Law of Sines

The law of sines says every triangle keeps a fixed exchange rate between sides and the sines of their opposite angles: bigger angle, proportionally bigger opposite side. (The common ratio is in fact the diameter of the triangle's circumscribed circle.) It is the tool of choice when you know an angle–side opposite pair: navigators and surveyors have leaned on it for centuries, since two sighted angles and one measured baseline pin down every other distance by triangulation. Example: a baseline b = 100 m with angles A = 40° and B = 65° gives a = 100 × sin 40° / sin 65° ≈ 70.9 m, no tape measure across the river required.

Solving for an angle takes care. It requires arcsin, and the principal branch cannot tell an acute angle from its obtuse supplement: the classic ambiguous SSA case, where two different triangles fit the same data. The solver returns the acute angle and says when the obtuse one fits too. The side rearrangements are single-valued closed forms with no such trap.