Algebra 101 - Unit 15

Lesson 96: Solving Basic Trigonometric Equations

Trigonometric equations involve sine, cosine, tangent, or other trig functions. Solving these equations often requires using identities, inverse functions, and understanding periodicity.

General Strategy

1. Isolate the trigonometric function if possible (e.g., sin θ = 1/2).
2. Use inverse trig functions or unit circle knowledge to find solutions in the desired interval.
3. Consider the periodic nature of trig functions to find all solutions.
4. Check for extraneous solutions when squaring or using identities.

Examples

Example 1 — Solve sin θ = 1/2:

θ = 30° or 150° (first and second quadrants). General solution: θ = 30° + 360°n or θ = 150° + 360°n, n ∈ ℤ

Example 2 — Solve cos x = −√2/2:

x = 135° or 225° (second and third quadrants). General solution: x = 135° + 360°n or x = 225° + 360°n, n ∈ ℤ

Example 3 — Solve 2 tan θ − 1 = 0:

2 tan θ − 1 = 0 → tan θ = 1/2 → θ = arctan(1/2) ≈ 26.57° General solution: θ ≈ 26.57° + 180°n, n ∈ ℤ

Practice Problems

1. Solve sin x = √3/2 for 0° ≤ x < 360°.
2. Solve cos θ = 0 for 0 ≤ θ < 2π.
3. Solve tan α = −1 for 0° ≤ α < 360°.
4. Solve 2 sin x − 1 = 0 for 0 ≤ x < 2π.
5. Solve cos 2x = 1/2 for 0 ≤ x < 2π.