Sine Rule: Formula, Proof, and Ambiguous Case Explained

In any triangle, the longest side sits opposite the largest angle. The sine rule turns that into an exact equation: a/sin A = b/sin B = c/sin C, where a is opposite angle A, b opposite B, and c opposite C. Remarkably, every ratio also equals 2R, twice the radius of the circle through all three vertices.

This rule matters because triangles are the basic measuring tool of the world. Surveyors, sailors, and astronomers measure angles from a safe spot and let the sine rule compute distances they cannot reach directly.

Key Concepts

• Sine rule: a/sin A = b/sin B = c/sin C = 2R, valid for every plane triangle.
• Circumradius R: the radius of the circumcircle through all three vertices.
• Opposite pairing: side a faces angle A, b faces B, c faces C. Mixing up the pairs is the most common setup error.
• SSA data: two sides and a non-included angle; it can give zero, one, or two triangles.
• Ambiguous case: when two different triangles both fit the given SSA measurements.

Formulas

• a/sin A = b/sin B = c/sin C = 2R. Angles may be in degrees or radians, as long as the sine uses the same unit.
• a = 2R sin A (likewise b = 2R sin B, c = 2R sin C).
• Ambiguity test: with angle A, adjacent side b, opposite side a, let h = b sin A. If a < h: no triangle. If a = h: one right triangle. If h < a < b: two triangles. If a ≥ b: one triangle.
• Mirror angles: if sin B = k (0 < k < 1), the candidates are B and 180° − B, since sin(180° − B) = sin B.

Worked Example

A triangle has angle A = 30°, adjacent side b = 10, and opposite side a = 6. Determine how many triangles fit and find them.

1. Altitude: h = b sin A = 10 × 0.5 = 5. Since 5 < 6 < 10, this is the ambiguous case with two triangles.
2. sin B = b sin A / a = 10 × 0.5 / 6 = 5/6 ≈ 0.8333.
3. Candidates: B₁ ≈ 56.4° and the mirror B₂ = 180° − 56.4° = 123.6°.
4. Triangle 1: C₁ = 93.6°, c₁ = 6 × sin 93.6° / 0.5 ≈ 11.98.
5. Triangle 2: C₂ = 26.4°, c₂ = 6 × sin 26.4° / 0.5 ≈ 5.34.

Both triangles fit: (B, C) ≈ (56.4°, 93.6°) or (123.6°, 26.4°).

Proof via the circumcircle: extend the diameter through A to meet the circle at D, so AD = 2R. Angle ABD is 90° (angle in a semicircle) and angle ADB equals angle C (same chord AB). In right triangle ABD, sin C = AB/AD = c/(2R), so c/sin C = 2R; the same construction gives 2R for the other ratios.

Common Misconceptions

• "SSA gives a unique triangle." Not always: sin B = sin(180° − B), so always test the mirror angle.
• "2R is just some constant." It is the circumcircle's diameter, shared by all three ratios.
• "Sides and angles pair arbitrarily." They do not: in a/sin A, side a must be opposite angle A, or the answer is nonsense.

Real-World Applications

• Surveying: the distance across a river is found from one baseline and two angles.
• Navigation: ships fix their position by triangulation from two known landmarks.
• Engineering: truss bridge members' loads are found from joint angles with the sine rule.

The sine rule says every side divided by the sine of its opposite angle equals the same value, the circumdiameter 2R. Its proof drops a diameter through one vertex of the circumcircle and reads the side off a right triangle. In the SSA ambiguous case, compare the opposite side with the altitude b sin A: below it no triangle, equal to it one right triangle, between it and b two triangles, beyond b one triangle.