Lecture Notes: Solving Trigonometric Equations

Overview

• Focus on solving trigonometric (trig) equations.
• Solutions are given in radians, not degrees.

Types of Solutions

1. General Solutions

   • Represents every angle that makes the equation true.
   • Infinite number of solutions due to co-terminal angles.
   • Expressed as: initial solution + 2πk, where k is an integer.

2. Specific Solutions

   • Solutions within a specific range, usually between 0 and 2π.
   • Only considers a finite number of solutions in one cycle around the unit circle.

Solving the Example Equation

Equation: Find angle θ such that sin(θ) = 1/2

• General Solutions

  • First solution: θ = π/6
  • Second solution: θ = 5π/6
  • General solution expressions:
    • θ = π/6 + 2πk
    • θ = 5π/6 + 2πk

• Specific Solutions (0 to 2π)

  • First cycle: θ = π/6, 5π/6
  • Incorporating co-terminal angles:
    • k = 0: Solutions are π/6 and 5π/6
    • k = 1: Solutions are 13π/6 and 17π/6
    • k = 2: Solutions are 25π/6 and 29π/6

General Formula for Solutions

• Write all solutions from 0 to 2π and add 2πk to each initial solution to express all possible solutions.
• For homework, express general solutions and list specific number of solutions (e.g., first six positive solutions).

Tangent Equations

• Special case where general solution requires adding πk instead of 2πk.
• Example Equation: tan(θ) = 1/√3
  • First solution: θ = 5π/6
  • General formula: θ = 5π/6 + πk
  • List first six positive solutions by plugging in values of k.

Important Considerations

• Understand co-terminal angles: Repeated cycles around the unit circle.
• For tangent, recognize specific identities and simplifications for efficient problem-solving.

Problem Solving Strategy

1. Identify type of trigonometric function (sin, cos, tan).
2. Solve for initial solutions within one cycle (0 to 2π) using the unit circle.
3. Determine general solutions using 2πk or πk as applicable.
4. For specific solutions, select valid angles within given range.
5. Practice with homework problems to solidify understanding and notation.