Derive Law of Sines (Inscribed Triangle)

This example derives the law of sines using a triangle inscribed within a circle.

1. Start with an inscribed triangle on the circle of radius with the three angles , and and the sides with lengths , and .

   

2. Observe that the inscribed angle theorem applies to the angle . The inscribed angle theorem states that the central angle is equal to two times the inscribed angle that shares the same circumference.

   

   This allows us to split the triangle into two equal right triangles. The opposite sides of the right triangles can be found by applying the definition of sine, plugging the hypotenuse of length into the equation and solving for the adjacent side.

   

   Solve for the length of the opposite side.

   This gives us the ratio of sides.

3. Then, repeat the process shown in step 2 for the angle at the point . Apply the inscribed angle theorem and find the lengths of the opposite sides of the right triangles.

   

   This gives us the ratio of the length over .

   Combine this ratio with the ratio found in step 2 since they both equal .

4. Repeat the same process for . This gives us the law of sines shown below.