Law of Sines: worked example
| Angle A (degrees) | 40 |
|---|---|
| Angle B (degrees) | 60 |
| Side a (opposite angle A) | 8 |
| Side b | 10.8 |
| Side c | 12.3 |
| Angle C (degrees) | 80 |
Law of Sines: formula
Law of Sines: Wolfram MathWorld · OpenStax Algebra and Trigonometry: law of sines
Enter two angles and the side opposite one of them. The third angle is 180 degrees minus the sum of the other two, and the law of sines gives the remaining sides: each side divided by the sine of its opposite angle is the same for all three. Angles are in degrees, and the side is in any unit.
Make sure the known side is opposite the known angle A, as the labels indicate.
Law of Sines: changing Angle A (degrees)
| Angle A (degrees) | Side b | Side c | Angle C (degrees) |
|---|---|---|---|
| 32 | 13.1 | 15.1 | 88 |
| 36 | 11.8 | 13.5 | 84 |
| 40 (example) | 10.8 | 12.3 | 80 |
| 44 | 10 | 11.2 | 76 |
| 48 | 9.3 | 10.2 | 72 |
Law of Sines: compare input ranges
| Input | Tested from | To | Side b: low input | Side b: high input | Swing |
|---|---|---|---|---|---|
| Side a (opposite angle A) | 7 | 9 | 9.4 | 12.1 | 2.7 (25%) |
| Angle A (degrees) | 36 | 44 | 11.8 | 10 | 1.8 (17%) |
| Angle B (degrees) | 54 | 66 | 10.1 | 11.4 | 1.3 (12%) |
Law of Sines: individual input comparisons
| Input | Value | Side b | Side c | Angle C (degrees) |
|---|---|---|---|---|
| Angle A (degrees) | 32 | 13.1 | 15.1 | 88 |
| 36 | 11.8 | 13.5 | 84 | |
| 40 | 10.8 | 12.3 | 80 | |
| 44 | 10 | 11.2 | 76 | |
| 48 | 9.3 | 10.2 | 72 | |
| Angle B (degrees) | 48 | 9.2 | 12.4 | 92 |
| 54 | 10.1 | 12.4 | 86 | |
| 60 | 10.8 | 12.3 | 80 | |
| 66 | 11.4 | 12 | 74 | |
| 72 | 11.8 | 11.5 | 68 | |
| Side a (opposite angle A) | 6 | 8.1 | 9.2 | 80 |
| 7 | 9.4 | 10.7 | 80 | |
| 8 | 10.8 | 12.3 | 80 | |
| 9 | 12.1 | 13.8 | 80 | |
| 10 | 13.5 | 15.3 | 80 |
Law of Sines: combined input changes
| Angle A (degrees) / Angle B (degrees) | 48 | 60 | 72 |
|---|---|---|---|
| 32 | 11.2 | 13.1 | 14.4 |
| 36 | 10.1 | 11.8 | 12.9 |
| 40 | 9.2 | 10.8 | 11.8 |
| 44 | 8.6 | 10 | 11 |
| 48 | 8 | 9.3 | 10.2 |
Law of Sines: which assumptions move the result?
Law of Sines: values and definitions
| Quantity and definition | Value used |
|---|---|
| Angle A (degrees) | 40 |
| Angle B (degrees) | 60 |
| Side a (opposite angle A) | 8 |
| Side b | 10.8 |
| Side c | 12.3 |
| Angle C (degrees) | 80 |
Law of Sines: half and double Side a (opposite angle A)
| Side a (opposite angle A) | Side b |
|---|---|
| 4 | 5.4 |
| 8 | 10.8 |
| 16 | 21.6 |
Common error: Entering two angles that add to 180 or more. The third angle would be zero or negative, so no triangle exists.
The two-sides-and-a-non-included-angle case can have two solutions, the ambiguous case, which this angle-angle-side tool avoids. Angles that sum to 180 degrees or more do not form a triangle. Measurement error in angles has a large effect on computed sides for thin triangles.
Use the law of cosines for two sides with the included angle.
Geometry calculators: Cube Calculator, Equilateral Triangle Calculator, Trigonometry Calculator, Golden Ratio Calculator, Midpoint Calculator, Rectangle Area & Perimeter Calculator.
In any triangle the ratio of a side to the sine of its opposite angle is the same for all three sides. Knowing two angles gives you the third, since they total 180, and one side sets the scale.
Match each side to its opposite angle. Side a sits across from angle A, and swapping them scrambles the whole ratio.
See the triangle third side calculator for the law of cosines, the right triangle and Pythagorean theorem pages for right angles, and the trigonometry calculator for sine values. The triangle area and equilateral triangle pages give areas, and the arc length page handles circular measures.
What is the ambiguous case?
When you know two sides and a non-included angle, there can be two different triangles that fit, so extra information is needed.
What is the law of sines?
Divide a side by the sine of its opposite angle. Repeat for the other sides. The ratios agree in any nondegenerate triangle, provided each side is paired with its own opposite angle and the angle units are consistent.
When do I use the law of sines instead of the law of cosines?
Two angles and a side suit the law of sines. Two sides and an opposite angle can also use it, with an ambiguity check. Two sides and their included angle suit the law of cosines.
The known angle’s position determines which setup applies.