Derive Sum of Two Angles (Ptolemy's Theorem)

This example derives the sum of two angle identities using the law of sines, the inscribed angle theorem and Ptolemy’s Theorem.

Steps

  1. Draw an inscribed quadrilateral on a circle of diameter , where one of the quadrilateral’s diagonals lays on the diameter of the circle. This gives us the diagram drawn below, where the length of the line is equal to .

    Derive Sum of Two Angles (Ptolemy's Theorem) step 1 a

    Observe that the inscribed angle theorem garuantees that the triangle is a right triangle with the angle and that the triangle is a right-triangle with an angle .

    Derive Sum of Two Angles (Ptolemy's Theorem) step 1 b

    Apply the definition of sine and cosine to find the lengths of and for the right triangle . The same process can be followed to find the lengths of and for the right triangle . This labeled lengths are shown below.

    Derive Sum of Two Angles (Ptolemy's Theorem) step 1 c
  2. Next, observe that the law of sines applies to the triangle highlighted below.

    Derive Sum of Two Angles (Ptolemy's Theorem) step 2 a

    The general form of the law of sines is given below:

    Substitute the labeled lines into the law of sines.

    Since the circle has a diameter of the common ratio is .

    Then, we can substitute the angle in for in the expression.

    Solve for the line .

    Label this length in the diagram.

    Derive Sum of Two Angles (Ptolemy's Theorem) step 3 c
  3. Finally, apply Ptolemy’s theorem to the lengths of the inscribed quadrilateral on the circle. Recall, Ptolemy’s theorem relates product of the diagonals to the sum of the products of the sides of an inscribed quadrilateral:

    Substitute the lengths from above into the equation.

    This gives us the equation for the first summation identity.

  4. Then, we can take advantage of symmetry of the trigonometric functions to define the sum of two angles cosine identity. Shown below is the symmetry of the two functions.

    Start with the sum of two angles (alpha) and (beta) as input to the cosine function.

    Apply the symmetry from the first equation.

    Change the associative groups of the addition.

    Apply the sum of two angles identity from the previous step.

    Apply the symmetry from the first and second equation.

    This gives us the sum of two angles cosine identity and completes the derivation of the two identities.