Derive Law of Sines (Inscribed Triangle)

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

Steps

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

    derive law of sines step 1
  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.

    derive law of sines step 2 a

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

    derive law of sines step 2 b

    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.

    derive law of sines step 3

    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.