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. The two identities are shown below:

Steps

  1. Observe that the angle (theta) forms the complex number on the unit circle given by Euler’s formula.

    This complex number is visualized by the figure below:

    A point on the unit circle in the complex plane.
    Figure 1: Complex Unit Circle
  2. Model the addition of the two angles (alpha) and (beta) as input to the exponential function.

    Distribute the complex constant for both angles.

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

  4. Apply Euler’s formula to both sides of the equation where the exponential function appears.

    Multiply the right-hand side of the equation.

    Substitute for in the expression.

  5. Then, taking the last expression from the step above we can separate real parts from the complex parts of the number.

    Dividing both sides of the second expression by gives us the two summation identities.

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