Equations with many answers
A trigonometric equation is true only for certain angles. Because trig functions repeat, there are usually several solutions in one cycle, and infinitely many overall.
Basic strategy
- Isolate the trig function.
- Find the reference angle using the inverse trigonometric function (use the positive value).
- Use the CAST rule to find which quadrants give the correct sign.
- Write all solutions in the required interval.
Example 1
Solve 2 sin θ − 1 = 0 for 0° ≤ θ < 360°.
- sin θ = ½
- Reference angle: 30°
- Sine is positive in I and II → θ = 30°, 150°
Example 2
Solve cos θ = −0.4 for 0° ≤ θ < 360°.
- Reference angle: cos⁻¹(0.4) ≈ 66.4°
- Cosine is negative in II and III
- θ ≈ 180 − 66.4 = 113.6° and 180 + 66.4 = 246.4°
Example 3: tangent
Solve tan θ = √3 for 0 ≤ θ < 2π.
- Reference angle π/3; tan positive in I and III
- θ = π/3, 4π/3
Quadratic trig equations
Factor like a quadratic.
Example: 2 sin²θ − sin θ − 1 = 0
- (2 sin θ + 1)(sin θ − 1) = 0
- sin θ = −½ → θ = 210°, 330°
- sin θ = 1 → θ = 90°
- Solutions: 90°, 210°, 330°
Using identities
Example: 2cos²θ + 3 sin θ = 3
- Replace cos²θ = 1 − sin²θ: 2 − 2sin²θ + 3 sin θ = 3
- 2sin²θ − 3 sin θ + 1 = 0 → (2 sin θ − 1)(sin θ − 1) = 0
- sin θ = ½ or 1 → 30°, 90°, 150°
Multiple angles
Example: sin 2θ = ½ for 0° ≤ θ < 360°
- 2θ ranges over 0° to 720°
- 2θ = 30°, 150°, 390°, 510°
- θ = 15°, 75°, 195°, 255°
General solutions
The general solution lists all solutions:
sin θ = ½ → θ = 30° + 360°k or θ = 150° + 360°k, k an integer.
Worked examples
Example 1: cos θ = 1, 0° ≤ θ < 360° → 0°.
Example 2: 3 tan θ + 3 = 0 → tan θ = −1 → 135°, 315°.
Example 3: sin θ = 2? No solution (sine is between −1 and 1).
Common mistakes
- Giving only the calculator’s answer and missing the second solution.
- Using a negative value to find the reference angle.
- Dividing by sin θ or cos θ, which can lose solutions. Factor instead.
Try it yourself
- sin θ = √2/2, 0° ≤ θ < 360°
- 2 cos θ + 1 = 0, 0° ≤ θ < 360°
- tan θ = 0.75, 0° ≤ θ < 360° (one decimal)
- cos²θ = ¼, 0° ≤ θ < 360°
- sin θ cos θ = 0, 0° ≤ θ < 360°
Answers: 1) 45°, 135° 2) 120°, 240° 3) 36.9°, 216.9° 4) 60°, 120°, 240°, 300° 5) 0°, 90°, 180°, 270°
Quick recap
- Isolate, find the reference angle, then use CAST for all quadrants.
- Factor quadratic equations; use identities to get one function.
- State all solutions in the interval, or write a general solution.