When to use the law of sines
The law of sines applies directly to two triangle-solving cases; other cases require the law of cosines instead.
| Known values | Best method |
|---|---|
| Two angles + any side (AAS or ASA) | Law of sines — unambiguous, single solution |
| Two sides + non-included angle (SSA) | Law of sines, but check for the ambiguous case (0, 1 or 2 triangles possible) |
| Three sides (SSS) | Law of cosines |
| Two sides + included angle (SAS) | Law of cosines |
- This calculator's AAS/ASA input mode (two angles and one side) always has exactly one valid solution, so the ambiguous SSA case does not arise here.
- The three angles of any triangle must sum to exactly 180°; if the two entered angles already total 180° or more, no valid triangle exists and the calculator returns no result.
What is the law of sines?
The law of sines states that in any triangle, the ratio of a side's length to the sine of the angle opposite that side is constant across all three sides: a / sin A = b / sin B = c / sin C. This makes it possible to find a missing side or angle whenever enough information is known.
Given two angles and the side opposite one of them (the AAS or ASA case used here), the triangle is fully and unambiguously determined — unlike the SSA (two sides and a non-included angle) case, which can sometimes produce two valid triangles, one valid triangle, or none, known as the ambiguous case.
How to use this law of sines calculator
- Enter angle A and the length of side a, the side opposite angle A.
- Enter angle B, the second known angle.
- The calculator requires angle A plus angle B to be less than 180°, since the third angle (C) must be positive.
- Read side b, angle C, side c and the triangle's area, all derived from the law of sines.
The law of sines formula
Once angle C is found from the angle sum (180° − A − B), the law of sines ratio a / sin A is applied to find both remaining sides.
Common mistakes
- Entering side a as if it were opposite angle B instead of angle A — the law of sines pairs each side strictly with its opposite angle.
- Entering two angles that sum to 180° or more, leaving no room for a valid third angle.
- Confusing the law of sines (ratios of sides to sines of opposite angles) with the law of cosines (used for SAS and SSS cases).
- Forgetting that angles must be entered in degrees, not radians.
よくある質問
What is the law of sines formula?
a / sin A = b / sin B = c / sin C, where each side is paired with the sine of the angle directly opposite it.
How do you find a missing side using the law of sines?
Set up the ratio with the known side and angle, then solve for the missing side: b = (a × sin B) / sin A. For A = 30°, a = 10 and B = 45°, side b = (10 × sin 45°) / sin 30° = (10 × 0.707107) / 0.5 ≈ 14.1421.
What is the third angle if A = 30° and B = 45°?
Since the three angles of a triangle sum to 180°, angle C = 180° − 30° − 45° = 105°.
What is the ambiguous case in the law of sines?
The ambiguous (SSA) case arises when two sides and a non-included angle are known — depending on the values, zero, one, or two distinct triangles can satisfy the given measurements. It does not arise in this calculator's AAS input mode, which always has exactly one solution.
How do you find the area of a triangle using the law of sines?
Once two sides and the included angle are known, area = ½ × side₁ × side₂ × sin(included angle). For a = 10, b ≈ 14.1421 and included angle C = 105°, area = ½ × 10 × 14.1421 × sin(105°) ≈ 68.3013 square units.
When should I use the law of sines instead of the law of cosines?
Use the law of sines when you know two angles and any side (AAS/ASA), or two sides and a non-included angle (SSA). Use the law of cosines when you know three sides (SSS) or two sides and the included angle (SAS), since the law of sines cannot solve those cases directly.
参考文献
- Weisstein, Eric W. "Law of Sines." MathWorld — A Wolfram Web Resource.
- Standard trigonometry textbook conventions (e.g. Larson, Trigonometry, Cengage Learning).