Summary of Inverse Trigonometric Functions

Key Concepts

• Inverse of a function exists if it is one-one and onto.
• Trigonometric functions are not one-one and onto over their natural domains and ranges.
• Restrictions on domains and ranges ensure the existence of inverses for trigonometric functions.

Basic Definitions

• Sine Function: $ext{sine}: ext{R} o [-1, 1]$
• Cosine Function: $ext{cosine}: ext{R} o [-1, 1]$
• Tangent Function: ext{tan}: ext{R} ackslash ig\{ x: x = (2n + 1) rac{ ext{π}}{2}, n ext{ in } ext{Z} \}
• Cotangent Function: ext{cot}: ext{R} ackslash ig\\{ x: x = n ext{π}, n ext{ in } ext{Z} \}
• Secant Function: ext{sec}: ext{R} ackslash ig\\{ x: x = (2n + rac{ ext{π}}{2}), n ext{ in } ext{Z} \} \to ext{R} ackslash (-1, 1)
• Cosecant Function: ext{cosec}: ext{R} ackslash ig\\{ x: x = n ext{π}, n ext{ in } ext{Z} \} \to ext{R} ackslash (-1, 1)

Properties of Inverse Trigonometric Functions

• The principal value branches of inverse trigonometric functions are defined as follows:
  • $y = ext{sin}^{-1} x$: Domain $[-1, 1]$, Range $[-\frac{\text{π}}{2}, \frac{\text{π}}{2}]$
  • $y = ext{cos}^{-1} x$: Domain $[-1, 1]$, Range $[0, \text{π}]$
  • $y = ext{cosec}^{-1} x$: Domain $\text{R} \backslash (-1, 1)$, Range $[-\frac{\text{π}}{2}, 0) \cup (0, \frac{\text{π}}{2}]$
  • $y = ext{sec}^{-1} x$: Domain $\text{R} \backslash (-1, 1)$, Range $[0, \frac{\text{π}}{2}) \cup (\frac{\text{π}}{2}, \text{π}]$
  • $y = ext{tan}^{-1} x$: Domain $\text{R}$, Range $(-\frac{\text{π}}{2}, \frac{\text{π}}{2})$
  • $y = ext{cot}^{-1} x$: Domain $\text{R}$, Range $(0, \text{π})$

Important Notes

• $ext{sin}^{-1} x$ should not be confused with $(\text{sin} x)^{-1}$.
• The principal value of an inverse trigonometric function lies within its defined range.

Inverse Trigonometric Functions

2.1 Introduction

• The inverse of a function f, denoted by f⁻¹, exists if f is one-one and onto.
• Trigonometric functions are not one-one and onto over their natural domains and ranges; hence their inverses do not exist without restrictions.
• This chapter discusses restrictions on domains and ranges of trigonometric functions to ensure the existence of their inverses.
• The inverse trigonometric functions are important in calculus and have applications in science and engineering.

2.2 Basic Concepts

Trigonometric Functions

• Sine Function:
  • Domain: R
  • Range: [-1, 1]
• Cosine Function:
  • Domain: R
  • Range: [-1, 1]
• Tangent Function:
  • Domain: R ackslash {x: x = (2n + 1)π, n ∈ Z}
  • Range: R
• Cotangent Function:
  • Domain: R ackslash {x: x = nπ, n ∈ Z}
  • Range: R
• Secant Function:
  • Domain: R ackslash {x: x = (2n + π/2), n ∈ Z}
  • Range: R ackslash (-1, 1)
• Cosecant Function:
  • Domain: R ackslash {x: x = nπ, n ∈ Z}
  • Range: R ackslash (-1, 1)

2.3 Properties of Inverse Trigonometric Functions

• The principal value branch of inverse trigonometric functions is defined as follows:
  • If y = sin⁻¹ x, then x = sin y and vice versa.
  • The principal value of sin⁻¹ x lies in the range of [-π/2, π/2].
  • Similar definitions apply for other inverse trigonometric functions.

Example Problems

1. Find the principal value of sin⁻¹(1/2):
   • Solution: sin⁻¹(1/2) = π/6
2. Find the principal value of cot⁻¹(1):
   • Solution: cot⁻¹(1) = π/4

2.4 Summary of Domains and Ranges

Common Mistakes and Exam Tips

Common Pitfalls

• Confusing Inverse Functions: Students often confuse inverse trigonometric functions (e.g., sin⁻¹x) with the reciprocal of trigonometric functions (e.g., (sin x)⁻¹). Remember that sin⁻¹x is the angle whose sine is x, while (sin x)⁻¹ is the cosecant of x.
• Ignoring Domain Restrictions: Inverse trigonometric functions have specific domains and ranges. For example, sin⁻¹x is defined for x in [-1, 1] and its range is [-π/2, π/2]. Not adhering to these can lead to incorrect answers.
• Misapplying Principal Values: When asked for the principal value of an inverse function, ensure you are within the specified range. For instance, the principal value of sin⁻¹(1/2) is π/6, not any other angle that has the same sine value.
• Incorrectly Solving Equations: When solving equations involving inverse trigonometric functions, ensure to apply the correct identities and properties. For example, if sin⁻¹x = y, then sin(y) = x must hold true.

Tips for Exam Success

• Review Graphs: Familiarize yourself with the graphs of inverse trigonometric functions, as they can help visualize the domain and range.
• Practice with Examples: Work through various examples to solidify your understanding of how to find principal values and solve equations involving inverse trigonometric functions.
• Check Your Work: After solving problems, check if your answers fall within the expected ranges for inverse functions. This can help catch mistakes early.
• Understand Properties: Make sure to understand the properties of inverse trigonometric functions, such as sin(sin⁻¹x) = x for x in [-1, 1]. This can be useful for verification.