Law of sines

In trigonometry, the law of sines (or sines law, sine formula) is a statement about arbitrary triangles in the plane.

If the sides of the triangle are A, B and C and the angles opposite to those sides are a, b and c, respectively, then the law of sines states that: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

Proof

Let a, b, and c be the sides of a triangle opposite the angles A, B and C to side b, perpendicular to each other. This divides the original triangle into two right triangles. We let X be the length of this dividing line.

Acute triangle ABC with altitude X drawn from B

Since sine is opposite divided by hypotenuse, then

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

Solving for X in the first two equations,

$X=c \sin A$ and $X=a \sin C$

By cross products,

$\frac{a}{\sin A}=\frac{c}{\sin C}$

But solving for a,

$a=\frac {\sin C}{X}$ and $a=\frac{\sin B}{b}$

Combining the resulting equations, we get

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