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:
Trigonometry is 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.
TODO: wild angle interactive
The degree angle system divides a full rotation into units called degrees. In the XY coordinate system it is a convention to measure angles starting from the rightward (positive ) direction with the counter-clockwise direction as positive.
TODO: show the degree angle system oriented in the -axis with zero on the top.
An acute angle is defined as an angle smaller than a perpendicular angle.
An angle greater than 90 degrees.
A perpendicular angle is equal to a quarter of a full rotation.
TODO: possibly replace above with table w/thumbnails and definitions
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 control points can be dragged around to change the shape of the triangle.
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.
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
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 that evenly divide a full rotation, their geometry appears in many special cases.
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?
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.
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 |
“SoaCaoToa” is a mnemonic that can be used to remember the different ratios that the function return.
Given the angle of a right triangle, the sine function returns the ratio of its opposite side over the hypotenuse.
Given an angle of a right triangle, the cosine function returns the ratio of the adjacent over the hypotenuse.
Given an angle of a right triangle, the tangent function returns the ratio of the opposite over the adjacent.
While calculators are everywhere and easily accessible in modern mathematics, historically, tables have been used to look up the ratios corresponding to an angle. This fact is why the trigonometric functions are also called trigonometric ratios.
Common ratios
| Angle | Sine | Cosine | Tangent |
|---|---|---|---|
0° |
0.000 |
1.000 |
0.000 |
5° |
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 |
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 identities. The sum of two angles and the difference of two angles.
The sum of two angles expresses 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.
The difference between 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.
- 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