This example demonstrates how to derive the double angle identities using the properties of complex numbers in the complex plane. The double angle 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 double angle (alpha) as the input to the exponential function.

Distribute the complex constant .

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.

Equate the real parts and complex parts of the two complex numbers.

This gives us the cosine double angle identity.

And the sine double angle identity.

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 double angle identities give the sine and cosine of a double angle in terms of the sine and cosine of a single angle.

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.