Trigonometry: Origins

Students in math classes from middle school to university often wonder, or actually ask, a valid and important question: “Why should we care?” For various reasons, some better than others, teachers rarely give this question the honest, thoughtful answers it deserves. I’m going to try to answer it with respect to some specific topics people have asked me about. This is going to be a learning experience on my part. The presentation will quick and dirty. I will ignore all the good pedagogical practice that I know and a great deal that I don’t. I appreciate your interest and your patience. This project has been burning a hole in my skull for weeks and I’m sick of putting it off trying to make it better.

What’s the point of studying trigonometry?

Trigonometry uses the deep relationships between circles and triangles to make highly precise measurements and designs and to analyze cyclical patterns in natural and manmade systems. In addition, like the other simple functions studied in high school, trigonometric functions are used to explore and approximate more complicated functions that arise throughout pure and applied mathematics. Whenever someone discusses distances, angles, or cycles in a mathematical context, they are using the language of trigonometry.

From a historical perspective, trigonometry was developed by astronomers and mathematicians from Eratosthenes to Newton to describe and predict the motions of the heavenly bodies and so to understand our place in the cosmos. The fact that the orbits are approximately circular is the reason that tides and shadows change their heights and lengths in a roughly sinusoidal way and that the Earth’s shadow on the moon changes so smoothly throughout the lunar phases.

Trigonometry also provides us with the techniques of surveying and cartography, which allowed European navigators of the eighteenth and nineteenth centuries to map the world and establish global transportation networks. Everywhere that rotation and distance occur in nature and technology — and our technology is based on the wheel — we are called to use trigonometry. It gives us a language for the most natural questions to ask about physical form.

In the next article, I’m going to talk about one of the deepest and most overlooked ideas relating to high-school math, an idea that uses the utter simplicity of circles and squares to analyze complicated relationships and store overwhelming amounts of data. The key tools here are trigonometric functions and another old friend, polynomials.