Law of Sines Formula
The Law of Sines relates triangle sides to their opposite angles.
Learn the formula, when to use it, ambiguous cases, and worked examples.
The Formula
The Law of Sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant throughout the triangle. This ratio equals the diameter of the triangle's circumscribed circle (circumdiameter = 2R). It applies to any triangle — acute, obtuse, or right.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| a, b, c | Side lengths of the triangle | any (m, cm, etc.) |
| A, B, C | Angles opposite to sides a, b, c | degrees or radians |
| R | Circumradius (radius of circumscribed circle) | same as sides |
Extended Law of Sines
The common ratio equals twice the circumradius R. This links the law of sines directly to circle geometry.
Example 1 — Finding an Unknown Side
Triangle with A = 35°, B = 80°, a = 10 cm. Find side b.
a / sin(A) = b / sin(B)
10 / sin(35°) = b / sin(80°)
b = 10 × sin(80°) / sin(35°) = 10 × 0.9848 / 0.5736
b ≈ 17.17 cm
Example 2 — Finding an Unknown Angle
Triangle with a = 7, b = 10, A = 40°. Find angle B.
sin(B) / b = sin(A) / a
sin(B) = 10 × sin(40°) / 7 = 10 × 0.6428 / 7 = 0.9183
B = arcsin(0.9183) ≈ 66.7° (or the ambiguous case: B ≈ 113.3°)
The Ambiguous Case (SSA)
When given two sides and a non-included angle (SSA), there may be 0, 1, or 2 valid triangles:
- If sin(B) > 1: No triangle exists
- If sin(B) = 1: Exactly one right triangle
- If sin(B) < 1 and the given angle A is acute: Two possible triangles (B and 180° − B)
- If sin(B) < 1 and the given angle A is obtuse: One triangle only
When to Use the Law of Sines
- AAS or ASA: Two angles and one side are known
- SSA (with caution): Two sides and a non-included angle — check for the ambiguous case
- Do NOT use for SAS or SSS — use the Law of Cosines instead