Algebra 101 - Unit 16

Introduction

Advanced trigonometric equations often involve multiple angles, products, sums, or powers of trigonometric functions. Mastery of identities and algebraic manipulation is key to solving these equations.

Strategy for Solving Advanced Equations

1. Isolate the trigonometric expression if possible.
2. Use appropriate identities to simplify: double-angle, half-angle, sum-to-product, or Pythagorean identities.
3. Factor expressions when possible.
4. Reduce to basic trigonometric equations (sin θ = k, cos θ = k, tan θ = k).
5. Consider all solutions within the given interval and include general solutions if required.
6. Check for extraneous solutions when squaring or manipulating expressions.

Examples

Example 1 — Multiple Angles:

Solve 2 sin² θ − 3 sin θ + 1 = 0 Factor: (2 sin θ − 1)(sin θ − 1) = 0 → sin θ = 1/2 or sin θ = 1 Solutions: θ = 30°, 150°, 90° + 360°n, n ∈ ℤ

Example 2 — Using Double Angle Identity:

Solve cos 2θ = ½ → cos 2θ = ½ → 2θ = 60°, 300° → θ = 30°, 150° + 180°n, n ∈ ℤ

Example 3 — Sum-to-Product:

Solve sin θ + sin 3θ = 0 → 2 sin 2θ cos θ = 0 → sin 2θ = 0 or cos θ = 0 Solutions: θ = 0°, 90°, 180°, 270°, etc.

Example 4 — Quadratic in Trig Function:

Solve 4 cos² θ − 4 cos θ − 3 = 0 → (2 cos θ − 3)(2 cos θ + 1) = 0 → cos θ = 3/2 (no solution), cos θ = −1/2 → θ = 120°, 240° + 360°n, n ∈ ℤ

Practice Problems

1. Solve 3 sin² θ − 2 sin θ − 1 = 0 for θ ∈ [0°, 360°).
2. Solve 2 cos² θ − cos θ − 1 = 0 for θ ∈ [0°, 360°).
3. Solve sin 2θ − √3/2 = 0 for θ ∈ [0°, 360°).
4. Solve 2 sin θ cos θ − sin θ = 0 for θ ∈ [0°, 360°).
5. Solve cos 3θ = 0 for θ ∈ [0°, 360°).