# Sine Law

## Sine Law

In a triangle, the name given to the relationship of proportionality between the measures of the sides and the sine values of the angles opposite each of these sides.

### Property

Consider the radius $$r$$ of any circumscribed circle of a triangle with sides $$a$$, $$b$$ and $$c$$ and angles A, B and C; this gives us the following relationship: $$\dfrac{a}{\sin(\textrm{A})} = \dfrac{b}{\sin(\textrm{B})} = \dfrac{c}{\sin(\textrm{C})} = 2r$$.

### Example

Consider triangle ABC in which side AB measures 15 cm and angles ABC and BAC measure 60° and 50° respectively. We can deduce that angle BCA measures 70°, since 180° − (60° + 50°), then we can calculate the measure a of side BC by posing $$\dfrac{a}{\sin(50)} = \dfrac{15}{\sin(70)}$$.

Which gives a ≈ 12.23 cm.