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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.

220px-Acute Triangle B.svg

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}

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