 4-5 hours; can break into segments for first-timers.
• Trigonometry basics will be recalled as needed.
• Practice sheet from last year's session included in the description.

Nature of the Chapter

• Initially easy but can become confusing later.
• Focus on the basics at the beginning to avoid difficulties later.
• Requires some prior knowledge of trigonometric basics.
• Values of inverse trigonometric functions are angles.

Important Formulae and Points

• Inverse Trigonometric Functions:
  • sin⁻¹(x) represents an angle whose sine is x.
  • cos⁻¹(x) represents an angle whose cosine is x.
  • tan⁻¹(x) represents an angle whose tangent is x.
• Inverse of trigonometric functions converts a trigonometric value back to an angle:
  • sin(sin⁻¹(x)) = x for x in [-1, 1].
  • cos(cos⁻¹(x)) = x for x in [0, π].
  • tan(tan⁻¹(x)) = x for all x.
• Inverse Values for Common Angles:
  • sin⁻¹(1) = π/2, cos⁻¹(1) = 0, tan⁻¹(1) = π/4.
  • sin⁻¹(1/2) = π/6, cos⁻¹(1/2) = π/3, tan⁻¹(1/√3) = π/6.
• Key Concept: Inverse values determine an angle.

Finding Domains and Ranges

• Domain: Values for which the inverse function is defined.
  • Example: sin⁻¹(x) is defined for x in [-1, 1].
• Range: Values of the inverse function.
  • Example: Range of sin⁻¹(x) is [-π/2, π/2].
• Apply transformations and check ranges using inequalities.
• Practice problems to understand domain and range criteria.

Converting Between Formulas

• Principal Domain: Specific range where inverse trigonometric functions are uniquely defined.
  • sin and cos must be within their principal domains.
• Use primary results and convert between forms.
  • Example: Convert sin⁻¹(x) to cos⁻¹ and vice versa for simplicity.
• Handle negative values carefully using identities:
  • sin⁻¹(-x) = -sin⁻¹(x)
  • cos⁻¹(-x) = π - cos⁻¹(x)

Composition of Functions

• Compositions like sin⁻¹(sin(x)) and cos⁻¹(cos(x)).
• Depends on whether the function and its argument are within respective domains.
• Ensure angles fall within the principal range for exact results.

General Formulas

• sin(sin⁻¹(x)) = x, cos(cos⁻¹(x)) = x, for valid x values.
• For negative values, transform using identities.
  • Example: cos⁻¹(-x) = π - cos⁻¹(x).
• Useful in simplifying complex expressions.

Summation of Series

• Handle summation questions for sequences involving inverse trigonometric functions.
• Apply known formulas for reducing summations.
  • tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a + b) / (1 - ab)) for ab < 1.

Simplification of Expressions

• Use standard forms and results to simplify nested functions.
• Often, reconvert nested functions into primary forms for easy simplification.
• Cross-check results for domains and ranges.

Solving Equations

• Solve equations involving inverse trigonometric functions by applying domain constraints.
• Apply inverse functions correctly, ensuring the argument falls within the acceptable range.
• Verify solutions by substituting back into the original equation.

Problems and Applications

• Practice problems are critical for mastery.
• Problems often test understanding of domain, range, simplification, and composition.
• Effective use of identities and properties simplifies complex problems.
• Ranges and domain checks ensure correct solutions.

Practice Sheet

• Download and complete the practice sheet from the description for a thorough understanding.
• Solutions are provided for reference and self-checking.

Conclusion

• Key concepts from inverse trigonometric functions are interconnected.
• Focus on understanding primary results and applying them correctly.
• Regular practice and problem-solving enhance comprehension.

Important: Always apply results within constraints of domains and ranges. Verify solutions thoroughly.