Trigonometry in the Complex Plane

Welcome to the trigonometry in the complex plane. This book introduces college level trigonometry using the geometry of the circle in the complex plane. Building upon the concepts in the previous book, this book discusses the geometric properties of complex arithmetic, defines the trigonometric functions and the exponential function, then explores Euler’s formula and Fourier Transformations. This book is the last of the three-part trigonometry series:

  1. Trigonometry Introduction
  2. Trigonometry Fundamentals
  3. Trigonometry in the Complex Plane

Introduction

The set of complex numbers, sometimes called by the unfortunate name of “imaginary numbers” are a tricky bit of math that can appear arcane and convoluted from the onset and then, after close study, suprisingly simple. The goal of this book is to try and get every reader to the latter and not get lost in the sauce as they say. All potatoes here in Wumbo land.

First the pertinent points are recapped from the radian angle system and the trigonometric functions using the geometry of the circle. Here we also define the trigonometric functions as infinite series using calculus as we will be using these definitions later on.

Then the definition of complex numbers and the geometry of complex number arithmetic is introduced. After that, the complex unit circle and trigonometric functions are discussed in relation to complex number arithmetic. This is where the usefulness of the complex number system begins to shine.

Lastly, we take a detour to define the exponential function, which is often written in shorthand as Euler’s number. Extending the function to complex numbers, a surprising connection to trigonometry appears.

Radians

Radian Angle System annotated using the circle constant τ (tau).
Figure 1: Radian Angle System

The radian angle system defines rotation as unitless measure based on the geometry of the circle. Up until this point in this series the claim has been made that the radian angle system is preferred over the degree angle system. Now we will finally see the benefit of using this system as we look again at the trigonometric functions and how to define them.

Trig Functions

This figure illustrates the trigonometric functions defined using the circle.
Figure 2: Circle Trigonometry

Sine

Given an angle in radians the sine function returns the ratio of over the radius . Geometrically, this ratio is equal to the component of the corresponding point on the unit circle. The plot of the sine function is shown below for angles from to (tau) radians.

Plot of Sine Function from 0 to τ (tau) radians.
Figure 3: Sine Plot

The sine function can be defined using a Taylor Series.

Cosine

Given an angle in radians the cosine function returns the ratio of the adjacent side over its hypotenuse. The plot of the cosine function is shown below for the domain to (tau) radians, where represents a full rotation. The numeric values for radians are shown on the top edge of the plot’s grid.

Plot of Cosine from 0 to τ (tau) radians.
Figure 4: Cosine Plot

In the same fashion a Taylor Series expansion can be used to define the cosine.

Derivatives

Here we see a relationship between the two functions that wouldn’t be true if we had instead used the degree angle system. If we take the derivative of sine, as shown below, we are left with the power series definition of cosine. In other words, we are able to explicitly show that the derivative of sine is equal to cosine.

Applying the same process we can show that the derivative of cosine is equal to negative sine.

Complex Plane

The complex number system extends the ordinary number system from a linear interpretation, the number line, to a two dimensional interpretation known as the complex plane.

Complex Numbers

The arithmetic properties of the complex number system naturally represent 2D geometry. The addition of two complex numbers is like adding two vectors together. The multiplication of two complex numbers has a scale and rotation transformation property.

Definition

A complex number is denoted in the form shown below, where and are real numbers and the variable i represents the square root of negative one.

Complex Number

A complex number is visualized as a point in 2D space, usually drawn as an arrow, with a horizontal component equal to the real part of the number and the vertical component equal to the complex part of the number. For example, the visualization of the number 3+2i is illustrated below.

This figure illustrates a number the complex plane. The horizontal axis represents the real part of a complex number and the vertical axis represents the complex part of a complex number.
Figure 5: Complex Number

Addition

The addition of two complex numbers can be visualized as two vectors in the complex plane placed tail to tail. The figure below illustrates the addition of the complex numbers 2+3i and 4-1i, resulting in the number 6+2i.

The visualization of complex number addition.
Figure 6: Complex Number Addition

Multiplication

The multiplication of a complex number is a bit more tricky to visualize, but it is extremely useful to have a mental model of what multiplying two complex numbers looks like.

Identity Property

Multiplying any complex number by results in . Visually, this correspond with stretching and rotating the number so that it lies on top of the complex number in the complex plane. Conceptually, you can imagine stretching and rotating all of the horizontal and vertical axis lines in the first plane.

This figure illustrates visual representation multiplying 1 by a complex number
Figure 7: Complex Plane (Identity Property)
Complex Property

The second property of complex number mutliplication is multiplying a number by is equivalent to rotating by . For example, the complex number multiplied by is equal to the rotation shown below:

This figure illustrates visual representation of multiplying a complex number by the complex number i.
Figure 8: Complex Plane (Complex Property)

This is where the usefulness of complex numbers starts to appear; the multiplication of two complex numbers can be geometrically related to rotation. Let’s verify the result we saw above using algebra and look at another example.

Multiplication Property

The last property of multiplication puts the first two together and allows us to visualize what is happening when any two complex numbers are multiplied together. For example, to multiply the complex numbers and together, first we can visualize the stretch and rotation of the coordinate system from the first number , then we can draw the complex number arrow using this new stretched grid to find the product.

This figure illustrates visual representation of multiplying two complex numbers together.
Figure 9: Complex Plane Multiplication

We can verify this result using algebra.

Complex Unit Circle

A point on the unit circle in the complex plane can be elegantly expressed in terms of the sine and cosine of the point’s angle. This is illustrated in the figure below.

A point on the unit circle in the complex plane.
Figure 10: Complex Unit Circle

Complex Numbers Trigonometry

Trigonometric Identities

Euler's Formula

Euler’s formula, sometimes referred to as “our gem” is a famous mathematical expression

Fourier Transform

Fourier transformation is a method to extract

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