Algebra 101 - Unit 16

Introduction

Solving trigonometric equations involves finding all angles that satisfy a given trigonometric expression. Solutions often include general forms to account for periodicity of sine, cosine, and tangent functions.

Strategy for Solving Trigonometric Equations

1. Isolate the trigonometric function (sin, cos, tan, etc.).
2. Use reference angles and the unit circle to find solutions within one period.
3. Include all solutions using the periodicity formulas:
• sin θ = k → θ = arcsin(k) + 2nÏ€ or θ = Ï€ − arcsin(k) + 2nÏ€
• cos θ = k → θ = arccos(k) + 2nÏ€ or θ = −arccos(k) + 2nÏ€
• tan θ = k → θ = arctan(k) + nÏ€
4. Check solutions in the original equation, especially if squared terms were involved.
5. Express solutions in radians or degrees, based on problem context.

Examples

Example 1 — Sine Equation:

Solve sin θ = ½. Reference angle: θ₀ = 30° → θ = 30° + 360°n or θ = 150° + 360°n, n ∈ ℤ

Example 2 — Cosine Equation:

Solve cos θ = −√2/2. Reference angle: θ₀ = 45° → θ = 135° + 360°n or θ = 225° + 360°n, n ∈ ℤ

Example 3 — Tangent Equation:

Solve tan θ = 1. Reference angle: θ₀ = 45° → θ = 45° + 180°n, n ∈ ℤ

Example 4 — Using Identities:

Solve 2 sin² θ − 1 = 0 → sin² θ = ½ → sin θ = ±√2/2 Solutions: θ = 45°, 135°, 225°, 315° + 360°n, n ∈ ℤ

Practice Problems

1. Solve sin θ = −½ for all θ.
2. Solve cos θ = ¼ for all θ.
3. Solve 1 − 2 cos² θ = 0 for all θ.
4. Solve 2 tan² θ − 3 = 0 for all θ.
5. Solve sin(2θ) = √3/2 for θ ∈ [0°, 360°).