Trigonometry Introduction

Welcome to the introduction to trigonometry. This webbook introduces middle-school and high-school level trigonometry using the degree angle system and the geometry of the triangle. This is the first book of a three-part series:

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

Introduction

Trigonometry is a the study of triangles and angles. The subject was originally inspired by astronomy, navigation and geometry and has since evolved to have broad applications in math, physics and computer science. This book introduces trigonometry using the degree angle system and basic geometry.

Angles

TODO: wild angle interactive

Note: The illustration above is interactive, meaning that the blue control points can be dragged around. These interactive appear throughout this website.

Degrees

The degree angle system divides a full rotation into units called degrees. In the XY coordinate system it is convention to measure angles starting from the rightward (positive ) direction with the counter-clockwise direction as positive.

This figure illustrates the standard for measuring angles in the XY coordinate plane.
Figure 1: Standard XY Angle

TODO: show degree angle system oriented in the z axis with zero on the top.

Measuring Angles

Types of Angles

Acute Angle

An acute angle is defined as an angle smaller than a perpendicular angle.

Obtuse Angle

An obtuse angle is defined as an angle greater than a perpendicular angle.

Perpendicular Angle

A perpendicular angle is equal to a quarter of a full rotation.

TODO: possibly replace above with table w/thumbnails and definitions

Triangle

A triangle is a three sided polygon. The shape has three vertices that correspond to the angles formed by the sides of the shape. There are various ways to construct a triangle. For example, the triangle illustrated in the interactive below is formed by the three points. The blue controls points can be dragged around to change the shape of the triangle.

Note: The illustration above is interactive, meaning that the blue control points can be dragged around. These interactive appear throughout this website.

The name triangle can be broken into the prefix “tri” meaning three and “angle”. The shape is simple, versatile and if you look closely enough has a lot to do with circles.

Acute Triangle

Obtuse Triangle

Right Triangle

The right triangle has three sides relative to the angle that defines its shape: the adjacent, opposite and hypotenuse.
Figure 2: Right Triangle

The right-triangle is a special case of the general triangle where one of the angles is equal to .

TODO: show different cases of dividing triangles

Special Triangles

There are two special right-triangles whose properties are useful and appear in many places: the 45 45 90 right triangle and the 30 60 90 right triangle. Because the triangles have angles which evenly divide a full rotation, their geometry appears in many special cases.

The 45 45 90 special right triangle is defined by the three angles: 45, 45 and 90 degrees. Its simple geometry makes it useful in applications.
Figure 3: 45 45 90 Triangle
The 30 60 90 special right triangle is defined by the angles: 30, 60 and 90 degrees.
Figure 4: 30 60 90 Triangle

TODO: Why does the 30-60-90 triangle appear in such a simple form? THe 45-45-90 follows naturally from dividing a square by the diagonal. Does the construction of a hexagon lead to a simple form of this special triangle?

Pythagorean Theorem

A unique property of the right triangle is defined in Pythagorean’s theorem which relates the squares of the sides together. This can be visualized by drawing the square area of each side next to the right triangle.

This figure illustrates the geometric interpretation of the Pythagorean theorem.
Figure 5: Pythagorean Theorem

Trig. Functions

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

The trigonometric functions return ratios of the sides of a right-triangle when given the angle of the right-triangle as input. Historically, before calculators and computers the angles corresponding to ratios would be stored in a table. Shown below are the right-triangle definitions of the trigonometric functions.

Function Equation
Sine
Cosine
Tangent

Mnenomic

“SoaCaoToa” is an mnemonic that can be used to remember the different ratios that the function return.

Sine

Given the angle of a right triangle the sine function returns the ratio of its opposite side over the hypotenuse.

Plot of Sine Function from 0 to 360 degrees
Figure 7: Sine Plot (Degrees)

Cosine

Given an angle of a right triangle the cosine function returns the ratio of the adjacent over the hypotenuse.

Plot of Cosine Function from 0 to 360 degrees
Figure 8: Cosine Plot (Degrees)

Tangent

Given an angle of a right triangle the tangent function returns the ratio of the opposite over the adjacent.

Ratio Table

While calculators are everywhere and easily accesible in modern mathematics, historically, tables have been used to lookup the ratios corresponding to an angle. This fact is why the trigonometric functions are also called the trigonometric ratios.

Common ratios

Angle Sine Cosine Tangent
0.000 1.000 0.000
0.087 0.996 0.087
10° 0.174 0.985 0.176
15° 0.259 0.966 0.268
20° 0.342 0.940 0.364
25° 0.423 0.906 0.466
30° 0.500 0.866 0.577
35° 0.574 0.819 0.700
40° 0.643 0.766 0.839
45° 0.707 0.707 1.000
50° 0.766 0.643 1.192
55° 0.819 0.574 1.428
60° 0.866 0.500 1.732
65° 0.906 0.423 2.145
70° 0.940 0.342 2.747
75° 0.966 0.259 3.732
80° 0.985 0.174 5.671
85° 0.996 0.087 11.430
90° 1.000 0.000 INFINITY

Trig Identities

The trigonometric identities are a set of useful equations that can be used to transform and manipulate math expressions. They represent a combination of different theorems and observations that relate to triangles. However, for brevity, this book looks at only two groups of identites. The sum of two angles and the difference of two angles.

Sum of Two Angles

The sum of two angles express the trigonometric ratios of sine and cosine in terms of the ratios of the individual angles. These identities can be visually expressed as a “proof without words” as shown in the figure below. To derive this proof, see this page.

This figure illustrates the "proof without words" of the sum of two angles identities
Figure 9: Sum of Two Angles Identities (Visual Proof)

Difference of Two Angles

The difference of two angles express the trigonometric ratios of sine and cosine in terms of the ratios of the individual angles. These identities can be visually expressed as a “proof without words” as shown in the figure below. To derive this proof, see this page.

This figure illustrates the "proof without words" of the difference of two angles identities
Figure 10: Difference of Two Angles Identities

Applications and Experiments

  • How to find the height of an object
  • Calculate unknown length given angle and/or side information (general)
  • Sun rise and sun set science experiment
  • Triangulation using a compass

Links