Euler's Formula

Euler’s formula takes in angle an input and returns a complex number that represents a point on the unit circle in the complex plane that corresponds to the angle. This section introduces complex number input and Euler’s formula simultaneously. Note, the math here gets a little tricky because of how many areas of math come together. The definition of Euler’s formula is shown below.

Euler’s formula can be visualized as, when given an angle, returning a point on the unit circle in the complex plane. This is shown in the figure below.

Euler's Formula in the Complex Plane
Figure 1: Euler's Formula
Expression Description
The exponential function
The symbol satisfies the expression
An angle provided in radians.
The cosine function.
The sine function.

Notation

It is worth mentioning that the convention of writing the exponential function in the shorthand form can be confusing. Instead, consider the form below which conceptually and in applications makes Euler’s Identity more approachable.

Appearance of Sine and Cosine

This realization can be rather shocking. We can show how the definitions of sine and cosine show up in the formula. Start by expanding the power series using our original definition of the funtion.

Then, let’s focus on the places where the symbol appears.

Here we can substitue into the power series.

Examples

Input of an Angle

TODO: interactive of an angle theta

Examples