Law of Sines

Given a triangle with sides of length a, b and c, opposite angles of measure A, B and C, respectively, and a circumcircle with radius R, $\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}=2R$ .

Proof of the Law of Sines

The formula for the area of a triangle is: $\displaystyle [ABC] = \frac{1}{2}ab\sin C$

Since it doesn't matter which sides are chosen as $a$ , $b$ , and $c$ , the following equality holds:

$\displaystyle \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B = \frac{1}{2}ab\sin C$

Multiplying the equation by $\displaystyle \frac{2}{abc}$ yeilds:

$\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Page

Toolbox

Search

Law of Sines

Proof of the Law of Sines

See also