A trigonometric equation such as sinθ = 0.5 has more than one solution in 0°–360°. Use your calculator's inverse function to find the first (principal) solution, then use the symmetry of the sine, cosine or tangent graph to find all others in the required range.
Step 1 — find the principal value
The principal value is the first solution your calculator gives when you press sin⁻¹, cos⁻¹ or tan⁻¹.
- For sinθ = k: calculator gives a value in −90° to 90°.
- For cosθ = k: calculator gives a value in 0° to 180°.
- For tanθ = k: calculator gives a value in −90° to 90°.
Example: sinθ = 0.5 → θ = sin⁻¹(0.5) = 30° (principal value)
Step 2 — use the symmetry rules to find all solutions in 0°–360°
Each trig function has a predictable symmetry pattern:
| Function | First solution | Second solution in 0°–360° |
|---|---|---|
| sinθ = k (k > 0) | θ = α | θ = 180° − α |
| sinθ = k (k < 0) | θ = 180° − α (where α < 0, so use 360° + α) | θ = 360° + α |
| cosθ = k (k > 0) | θ = α | θ = 360° − α |
| cosθ = k (k < 0) | θ = α (180° < α < 180°) | θ = 360° − α |
| tanθ = k | θ = α | θ = 180° + α |
The key fact: sin is symmetric about 90°, cos is symmetric about 0° (and 360°), and tan repeats every 180°.
How do you solve sinθ = k in 0°–360°?
Worked example 1: Solve sinθ = 0.5 for 0° ≤ θ ≤ 360°.
- Principal value: θ₁ = sin⁻¹(0.5) = 30°.
- Symmetry rule for positive sin: θ₂ = 180° − 30° = 150°.
- Solutions: θ = 30° or θ = 150°.
Check: sin 30° = 0.5 ✓; sin 150° = sin(180°−30°) = sin 30° = 0.5 ✓.
Worked example 2: Solve sinθ = −0.6 for 0° ≤ θ ≤ 360°.
- sin⁻¹(0.6) = 36.87° (the positive reference angle).
- sin is negative in the 3rd and 4th quadrants.
- θ₁ = 180° + 36.87° = 216.87°; θ₂ = 360° − 36.87° = 323.13°.
- Round to the precision asked: typically 1 decimal place.
How do you solve cosθ = k in 0°–360°?
Worked example 3: Solve cosθ = 0.5 for 0° ≤ θ ≤ 360°.
- Principal value: θ₁ = cos⁻¹(0.5) = 60°.
- Symmetry rule: θ₂ = 360° − 60° = 300°.
- Solutions: θ = 60° or θ = 300°.
Check: cos 60° = 0.5 ✓; cos 300° = cos(360°−60°) = cos 60° = 0.5 ✓.
Worked example 4: Solve cosθ = −0.5 for 0° ≤ θ ≤ 360°.
- cos⁻¹(0.5) = 60° (reference angle).
- cos is negative in 2nd and 3rd quadrants: θ₁ = 180° − 60° = 120°; θ₂ = 180° + 60° = 240°.
How do you solve tanθ = k in 0°–360°?
Tangent repeats every 180°, so there are always exactly two solutions in 0°–360°.
Worked example 5: Solve tanθ = 1 for 0° ≤ θ ≤ 360°.
- Principal value: θ₁ = tan⁻¹(1) = 45°.
- Add 180°: θ₂ = 45° + 180° = 225°.
- Solutions: θ = 45° or θ = 225°.
Worked example 6: Solve tanθ = −2 for 0° ≤ θ ≤ 360°.
- tan⁻¹(2) = 63.43° (reference angle, ignoring the sign).
- tan is negative in 2nd and 4th quadrants: θ₁ = 180° − 63.43° = 116.57°; θ₂ = 360° − 63.43° = 296.57°.
What if the question uses exact values?
Some questions expect exact answers using the special triangles:
| θ | sinθ | cosθ | tanθ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Worked example 7: Solve sinθ = √3/2 for 0° ≤ θ ≤ 360°. Give exact answers.
sinθ = √3/2 → principal value = 60°. Second solution = 180° − 60° = 120°.
θ = 60° or θ = 120°.
Frequently asked questions
What does the CAST diagram tell me?
CAST is a memory aid showing which trig functions are positive in each quadrant. Starting in the 4th quadrant and going anticlockwise: Cos (4th), All (1st), Sin (2nd), Tan (3rd). If your equation has a positive k, the solutions are in the two quadrants where that function is positive. If k is negative, solutions are in the two quadrants where it is negative.
How do I know how many solutions to expect?
In 0°–360°, each equation sinθ = k, cosθ = k or tanθ = k has exactly two solutions (unless the value is ±1 for sin/cos or the range is narrowed). If a question asks for solutions in a different range such as −180° to 180°, apply the same symmetry ideas relative to that range.
What if the equation has sin2θ or sinθ/2?
First solve for the substituted variable. For sin2θ = 0.5, let u = 2θ and solve sinu = 0.5 in the range 0° ≤ u ≤ 720° (double the range), then halve all solutions. So u = 30°, 150°, 390°, 510° → θ = 15°, 75°, 195°, 255°.
Can I use the graph instead of the symmetry rules?
Yes — sketching the sine or cosine graph and drawing a horizontal line at y = k is a reliable visual method. The x-coordinates of the intersections give all solutions. For exam questions, sketch the graph, mark the intersections, and read off the values. This approach is especially helpful when k is negative and the symmetry rules feel confusing.
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