Lesson 1: Solving Basic Trig Equations
Estimated time: 30-35 minutes
Learning Objectives
- Isolate the trig function in a basic equation
- Find all solutions on [0, 2π) using the unit circle
- Write the general solution using periodicity
- Solve linear and quadratic trig equations
Strategy for Solving Trig Equations
Steps: (1) Isolate the trig function. (2) Find the reference angle. (3) Use the unit circle to find all solutions in [0, 2π). (4) For general solution, add the appropriate period.
Worked Examples
Example 1: 2 sin θ − 1 = 0
Step 1: sin θ = 1/2
Step 2: Reference angle: π/6. Sine is positive in QI and QII.
Step 3: θ = π/6 or θ = 5π/6 on [0, 2π).
General: θ = π/6 + 2nπ or θ = 5π/6 + 2nπ (n integer).
Example 2: 2 cos²θ − cos θ − 1 = 0
Factor: (2cosθ + 1)(cosθ − 1) = 0
cosθ = −1/2 ⇒ θ = 2π/3, 4π/3
cosθ = 1 ⇒ θ = 0
Solutions on [0, 2π): θ = 0, 2π/3, 4π/3
Example 3: tan θ = −√3
Reference angle: π/3. Tan is negative in QII and QIV.
θ = π − π/3 = 2π/3 or θ = 2π − π/3 = 5π/3
General: θ = 2π/3 + nπ (since tan has period π).
Important Notes
- Trig equations can have infinitely many solutions (general form) or a finite number on a restricted interval.
- For equations involving squared trig functions, treat them like quadratic equations.
- Never divide both sides by a trig function — you may lose solutions. Factor instead.
Check Your Understanding
1. Solve 2cosθ + √3 = 0 on [0, 2π).
cosθ = −√3/2. θ = 5π/6, 7π/6.
2. Solve sin²θ = 1/4 on [0, 2π).
sinθ = ±1/2. θ = π/6, 5π/6, 7π/6, 11π/6.
3. Write the general solution of sinθ = 0.
θ = nπ for all integers n.
Key Takeaways
- Isolate the trig function, then use the unit circle to find solutions.
- For general solutions, add 2nπ for sin/cos or nπ for tan.
- Factor quadratic trig equations — never divide by a trig function.
- Check how many solutions exist in the specified interval.