Understanding Restrictions on Trigonometric Identities and Expressions

Overview

• Non-Permissible Values (PVs): Restrictions or non-permissible values occur in trig identities and expressions due to algebraic reasons or intrinsic properties of trig functions.
• Common Causes of Restrictions:
  • Denominators that cannot be zero.
  • Certain values for trig functions are undefined.

Examples and Analysis

Example 1: Expression with Sine and Cosine

• Expression: Involves sin(x) and cos(x).
  • Sine and Cosine: Defined for all x. No inherent restrictions from these functions.
  • Denominator: When it includes a variable, it cannot be zero.
    • For cos(x) - 1 = 0, solve for restrictions: cos(x) ≠ 1.
    • Values of x: Where cos(x) = 1, i.e., x = 0, 2π, 4π, ..., x = 2πn where n is an integer.

Example 2: Identity Involving Secant and Tangent

• Identity: Requires looking at both sides of the equation.

Trig Functions Involved

• Secant and Tangent:
  • Tangent (tan(θ)): Defined as sin(θ)/cos(θ). Undefined where cos(θ) = 0.
    • Values of θ: θ = π/2, 3π/2, 5π/2, ..., expressed as θ = π/2 + πn.
  • Secant (sec(θ)): Defined as 1/cos(θ). Same restrictions as tan(θ).

Expressions in Denominators

• Denominator Restrictions:
  • 1 + tan(θ) ≠ 0: Solve for tan(θ) ≠ -1.
    • Where tan(θ) = -1: Use graph or angle properties.
    • Values of θ: θ = 3π/4, 7π/4, ..., θ = 3π/4 + πn.
  • sin²(θ) ≠ 0:
    • Where sin(θ) = 0: θ = 0, π, 2π, ..., θ = πn.

Combining Restrictions

• Combine Different Restrictions:
  • From Tangent and Secant: θ ≠ π/2 + πn.
  • From 1 + tan(θ): θ ≠ 3π/4 + πn.
  • From sin²(θ): θ ≠ πn.
  • Simplified Expression: Combine all into simplest form.
    • Overlap and fill gaps resulting in a restriction like θ = π/2n and θ = 3π/4 + πn.

Conclusion

• Expressing Restrictions:

  • Essential for avoiding undefined or problematic values in trig identities and expressions.
  • Use graphical intuition and algebraic manipulation to determine these restrictions efficiently.

• Key takeaway: Understanding and expressing restrictions helps maintain the integrity and correctness of mathematical analysis involving trigonometric identities and expressions.