Law of sines

The formula (the core equation(s), each with a one-line reading of what it says)

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

Reads: sides and the sines of their opposite angles keep a fixed ratio — the biggest side always faces the biggest angle, in exact proportion.

a / sin(A) = 2·R

Reads: that common ratio isn't arbitrary — it equals the diameter of the circle passing through all three corners. For a 3-4-5 right triangle, 5/sin(90°) = 5, so R = 2.5 and the hypotenuse is the circle's diameter.

Solving for a side:    a = c · sin(A) / sin(C)
Solving for an angle:  sin(B) = b · sin(A) / a

Reads: the first form is safe; the second is where people get burned, because asin only hands back angles up to 90° and the triangle may want the obtuse one.

Where you meet it (2-4 concrete engineering situations, specific: bench, test stand, review board)

Locating a point you can't tape to. Two survey stations 100 m apart, each shooting a bearing to a tower on the far side of a creek. The angles at the two ends of the baseline are 68° and 74°, so the angle at the tower is 38°, and the station-to-tower distances are 100·sin(74°)/sin(38°) = 156.13 m and 100·sin(68°)/sin(38°) = 150.60 m. One measured length, two measured angles, no wet boots. Every total-station intersection shot and every resection is this triangle.

The linkage bench. Position analysis of a four-bar or a hydraulic-cylinder linkage keeps producing the same subproblem: two known link lengths, one known angle, find the rest. That is the SSA case, and its famous "ambiguity" is not a textbook curiosity — the two mathematical solutions are the open and crossed assembly configurations of the mechanism. If your kinematics spreadsheet snaps to the wrong branch mid-stroke, this is why.

Antenna baseline work. Two receive sites on a known baseline each measure an angle of arrival to an emitter; the law of sines converts the baseline plus two bearings into ranges. Same math as the surveyor's intersection, just at RF, and the same conditioning rule applies: a target nearly in line with the baseline makes a skinny triangle and a garbage fix.

How it works (the real substance — behavior, gotchas, limits of validity, the mistake people make)

The mechanism is one altitude. Drop a perpendicular of height h from corner C to side c: the two right triangles it creates give h = b·sin(A) and h = a·sin(B). Same altitude, so a/sin(A) = b/sin(B). Drop a different altitude and the third ratio joins. The 2R version comes from the inscribed-angle theorem: every angle subtending side a from the circumscribed circle sees the same arc, so the ratio is a property of the circle, not of which triangle you drew inside it.

What it actually solves, sorted by safety:

ASA / AAS — safe. Two angles pin the third (A + B + C = 180°), then every side comes out of the ratio. This is the surveyor's case and it is unambiguous.
SSA — the ambiguous case. Given a = 7, b = 10, A = 40°: sin(B) = 10·sin(40°)/7 = 0.9183, and two angles have that sine — B = 66.67° or B = 113.33°. Both make legal triangles, with third sides c = 10.43 and c = 4.89. The math is not being difficult; there genuinely are two triangles, and only outside knowledge (which linkage branch, which side of the baseline) picks one.
SAS / SSS — wrong tool. The known angle isn't opposite a known side, or there's no angle at all. Start with the law of cosines, then come back.

The mistake people make, in order of frequency:

Trusting asin for a possibly obtuse angle. sin(113.33°) = sin(66.67°), and the calculator always returns 66.67°. Rule of thumb: use the law of sines to find small angles (anything opposite a shorter side is guaranteed acute) and get the largest angle by subtraction or from the law of cosines, whose acos distinguishes acute from obtuse on its own.
Solving skinny triangles and believing the answer. The ratio's sensitivity lives in the sine. Near 2°, a 0.1° angle error moves sin(θ) by 5.0%; near 88°, the same 0.1° moves it by 0.006%. That is nearly a thousandfold difference in how hard your angle-measurement error hits the computed distance. Surveyors keep intersection angles roughly between 30° and 150° for exactly this reason, and an emitter-location geometry with the target near the baseline extension deserves the same suspicion.
Degrees and radians. The formula doesn't care, but your sin() function does. A 68° argument fed to a radians-mode function fails loudly; a 1.2-radian argument fed as degrees fails quietly, which is worse.

Limit of validity: this is flat-plane geometry. Over machine-shop and building-site distances the Earth's curvature is invisible, but on long geodetic or instrumentation-range baselines the triangle lives on a sphere, the angles sum to more than 180°, and the plane law drifts. The spherical replacement keeps the same shape with arc lengths inside the sines — sin(a)/sin(A) = sin(b)/sin(B) = sin(c)/sin(C) — and it is the older of the two laws by three centuries.

History (who derived it and when, told as a short story with inline [n] citations)

The spherical version came first, because the customers were astronomers and the sky is a sphere. In the late 900s the rule surfaced among a cluster of scholars in the Islamic world — Abu al-Wafa in Baghdad, his student Abu Nasr Mansur in Khwarazm, and the astronomer al-Khujandi — and the priority fight was never settled; Mansur's proof survives in several of his treatises, but he may have learned the result from his teacher ^[1]^[2]. In eleventh-century al-Andalus, Ibn Muʿadh al-Jayyani of Córdoba put the general sine law into The Book of Unknown Arcs of a Sphere, regarded as the first treatise devoted to spherical trigonometry as its own subject ^[2]^[3].

The plane version — the one on your fixture drawing — is credited to Nasir al-Din al-Tusi (1201–1274), working in thirteenth-century Persia. His Treatise on the Quadrilateral stated and proved the sine law for plane triangles and, more consequentially, treated trigonometry as an independent branch of mathematics rather than a bag of astronomer's tricks ^[2]^[4].

Europe caught up through Regiomontanus, who completed De triangulis omnimodis ("On Triangles of All Kinds") in 1464 — five books modeled on Euclid, with Book II presenting the sine law and putting it to work solving triangles ^[5]^[6]. The payoff for engineers arrived in 1615, when Willebrord Snellius ran a chain of triangles — fourteen stations, mostly church spires, 53 triangle measurements — from Alkmaar to Bergen op Zoom to measure roughly 130 km of the Netherlands with a two-foot quadrant, publishing the method in Eratosthenes Batavus (1617) ^[7]^[8]. That is the birth of triangulation: measure one baseline carefully, measure angles everywhere else, and let the law of sines carry the length through the whole network. Every geodetic survey until the GPS era ran on that idea.

Related tools (bullet list of HE calculator slugs that use or neighbor this topic, as /tools/ links)

/tools/convert-angle — degrees, radians, and mils for the measured angles before they hit sin()
/tools/radar-horizon — the line-of-sight geometry that decides whether your baseline stations can see the target at all
/tools/fresnel-zone — link-path geometry that starts from the same surveyed endpoint distances
/tools/projectile-range — a neighboring dose of sines doing trajectory work

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →