Law of sines: statement, derivations, uses and the ambiguous case

Statement and common forms

The most common algebraic statement is: strong>a / sin A = b / sin B = c / sin C = 2Rstrong>. Equivalently, one may write: strong>sin A / a = sin B / b = sin C / cstrong>. Both forms are equivalent and are chosen according to which unknowns are convenient to isolate. The appearance of the circumcircle radius R is a useful geometric interpretation: each side is a chord of the circumcircle and chord length equals 2R sin(angle subtended at the centre/2), which simplifies to the form above.

Derivations

There are several short and instructive derivations. One uses triangle area: the area T can be written in three ways using different pairs of sides and the included angle, for example T = 1/2 ab sin C = 1/2 bc sin A = 1/2 ca sin B. Equating any two of these expressions and rearranging gives a / sin A = b / sin B = c / sin C. Another derivation comes from the circumcircle: a chord subtending angle A at the circumference has length a = 2R sin A, so dividing both sides by sin A yields a / sin A = 2R, and similarly for the other sides.

Applications and examples

The law of sines is used to determine unknown sides or angles in non-right (scalene) triangles when either (A) two angles and one side (AAS or ASA) are known, or (B) two sides and a non-included angle (SSA) are known. Example: given A = 30°, a = 5 and B = 45°, one finds b = (sin 45° / sin 30°) * 5 = (√2/2 ÷ 1/2) * 5 = 5√2 ≈ 7.07. Such computations appear across surveying, astronomy and navigation where indirect measurements are common.

Ambiguous case (SSA) and resolutions

When two sides and an angle not between them are given (the SSA configuration) the law of sines can lead to zero, one, or two valid triangle solutions. This arises because sin x = sin(180° − x), so an angle computed from the inverse sine may correspond to two different interior angles. Practical resolution requires checking whether the computed height fits the known side lengths and using angle-sum constraints to reject impossible alternatives.

Related results and historical notes

The law of sines complements the law of cosines; together they provide complete tools for solving any triangle. Historically, versions of the law trace back to ancient Indian and Greek mathematics and were developed further in medieval and early modern trigonometry. For more details on geometric proofs, historical development, and worked problems consult introductory trigonometry texts or referenced resources: diagram and statement, trigonometry overview, and collections of proofs at proofs and examples.

Notable facts: the ratio equals the diameter of the circumcircle, so a / sin A = 2R; this gives a quick way to compute the circumradius when side lengths and angles are known. For computational use, modern calculators and software implement the same relations but attention to degrees vs radians and the ambiguous SSA case remains important.