Law of Sines: Solving Any Triangle

The Law of Sines states that in any triangle with sides a, b, c and opposite angles A, B, C respectively: a / sin A = b / sin B = c / sin C.

This constant ratio equals the diameter of the circumscribed circle (the circle passing through all three vertices), providing a beautiful geometric interpretation: the larger the side, the larger the opposite angle — and the Law of Sines quantifies exactly that relationship.

The intuition is direct: if angle A is large, side a (facing it) is also large. The law tells us in what proportion these growths are related, allowing us to calculate the unknown from the known.

The Law of Sines applies when you know: (AAS) two angles and any side — you can find the other two sides; (ASA) two angles and the side between them — same; (SSA) two sides and the angle opposite one of them — a situation that requires special attention, the ambiguous case.

It does not apply directly when you know only the three sides (SSS) or two sides and the angle between them (SAS) — in those cases, the Law of Cosines is the right tool.

A common mistake is trying to use the Law of Sines when there is no complete pair (side + opposite angle). Always verify that you have at least one side-angle pair before applying the formula.

Example AAS: A = 40°, B = 75°, a = 10. First, C = 180° − 40° − 75° = 65°. Then, b / sin 75° = 10 / sin 40°, so b = 10 × sin 75° / sin 40° ≈ 10 × 0.966 / 0.643 ≈ 15.02. Similarly, c = 10 × sin 65° / sin 40° ≈ 14.10.

The process has three steps: (1) calculate the missing angle using A + B + C = 180°; (2) set up the Law of Sines ratios; (3) isolate the unknown side or angle and calculate.

Always verify that the sum of the angles found equals 180°. Small rounding errors can accumulate, so keep at least 4 decimal places in intermediate calculations.

In the SSA case (knowing a, b, and A), there may be zero, one, or two solutions for the triangle. This happens because, when drawing side a from vertex B with a fixed length, the far end may not reach side b (no solution), touch it at a single point (unique solution), or cross it at two points (two solutions).

Practical rule: if A is obtuse (greater than 90°), a solution exists only if a > b. If A is acute, compare a with b × sin A (the triangle's altitude): if a < altitude, no solution; if a = altitude, right triangle; if altitude < a < b, two solutions; if a ≥ b, unique solution.

When there are two solutions, the two possible angles for B are supplementary: B₁ and B₂ = 180° − B₁. Each produces a different triangle. In practical problems, context usually indicates which one makes sense.

In surveying, the Law of Sines allows measuring inaccessible distances. Measure two points A and B on the ground (baseline), observe the angle to an inaccessible point C from each end, and calculate the distance AC or BC.

In navigation, pilots and sailors triangulate positions: they observe two known landmarks and measure the bearing angles to determine the exact position of the vessel.

The law also appears in physics (geometric optics, analysis of forces in equilibrium) and in computer graphics, for interpolating positions in three-dimensional scenes.