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: .

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

and and

Solving for X in the first two equations,

and

By cross products,

But solving for a,

and

Combining the resulting equations, we get

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