Law of Sines

Revision as of 20:20, 29 June 2006 by Xantos C. Guin (talk | contribs) (Added proof for the non-extended 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:

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

See also

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