The sine rule

The second great tool for solving general triangles relates each side to the angle that lies opposite it.

Theorem — The sine rule

In a triangle $ABC$ with sides $a,b,c$ opposite the vertices $A,B,C$ and a circumscribed circle of radius $R$: $\frac{a}{\sin\widehat{A}} = \frac{b}{\sin\widehat{B}} = \frac{c}{\sin\widehat{C}} = 2R.$

The ratio of a side to the sine of the opposite angle is therefore the same for all three sides, and it equals the diameter $2R$ of the circumscribed circle.

Links

Topics: Triangle trigonometry
Concepts: Circumscribed circle · Triangle solving · Sine rule
Functions: Sine
Methods: Triangle solving
Skills: Using formulae