Solving a Triangle

The three sides and three angles of a triangle are together called its elements. Given any three of the six elements, provided at least one of them is a side, the remaining three elements can be found uniquely — this process is called solving the triangle. (Knowing only the three angles is not enough: infinitely many similarly-shaped triangles of different sizes share the same three angles, so at least one side is essential to fix the actual size.) In $\triangle ABC$ we use the usual notation: $l(BC)=BC=a$, $l(CA)=AC=b$, $l(AB)=AB=c$ — that is, each lower-case letter denotes the side opposite the vertex of the same (capitalised) name. The rest of this section develops the standard relations …