Trigonometry: A Very Short Introduction

Very Short Introductions #626

by Glen van Brummelen

Born of the desire to understand the workings of motions of the heavenly bodies, trigonometry gave the ancient Greeks the ability to predict their futures. Most of what we see of the subject in school comes from these heavenly origins; 15th century astronomer Regiomontanus called it "the foot of the ladder to the stars."

In this Very Short Introduction Glen Van Brummelen shows how trigonometry connects mathematics to science, and has today become an indispensable tool in predicting cyclic patterns like animal populations and ocean tides. Its historical journey through major cultures such as medieval India and the Islamic World has taken it through disciplines such as geography and even religious practice. Trigonometry has also been a major player in the most startling mathematical developments of the modern world. Its interactions with the concept of infinity led to Taylor and Fourier series, some of the most practical tools of modern science. The birth of complex numbers led to a shocking union of exponential and trigonometric functions, creating the most beautiful formulas and powerful modelling tools in science. Finally, as Van Brummelen shows, trigonometry allows us to explore the strange new worlds of non-Euclidean geometries, opening up bizarre possibilities for the shape of space itself. And
indeed, one of those new geometries - spherical - takes us full circle back to ancient Greek astronomers and European navigators, who first used it to chart their ways across the heavens and the earth.


ABOUT THE
The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Genres: Mathematics, Nonfiction, Science

192 pages, Paperback

Published March 2, 2020

Reviews

[Bojan Tunguz · Rating 5 out of 5]

When I was first exposed to trigonometry back in high school, I found it very counterintuitive and mysterious. After many years of working in science, it had become almost instinctive to use and think about it. In fact, it had almost dulled me to appreciating how radically different trigonometry was from its mathematical forerunners, and how transformative it was for all of science and engineering.

This very short introduction manages to get me to think about trigonometry anew, with a new pair of eyes. The book takes you through all the steps in deriving some of the fundamental trigonometric results and identities, and makes some of those derivations seem rather intuitive. It was fun to go through the algebra of deriving basic trigonometric table values, using nothing else but basic trigonometric identities and some algebra. One of the biggest surprises in this book was the demonstration that calculators (and presumably all digital calculating machines) do not use Taylor series when calculating trigonometric values - the series converges far too slowly to be useful for numerical calculations. Instead, they seem to be suing some rather ingenious tricks that speed up calculations considerably.

One of the things that I appreciated the most about this book was the sheer palpable enthusiasm that its author, Glen Van Brummelen, brings to the subject matter. Glen is fascinated by this subject, and goes to great length to convey his enthusiasm on every page of the book. He must be a great teacher to have, and part of me really envies his students.

Excellent little book. Highly recommended.

[John Isles · Rating 5 out of 5]

The Oxford series of Very Short Introductions includes several on mathematical topics, of which this is one of the best. Although it reviews what we may once have learned in school, which might have proved deadly dull reading, it interlards it with enough humor and historical anecdotes to maintain interest throughout. Included are short introductions to spherical and non-Euclidean trigonometry.

[Maurizio Codogno · Rating 4 out of 5]

La trigonometria appare in modo naturale

Ho trovato il primo capitolo del libro, seppure partisse da un assunto interessante su come nacquero le idee che portarono allo sviluppo della trigonometria, piuttosto noioso, e ho lasciato per un po' il libro a sedimentare. Devo però dire che il resto del libro è molto più interessante. Lo so, molti di voi diranno "che cosa può esserci di interessante sulla trigonometria?" Beh, van Brummelen ha scelto di mostrare come tante parti della matematica hanno a che fare con la trigonometria, e devo dire che alcune corrispondenze sono state inaspettate anche per me. Poi naturalmente c'è il grande vantaggio dei libretti della collana Very Short Introduction: che sono appunto brevi. Insomma, può valer la pena di leggerlo, al limite saltando il primo capitolo!

[Kayla · Rating 4 out of 5]

This book was very well written, and even included some personal bits such as his cat graphed in R2.

He included very important aspects in trigonometry in a very concise way, while including examples, and explanations to other topics like linear algebra for those with at least a first year level understanding of mathematics.

[Manuel Del Río Rodríguez · Rating 4 out of 5]

Today’s book is part of a very nice series by Oxford University Press which publishes very short introductions on different topics for a lay audience. Van Brummelen’s volume is devoted to Trigonometry, a branch of math that most of us had to deal with to at least some degree in high school and which we likely considered, back then, a useless mess of weirdly named object connected to even weirder lists of formulas that were employing in solving problems with triangles.

This volume is an attempt at giving the subject a second chance, with a heavy focus on historical aspects (how and why and where trigonometry was invented), all the way from ancient Greece and its attempts at measuring the heavens, though India and the use of taut strings to get the first trigonometric ratios, all through the composition o lengthy tables of sines and cosines in the Modern period to help with navigation, and through, calculus, complex numbers and non-Euclidean geometries. Along the way, we see how a discipline that was clearly geometric in its origins acquires an algebraic flavor and ends up being used in infinite processes and formulae. Besides the history we also get some of the math, some of the calculations, and many identities mostly in the service of solving relatively easy problems and to illuminate practical and mathematical uses of trig. The last, seventh chapter, delves a bit in a weird and little looked into branch of trigonometry which was very popular in the past but has now became a rarity: spherical trigonometry, the science used for understanding and predicting movements in the ‘celestial sphere’.

The book is, as its title promises, quite short (the reading part is a little more than a hundred, rather small, pages, and it’s chock-full of charts and pictures). I think it is quite accessible even for not overtly mathy people, although you do need to go back to the charts and think a bit from time to time. It is a noble effort, but I still feel only people already invested in curiosity towards the topic will end up picking this up and giving it a read.

[Wing · Rating 4 out of 5]

This small book is more than a historical account of the ingenious development of trigonometry, it also guides the reader to appreciate the transformation of the synthetic aspect of the discipline to an analytic one. In other words, how geometry becomes algebraic is very well described. The author claims that only an elementary background in mathematics is required to read this book. To give an example, the Taylor series of sine is explained using geometry without calculus. To an extent he is honest. As a lay reader (by trade I am a physician) I do find the last chapter on spherical and non-Euclidean trigonometry difficult - I can't follow one of the proofs. Otherwise the book is pretty easy to follow. Four stars.

[RAD · Rating 5 out of 5]

(Review forthcoming).

[Bolivar Morales · Rating 5 out of 5]

This book will not teach you the fundamentals of trig . It will teach you the history behind it. Is a quick read 📚 and good one