# Sine Rule The **sine rule** is a relationship between the sides and angles of an arbitrary triangle. Given a triangle as follows: [[StandardTriangle.png:pic]] 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.