Trigonometric Equations

Trigonometric Equations: Basics

• Equations involving trigonometric functions of an unknown angle are called trigonometric equations.
• A solution is a value of θ that satisfies the equation.
• Throughout this chapter, n ∈ Z (n is an integer).

Periodic Functions

• A function f(x) is periodic if f(x + T) = f(x) for all x.
• The smallest positive T is called the fundamental period.

Periods of Standard Functions

• sin(ax + b), cos(ax + b), sec(ax + b), cosec(ax + b): 2π / a
• tan(ax + b), cot(ax + b): π / a
• |sin(ax + b)|, |cos(ax + b)|, |sec(ax + b)|, |cosec(ax + b)|: π / a
• |tan(ax + b)|, |cot(ax + b)|: π / (2a)

Standard Trigonometric Equations & General Solutions

• sin θ = 0 ⇒ θ = nπ
• cos θ = 0 ⇒ θ = (2n + 1)π / 2
• tan θ = 0 ⇒ θ = nπ
• sin θ = 1 ⇒ θ = (4n + 1)π / 2
• cos θ = 1 ⇒ θ = 2nπ

Equations of the Form f(θ) = f(α)

• sin θ = sin α ⇒ θ = nπ + (−1)ⁿα
  (α ∈ [−π/2, π/2])
• cos θ = cos α ⇒ θ = 2nπ ± α
  (α ∈ (0, π])
• tan θ = tan α ⇒ θ = nπ + α
  (α ∈ (−π/2, π/2])

Equations Involving Squares

• sin²θ = sin²α ⇒ θ = nπ ± α
• cos²θ = cos²α ⇒ θ = nπ ± α
• tan²θ = tan²α ⇒ θ = nπ ± α

Simultaneous Trigonometric Equations

• If sin θ = sin α and cos θ = cos α, then θ = 2nπ + α
• If sin θ = sin α and tan θ = tan α, then θ = 2nπ + α
• Rule: Find common solutions in [0, 2π] and then add 2nπ

General Solution of a cosθ + b sinθ = c

• Let a = r cosφ, b = r sinφ
• Where r = √(a² + b²) and tanφ = b / a
• Then the equation becomes: cos(θ − φ) = c / r
• If |c| > √(a² + b²) ⇒ No solution
• If |c| ≤ √(a² + b²), let cosα = |c| / √(a² + b²)
• General solution: θ = 2nπ ± α + φ

Important Notes

• If α is the least positive solution, then general solution is θ = 2nπ + α
• Always check the domain of the given equation