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The Law of Sines (also called the Sine Rule) is one of the two universal triangle-solving tools in trigonometry — together with the Law of Cosines, the two laws cover every general triangle, not just right ones. The Law of Sines states that the ratio of a side's length to the sine of its opposite angle is the same for all three sides of any triangle. This tutorial walks through what the law says, when to use it (vs. the Law of Cosines), how to solve a triangle in each applicable case, and the infamous "ambiguous case" of SSA.

The statement of the law

For any triangle with sides a, b, c and the angles A, B, C opposite those sides:

a / sin(A) = b / sin(B) = c / sin(C)

Every ratio in this chain equals the same constant — geometrically, that constant is the diameter of the triangle's circumscribed circle (the unique circle passing through all three vertices). So a / sin(A) = 2R, where R is the circumradius. This gives a fourth, less-used form of the law:

a = 2R · sin(A), b = 2R · sin(B), c = 2R · sin(C)

When to use the Law of Sines

The Law of Sines applies whenever you have a matched side-angle pair — that is, a side and the angle opposite to it. You need that pair to set up the ratio. From there, you can find any other side if you know its opposite angle, or any angle if you know its opposite side.

Specifically:

• ASA (Angle-Side-Angle): two angles and the side between them. The third angle follows from A + B + C = 180°. Then Law of Sines gives the remaining two sides.
• AAS (Angle-Angle-Side): two angles and a side opposite one of them. Same approach.
• SSA (Side-Side-Angle): two sides and an angle opposite one of them. This is the ambiguous case — see below.

When to use the Law of Cosines instead: SSS (three sides) and SAS (two sides + included angle). In those cases, no side-angle pair is yet known, and the Law of Cosines is the correct entry point.

Worked example — ASA

Given A = 50°, B = 60°, and the side c between them = 12. Find the other two sides.

Third angle: C = 180° − 50° − 60° = 70°.

By the Law of Sines: a/sin(50°) = 12/sin(70°). Solve for a: a = 12 · sin(50°)/sin(70°) ≈ 12 · 0.766/0.940 ≈ 9.78.

Similarly: b = 12 · sin(60°)/sin(70°) ≈ 12 · 0.866/0.940 ≈ 11.06.

Worked example — AAS

Given A = 35°, B = 45°, a = 7. Find c.

By the Law of Sines: 7/sin(35°) = c/sin(C). First compute C: C = 180° − 35° − 45° = 100°. Then c = 7 · sin(100°)/sin(35°) ≈ 7 · 0.985/0.574 ≈ 12.02.

The ambiguous case — SSA

This is the most-asked-about scenario. Given two sides and the angle opposite one of them, there may be zero, one, or two valid triangles.

Setup: sides a and b given, angle A given (opposite side a). The Law of Sines gives sin(B) = b · sin(A) / a. There are three cases for the result:

• sin(B) > 1: impossible. No triangle exists. Side a is too short to "reach" the third vertex.
• sin(B) = 1: exactly one triangle, with B = 90°. The unique right-triangle case.
• sin(B) < 1: two candidate angles: B₁ = arcsin(sin(B)) (acute) and B₂ = 180° − B₁ (obtuse). Both might be valid — check whether A + B₂ < 180° in each candidate. If both A + B₁ < 180° AND A + B₂ < 180°, you have two valid triangles. If only one passes the check, you have one triangle.

This is exactly the SSA branch implemented in the Triangle Solver. When two solutions exist, both are reported with an "ambiguous_note" flag.

Worked example — SSA with two solutions

Given a = 6, b = 8, A = 35°. Find B.

sin(B) = 8 · sin(35°) / 6 = 8 · 0.5736 / 6 ≈ 0.7648.

B₁ = arcsin(0.7648) ≈ 49.886°. Acute candidate.

B₂ = 180° − 49.886° ≈ 130.114°. Obtuse candidate.

Check A + B for each: A + B₁ = 35° + 49.886° = 84.886° (less than 180°, valid). A + B₂ = 35° + 130.114° = 165.114° (also less than 180°, valid). Both are valid — two triangles exist with the given measurements.

The acute triangle has C = 180° − 84.886° = 95.114°. The obtuse triangle has C = 180° − 165.114° = 14.886°. The two triangles share sides a and b, share angle A, but differ in B and C and in the length of c.

The relationship to the circumcircle

The constant ratio a/sin(A) equals the diameter of the circumscribed circle. This gives a quick way to find the circumradius R = a / (2 sin(A)) once any side and its opposite angle are known.

Conversely, if a triangle is inscribed in a circle of known radius R, then for any vertex angle θ, the opposite chord (side) has length 2R · sin(θ). The Law of Sines is fundamentally a statement about circles, made via the Inscribed Angle Theorem.

Common mistakes

• Confusing sin(A) with A. The ratio a/sin(A) uses the SINE of the angle, not the angle itself. Forget to take the sine and your numbers will be nonsense.
• Mode mismatch (degrees vs radians). Our calculator expects degrees. If your textbook is in radians, convert. sin(60°) ≈ 0.866 but sin(60 radians) ≈ −0.305.
• Trying to use Law of Sines on SSS or SAS. Those cases have no known side-angle pair. Use Law of Cosines to get the first angle, then switch to Law of Sines.
• Ignoring the ambiguous case. When given SSA, always check whether two solutions exist. Many textbook problems expect both to be reported.
• Forgetting that AAA does not determine size. Three angles give shape but not scale. You always need at least one side.