Lesson 2: Solving Equations Using Identities
Estimated time: 35-40 minutes
Learning Objectives
- Use Pythagorean identities to convert equations to one trig function
- Apply double-angle identities to solve equations
- Use factoring after identity substitution
- Handle equations that require multiple steps
Using Pythagorean Identities
When an equation contains two different trig functions, use an identity to rewrite in terms of one function, then solve the resulting equation.
Example 1: Using sin² + cos² = 1
Solve 2sin²θ + cosθ − 1 = 0 on [0, 2π).
Step 1: Replace sin²θ = 1 − cos²θ: 2(1−cos²θ) + cosθ − 1 = 0
Step 2: Simplify: −2cos²θ + cosθ + 1 = 0, or 2cos²θ − cosθ − 1 = 0
Step 3: Factor: (2cosθ + 1)(cosθ − 1) = 0
Step 4: cosθ = −1/2 ⇒ θ = 2π/3, 4π/3. cosθ = 1 ⇒ θ = 0.
Solutions: {0, 2π/3, 4π/3}
Using Double-Angle Identities
Example 2: Double Angle
Solve cos2θ + cosθ = 0 on [0, 2π).
Replace cos2θ = 2cos²θ − 1: (2cos²θ−1) + cosθ = 0
2cos²θ + cosθ − 1 = 0 ⇒ (2cosθ−1)(cosθ+1) = 0
cosθ = 1/2 ⇒ θ = π/3, 5π/3. cosθ = −1 ⇒ θ = π.
Solutions: {π/3, π, 5π/3}
Example 3: sin2θ = cosθ
2sinθcosθ = cosθ ⇒ cosθ(2sinθ−1) = 0
cosθ = 0 ⇒ θ = π/2, 3π/2. sinθ = 1/2 ⇒ θ = π/6, 5π/6.
Solutions: {π/6, π/2, 5π/6, 3π/2}
Common Mistakes
- Do not divide by trig functions — you lose solutions where the divisor is zero.
- Always check solutions in the original equation (especially after squaring).
- Choose the right form of cos2θ based on the other function in the equation.
Check Your Understanding
1. Solve 2cos²θ + sinθ = 1 on [0, 2π).
2. Solve sin2θ = sinθ on [0, 2π).
3. Solve tan²θ − 1 = 0 on [0, 2π).
Key Takeaways
- Use identities to reduce equations to one trig function.
- Choose the Pythagorean or double-angle form that matches the functions in the equation.
- Always factor rather than divide by trig functions.