Notes
sine rule

Sine Rule

The sine rule is a relationship between the sides and angles of an arbitrary triangle. Given a triangle as follows:

StandardTriangle.png

the sine rule states that the following equations hold:

\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}

Equivalently,

\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

It can be deduced from the sine formula for the area of a triangle. Starting from:

\Area = \frac{1}{2} a b \sin C = \frac{1}{2} b c \sin A = \frac{1}{2} c a \sin B

Multiply through by $2$ and divide by $a b c$ to get:

\frac{2 \Area}{ a b c} = \frac{2 a b \sin C}{2 a b c}= \frac{2 b c \sin A}{2 a b c}= \frac{2 c a \sin B}{2 a b c}

and the sine rule then follows by cancellation.

Created on April 12, 2021 15:30:34 by Andrew Stacey