This example derives the law of sines by dividing the general case of the triangle into two right triangles, applying the definition of sine to the share opposite side and equating the expressions together.

Observe that in the general case of the triangle the two angles and share the same opposite side.

Applying the definition of sine we get the two trigonometric ratios.

Solve both equations for and set them equal to each other.

Divide both sides by .

Divide both sides by .

Simplify the equation.

This gives us the first relation of the law of sines.

Not shown here, but this same process can be applied to either the opposite side shared by the angles and

or the opposite side shared by the angles and .