# How do I use the law of sines?

Solves a triangle with the law of sines from two angles and a side (AAS or ASA), or from two sides and an opposite angle (SSA), showing both triangles in the ambiguous case.

- Page: https://www.acalculator.org/math/law-of-sines-calculator
- JSON spec: https://www.acalculator.org/math/law-of-sines-calculator.json
- Version: 1a7f810057d6

## Default answer

Example with the default inputs (What do you know? Two angles and a side, Angle A 50 °, Angle B 100 °, Which side do you know? Side a (opposite A), Length of that side 10): The triangle has sides a = 10, b = 12.8558, c = 6.52704 and angles A = 50°, B = 100°, C = 30°.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | What do you know? | Two angles and one side, or two sides and the angle opposite the first of them. |
| A | Angle A | The angle opposite side a. |
| B | Angle B | The angle opposite side b. |
| side | Which side do you know? | Side a (opposite A), side b (opposite B), or side c (between angles A and B). |
| len | Length of that side | The length of the side you know, in any unit. |
| a | Side a | The side opposite angle A, in any unit. |
| b | Side b | The other known side, opposite angle B, in the same unit. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| c | Side c | The side opposite angle C. |
| a | Side a | The side opposite angle A. |
| b | Side b | The side opposite angle B. |
| A | Angle A | The angle opposite side a. |
| B | Angle B | The angle opposite side b. |
| C | Angle C | The angle opposite side c: 180° − A − B. |
| area | Area | Half of a × b × sin C, in square units of the sides. |
| triangles | Triangles | How many triangles fit the given parts: one, or two in the ambiguous case (SSA). |
| B2 | Second triangle: angle B | In the ambiguous case, the other angle B: 180° minus the first. |
| C2 | Second triangle: angle C | In the ambiguous case, 180° − A − the second angle B. |
| c2 | Second triangle: side c | In the ambiguous case, side c of the second triangle. |
| area2 | Second triangle: area | In the ambiguous case, the area of the second triangle. |

## Method

Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C. Two angles and a side: C = 180° − A − B, then each side = known side ÷ sin(its angle) × sin(the opposite angle). Two sides and angle A: sin B = b sin A ÷ a; B = arcsin, and also 180° − arcsin when A is acute and b sin A < a < b.

## Assumptions

- Angles are typed in degrees or radians, each more than 0° and less than 180°.
- Sides are in any one unit, from 10⁻⁹ to 10⁹; the area is in that unit squared.
- Sines at multiples of 15° are exact, and an angle within one part in 10¹² of a multiple of 15° (other than 0°) counts as that multiple.
- In the ambiguous case (two sides and an angle), sin B is taken as 1 when it is within 10⁻¹² of 1, giving one right triangle.

## Worked examples

1. mode = aas, A = 0.872665, B = 1.745329, side = a, len = 10 gives C = 0.523599, b = 12.855752, c = 6.527036, area = 32.13938. Source: OpenStax, Algebra and Trigonometry 2e, §10.1 Non-right Triangles: Law of Sines, Example 1 (b ≈ 12.9, c ≈ 6.5), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-1-non-right-triangles-law-of-sines.
2. mode = ssa, A = 0.610865, a = 6, b = 8 gives triangles = Two triangles, B = 0.870682, C = 1.660045, c = 10.419047, B2 = 2.270911, C2 = 0.259817, c2 = 2.687386. Source: OpenStax, Algebra and Trigonometry 2e, §10.1, Example 2 (β ≈ 49.9°, γ ≈ 95.1°, c ≈ 10.4; or β ≈ 130.1°, γ ≈ 14.9°, c ≈ 2.7), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-1-non-right-triangles-law-of-sines.
3. mode = ssa, A = 1.48353, a = 12, b = 9 gives triangles = One triangle, B = 0.843758, C = 0.814305, c = 8.760321. Source: OpenStax, Algebra and Trigonometry 2e, §10.1, Example 3 (b = 9, c = 12, γ = 85°: β ≈ 48.3°, α ≈ 46.7°, a ≈ 8.8), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-1-non-right-triangles-law-of-sines.
4. mode = ssa, A = 0.523599, a = 4, b = 8 gives triangles = One right triangle, B = 1.570796, C = 1.047198, c = 6.928203. Source: OpenStax, Algebra and Trigonometry 2e, §10.1 (one triangle when the side equals the altitude), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-1-non-right-triangles-law-of-sines.
5. mode = aas, A = 0.785398, B = 1.047198, side = c, len = 10 gives C = 1.308997, a = 7.320508, b = 8.965755. Source: OpenStax, Algebra and Trigonometry 2e, §10.1, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-1-non-right-triangles-law-of-sines.

## FAQ

### What is the law of sines?

In any triangle, each side divided by the sine of its opposite angle gives the same number: a ÷ sin A = b ÷ sin B = c ÷ sin C. Side a is opposite angle A, side b opposite B, and side c opposite C.

### When do I use the law of sines instead of the law of cosines?

Use the law of sines when you know an angle and the side opposite it, plus one more part: two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA). With two sides and the angle between them (SAS) or three sides (SSS), use the law of cosines.

### What is the ambiguous case?

With two sides a and b and the angle A opposite a (SSA), the parts may fit no triangle, one, or two. If A is acute and a is between the height b × sin A and b, side a can swing to two places, giving two triangles. The calculator shows both.

### Why does the calculator say there is no triangle?

In the SSA case, side a may be shorter than the height b × sin A, so it cannot reach the third side; or angle A is 90° or more and a is not the longest side. With two angles, they may add up to 180° or more, leaving no room for the third angle.

### Does the law of sines work for right triangles?

Yes. With C = 90°, sin C = 1, so a ÷ sin A = c, which gives sin A = a ÷ c, the usual opposite ÷ hypotenuse.

### How is the area found?

Area = ½ × a × b × sin C, half the product of two sides and the sine of the angle between them. For the 50°, 100°, 30° triangle with a = 10, the area is about 32.14 square units.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §10.1 Non-right Triangles: Law of Sines (the law, the ambiguous case, Examples 1 to 3, and the area formula). https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-1-non-right-triangles-law-of-sines (retrieved 2026-10-01)
- OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin gives an angle from −90° to 90°). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions (retrieved 2026-10-01)