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:

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:

Model the addition of the two angles (alpha) and (beta) as input to the exponential function.

Distribute the complex constant for both angles.

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

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.

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.

The exponential function models exponential growth. The output of the function at any given point is equal to the rate of change at that point. For real number input, the function conceptually returns Euler's number raised to the value of the input.

The sum of two angles identities express the cosine and sine of the sum of two angles in terms of their individual cosine and sine components.

A complex number is an extension of the real number line where in addition to the "real" part of the number there is a complex part of the number. The properties of complex numbers are useful in applied physics as they elegantly describe rotation.

Euler's Formula returns the point on the unit circle in the complex plane when given an angle.