Trigonometry Fundamentals

Welcome to the fundamentals of trigonometry. This webbook introduces high-school and college level trigonometry through the geometry of the circle. The content is intended to bridge the gap between basic trigonometry and advanced trigonometry. While some exposure to trigonometry is useful, it is not a prerequisit of this book. This book is the second of a three-part series:

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

Introduction

Trigonometry is a translational subject. It connects and is used by many different areas of math and other disciplines. The subject provides a medium and set of tools to approach concepts such as triangles, circles, waves, ellipsis and hyperbolas; ideas that play a formative role throughout math and physics.

"You think it's about triangles, but really it's about circles."

Unknown

Trigonometry traditionally focuses on the properties of triangles, especially that of the right triangle. The triangle is simple, versatile and appears everywhere. However, as we will see in this book, the geometry of the triangle and the circle are intimately related; Understanding the circle and why it is so important to trigonometry is vital to approach more complicated ideas in math and physics. You think it’s about triangles, but really it’s about circles.

Trigonometry is about circles
Figure 1: Trigonometry (Circles)

This book approaches trigonometry primarily through the lens of the circle in the cartesian plane as illustrated in the figure above. The radian angle system, circle constant and trigonometric functions are defined using the geometry of the circle.

Angles

An angle is defined as the amount of rotation between two rays. The point where the two rays start is called the vertex. Angles are a fundemental building block of trigonometry as every triangle’s shape is formed by three angles.

By convention, in the Cartesian Plane, an angle is measured starting at the positive direction with positive rotation in the counter-clockwise direction as shown in figure below. The variable (theta) is often used to represent the angle. Of course, in the real world, angles come in all different orientations. This convention is so the study of the angle can be formalized.

This figure illustration that an angle by itself is measured from the positive rightwards direction on the circle with the positive in the counter-clockwise direction.
Figure 2: Standard Angle Convention

It is also convention to measure an angle starting from the upwards direction down, just as clocks and other devices are oriented. This is more common when an angle is measured starting from the axis in three dimensions.

Radians

There are two systems in math used to measure angles: degrees and radians. The degree system is rooted in the origins of trigonometry, time keeping and astronomy. The degree system has been used for hundreds of years, but as math has evolved, the radian system has replaced degrees as the preferred system[1]. This is because, in short, the radian system has properties that lead to nice properties and elegance in later math.

This figure illustrates the radian angle system annotated using the circle constant τ (tau).
Figure 3: Radian Angle System

A full rotation in the cartesian plane is animated below. Note, when referring to the “cartesian plane” it is implied that this is the standard “XY” coordinate plane.

Circle Constant

The circle constant (tau) is a geometric constant defined by the circumference of a circle divided by its radius. The circle constant makes using radians more practical. Shown below is the definition of the circle constant in terms of any circle.

Definition of the circle constant τ (tau)
Figure 4: τ (tau) Definition
Note: This website uses the constant (tau) instead of (pi) as the default circle constant. The substitution can be used to translate between the two constants.

The Circle

The significance of a circle in the Cartesian plane

A circle of radius one placed at the origin of the coordinate system.
Figure 5: Unit Circle

Continuing the theme of looking at an angle in isolation, let’s introduce the right triangle.

Trigonometric Functions

The trigonometric functions are periodic wave functions that are used throughout math and physics. The three main functions are introduced using the geometry of the right-triangle as functions that take an angle as input and return the ratio of two sides of the corresponding right triangle:

The right triangle has three sides relative to the angle that defines its shape: the adjacent, opposite and hypotenuse.
Figure 6: Right Triangle

The definitions of these functions are extended to all angles representing the domain of all real numbers using the geometry of the circle. This is shown in the figure and definitions below. Note, the geometry of the right-triangle is still very much there, but these definitions have some real advantages when thinking about the functions.

This figure illustrates the geometry used to define the circle definitions of the trigonometric functions.
Figure 7: Circle Functions
Note: Nowadays these functions appear in calculators and programming languages. Historically, these functions were called trigonometric ratios and would be looked up in a table. The functions can also be defined using calculus, but that is not discussed until the next book in this series.

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 8: Sine Plot

Since the function returns a ratio, all possible output of the function can be visualized by constraining the hypotenuse to be of length , which forms the shape of the unit circle. This is shown in the interactive below.

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 9: Cosine Plot

Since the function returns a ratio, all possible output of the function can be visualized by constraining the hypotenuse to be of length , which forms the shape of the unit circle.

Tangent

Given an angle in radians the tangent function returns the tangent line to the point on the unit circle. Since the component of a point goes to zero at a quarter turn and three-quarters turn, the function diverges at those angles.

Plot of the tangent function from 0 to τ (tau) radians.
Figure 10: Tangent Function

Unit Circle

The unit circle is a circle of radius one, placed at the center of the cartesian coordinate system visualizes the output of the trigonometric functions. Besides being a nice visualization for the trigonometric functions, the unit circle’s properties also make it useful in later mathematics.

A circle of radius one placed at the origin of the coordinate system.
Figure 11: Unit Circle

The interactive below demonstrates the geometry of the unit circle and the output of the trigonometric functions.

Unit Circle Chart

The unit circle chart(s) are common angles and their corresponding pairs found on the unit circle. As a result, teachers often use the chart as a way to test student’s knowledge of the trigonometry of the unit circle. For example, the chart below shows the angles and values derived from dividing the unit circle into twelve equal parts.

Unit Circle Chart 4

The Unit Circle Divided by 4 measured using the circle constant.
Figure 12: Unit Circle Chart 4

Unit Circle Chart 8

The Unit Circle Divided by 8 measured using the circle constant.
Figure 13: Unit Circle Chart Eighths τ (tau)

Unit Circle Chart 12

The Unit Circle Divided by 12 measured using the circle constant.
Figure 14: Unit Circle Chart Twelfths τ (tau)

Here is another chart that shows the angles and values from dividing the unit circle into eight equal parts. Really a unit circle chart is just a set of common or simple points found on the unit circle.

Since the angles on the charts are common, the points can be solved for using the properties of the special triangles and similar triangles as shown on this page. Of course, more generally, the charts can be

Trigonometric Identities

The trigonometric identities are a set of equations that relate the trigonometric functions together. This section discusses the trigonometric identities and how to derive them using the unit circle.

Applications

  • How to find line perpendicular to another
  • How to derive the trigonometric identities (geometry, algebra & trig, and complex plane)
  • How to implement a shading algorithm in 3D using a triangle mesh

Vectors

  • Decompose velocity vector into individual components
  • Rotate the direction of a vector by an angle
  • Find the angle between two vectors
  • How to model 2D rotation in a coordinate system (image rotation problem) (linear algebra or complex numbers)
  • How to implement a simplified 2D “global” positioning system

Wave Forms

  • How to model our solar system

Links

References

  1. Radians Versus Degrees
    Wumbo (internal)