This webpage is an introduction to high-school and college level trigonometry. The content is intended to bridge the gap between basic understanding of trigonometry and some of the more notorious concepts that trigonometry plays a pivital role in.
Trigonometry is the study of triangles. The subject focuses on the properties of triangles, especially of the right triangle. From the name, you might guess that trigonometry is purely about triangles and to an extent that is true. A triangle is a simple, versatile shape that appears everywhere in mathematics and physics.
However, the further one goes into the study of trigonometry, it becomes apparent that the triangle is really a medium through which one approaches other ideas, such as circles, ellipsis and hyperbolas; triangles connect concepts together and decompose complicated ideas into tangible lengths and angles.
"You think it's about triangles, but really it's about circles."Unknown
For example, modern mathematics heavily utilizes the concept of a coordinate system, where points in space are defined relative to an origin. The two ubiquitous coordinate systems, the Cartesian Coordinate System and the Polar Coordinate System, utilize the trigonometry of the right-triangle to define a point in space. Since these coordinate systems appear everywhere in applications, trigonometry plays a pivotal foundational and translational role.
Before diving into the fundementals of trigonometry, here are some high-level motivating questions and applications of trigonometry.
- Coordinate Systems. Modern mathematics heavily utilizes the concept of a coordinate system. A point in the cartesian coordinate system forms the shape of a right-triangle, defined by the length of its adjacent and opposite sides. A point in the polar coordinate system forms the shape of a right-triangle, defined by the angle relative to the positive direction and the length of its hypotenuse. There are advantages to thinking both in terms of a Cartesian Coordinate System and a Radian Coordinate System.
- GPS (Global Positioning Systems) Triangulation forms the basis of Global Positioning Systems (GPS) as well as is a neat application of finding where you are on a maping using angles and landmark locations.
- Vectors. A right triangle forms the shape and components of a 2D vector which is a conceptual model used throughout physics.
- Waveforms. The sine function describes a sinusoidal waveform used throughout physics and math. Waves model a large amount of human’s understanding about light, color, sound and much more.
- Astronomy Planetary orbits can be defined in terms of triangles (see the construction of an ellipse) Estimate the curvature of the earth using two obelisks (how tall is an object given the length of its shadow)
- 2D and 3D modeling. Computer science often uses trigonometry to transform, rotate, translate and otherwise manipulate points in a computer system. A particularly relevant application is using 3D triangular meshes to compose complex shapes in three dimensions. (Shading, rendering, video-app filters)
This section of the book introduces the foundations of trigonometry such as key terms and concepts.
An angle is the amount of rotation between two rays. The point where the two rays meet is called the vertex. Angles are a fundemental building block of trigonometry as every triangle’s shape is formed by three angles.
An angle is often thought of in terms of the circle. The angle is measured relative to the fixed ray formed the point at the center of the circle and the point on the rightmost point of the circle. By convention, the positive direction is the counter-clockwise direction.
There are two conventions to measuring the rotation of an angle: degrees and radians. Both are reasonable ways to think about rotation and calculators allow for functions to evaluate angles in both units of measure. It is useful to be comfortable using both radians and degrees.
A triangle is a three sided polygon. The shape can be defined by three points as shown in the 2D interactive below. Try clicking and dragging some of the blue controls points points around.
There is more to discuss about triangles, but in the interest of laying ground-work for the study of trigonometry instead let’s first look at a list of motivating problems that hopefully explain why this subject is interesting and useful.
A right triangle is a special case of a triangle where one of the angles is a perpendicular angle.
The diagonal side is called the hypotenuse, the horizontal side is called the adjacent side, and the vertical side is called the opposite side.
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 triangle defined by the three angles: 45 degrees, 45 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.
The triangle defined by the three angles: 30 degrees, 60 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.
The trigonometric functions relate the angle of the right triangle to different ratios of its sides. These are very useful and found throughout mathematics and physics. We will start with the definitions of cosine, sine, and tangent.
A circle of radius one, placed at the center of the cartesian coordinate system is a useful way to visualize the properties of the right triangle and the trigonometric functions. Placing the triangle in the coordinate plane relates the adjacent, opposite, and tangent sides of the triangle to the x and y dimensions of a two dimensional plane.
By placing the unit circle at the origin of the cartesian coordinate system, observe that the dimensions of the triangle correspond with any point along the circumference of the circle.
This brings us back to the trigonometric functions and gives new meaning to their input and output.
Two logical concepts to study next are the the trigonometric identities and complex numbers. However, while these concepts are useful to study and do show up in later courses, a disclaimer is necessary before continuing. Trigonometry appears naturally in Physics, Linear Algebra, Geometry, Computer Science and Calculus. The trig. identities and the formulations of complex numbers are “nice-to-have” knowledge not necassarily “need-to-know”. They represent a rough approximation of the most useful information a student can have later on.
The student is at risk of finding these arbitrary, confusing and disengaging. By the time the applications of trigonometry resurface in calculus, college, or possibly never - they’ve been exposed to substantially complex ideas and pushed forward through the school system without firmly understanding the motivating questions behind much of the math. As a result, this part of the book represents a momentary stoppping point. There are mountains to be climbed, valleys to bush-whack through and a personal journey where further study may or may not be applicable to the student.
So, again, here are some motivating questions for why you might continue.
- How to define the trigonometric functions (computer science & calculus)
- How to derive the trigonometric identities (geometry, algebra & trig, and complex plane)
- 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
- How to implement a shading algorithm in 3D using a triangle mesh
- How to model our solar system
Useful concepts we acquire and learn about along the way.
- Trigonmetric Identities
- Taylor Series
- Exponential Function
- Complex Numbers
- Euler’s Formula
- Fourier Series
- Fourier Transform
- Polynomial Interpolation
Finally, the show must go on and so the rest of this book chooses two deep cuts of trigonometry as it is found in mathematics to study.
- Trigonometric Identities
- Complex numbers and Euler’s Formula.
- Waveforms and Fourier Transforms.
This section discusses the trigonometric identities and how to derive them using two strategies.