The law of sines hides a circle behind every triangle
Divide any side of a triangle by the sine of the angle facing it and you get the same answer for all three sides. That shared number is no coincidence: it equals the diameter of the circle passing through the triangle's three corners, a fact that goes back to Ptolemy.
The rule says side and opposite angle stay in fixed proportion around the whole triangle. A quick proof uses height. Put one side along the bottom and the triangle's altitude can be computed from either neighbouring side times the sine of the angle at its far end; setting the two expressions equal gives the law directly. The circle connection comes from the inscribed angle theorem, since angles resting on the same chord of a circle are equal, so every ratio collapses to twice the circle's radius.
Its everyday job is triangulation. Know two angles and one side, and the remaining sides follow, which is how surveyors fix distances they cannot walk. Together with its partner, the law of cosines, it handles lopsided scalene triangles that have no right angle to lean on. Plugged into the half-base-times-height formula for area, it also connects area with sines.
There is a trap. Given two sides and an angle not between them, the data can fit two different triangles, the so-called ambiguous case. That happens when the side opposite the known angle is shorter than the other given side but longer than the altitude, allowing it to swing into either of two positions. Solvers must check which answer, or both, makes sense.
A worked example shows the check. With sides of 20 and 24 and an angle of 40° opposite the longer one, the arcsine offers an obtuse candidate of 147.61°, but that is rejected because the triangle's angles would then add up to more than 180°. The law also extends beyond flat paper to curved surfaces of constant curvature.
Source: Law of sines
