Derive Sum of Two Angles Identities (Complex Plane)

This example derives the sum of two angles identities using the properties of complex numbers and Euler’s formula.


  1. The sum of two angles (alpha) and (beta) form a complex number on the unit circle in the complex plane. From Euler’s formula, we know this complex number can be expressed using cosine and sine.

    This complex number can be visualized as a point in the complex plane, where the horizontal coordinate is equal to and the vertical coordinate is equal to .

    Sum of two angles complex unit circle.
  2. Distribute the complex constant for both angles.

    Apply the addition property of the exponential function to expand the right-hand side.

  3. Apply Euler’s formula to both sides of the equation.

    Multiply the right-hand side of the equation.

    Substitute for on the right-hand side.

    The geometry of the expressions on the right-hand can side can be visualized by drawing the trigonometry of the two angles on the unit circle.

    Sum of two angles complex geometry showing the stretch and rotation caused by multiplying two complex numbers together.

    These lengths can be found using trigonometry, as shown in this example, but here they are the natural result of the multiplication of two complex numbers.

  4. Equate the real part and complex parts together.

    Divide both sides of the second expression by to get the two summation identities.