Trigonometry

Gelfand Correspondence Program in Mathematics

by Israel M. Gelfand

In a sense, trigonometry sits at the center of high school mathematics. It originates in the study of geometry when we investigate the ratios of sides in similar right triangles, or when we look at the relationship between a chord of a circle and its arc. It leads to a much deeper study of periodic functions, and of the so-called transcendental functions, which cannot be described using finite algebraic processes. It also has many applications to physics, astronomy, and other branches of science. It is a very old subject. Many of the geometric results that we now state in trigonometric terms were given a purely geometric exposition by Euclid. Ptolemy, an early astronomer, began to go beyond Euclid, using the geometry of the time to construct what we now call tables of values of trigonometric functions. Trigonometry is an important introduction to calculus, where one stud­ ies what mathematicians call analytic properties of functions. One of the goals of this book is to prepare you for a course in calculus by directing your attention away from particular values of a function to a study of the function as an object in itself. This way of thinking is useful not just in calculus, but in many mathematical situations. So trigonometry is a part of pre-calculus, and is related to other pre-calculus topics, such as exponential and logarithmic functions, and complex numbers.

Genres: Mathematics, Science, Geometry, Nonfiction, Education, Textbooks

239 pages, Paperback

First published June 1, 2001

About the author

Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (Yiddish: ישראל געלפֿאַנד‎, Russian: Израиль Моисеевич Гельфанд ; 2 September [O.S. 20 August] 1913 – 5 October 2009) was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the Wolf Prize, he was a Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University.

Reviews

[Jack Kinsella]

Terse, beautiful, and rigorous treatment of trigonometry written by a prominent soviet mathematician (three times winner of the Order of Lenin).

Despite hating trig in school, I loved working through this book. There is more math in its ~200 small pages than in 700 pages of any modern, glossy book written by some no-name pedagogue.

The book includes no solutions to its problems. I initially thought this would be an issue, but it turned out to be a blessing in disguise: It prompted me to figure out how to verify solutions on my own (and with help from https://www.wolframalpha.com/ )

[David Shuster · Rating 5 out of 5]

AMAZING BOOK. Possibly my favorite math textbook I have ever encountered.

Pros:
1) No glossy, irrelevant pictures, no bloated CD advertisements, etc. Not hopelessly daunting due to sheer size.
2) Gelfand & Saul give a logical flow to the text, clear and mathematically motivated explanations, as well as artful foreshadowing of future topics. Very readable book
3) Examples well-chosen and worked out
4) Good quantity of rote and thought-provoking exercises, and a good amount such that it feels feasible to work through all of them if you want, but not too few to get the ideas down either.
5) The authors' love for trig's elegance comes through clearly

Cons:
1) No solutions to exercises, which is a bit unideal for someone new/tentative with trig. (Also a pro, because you have to convince yourself your answers are correct, which is great.)
2) Could be nice to have a few appendices going a little more into applications such as sound waves or navigation, but its focus on straight-up math-exposition is a perfectly nice approach too.

[Mark Moon · Rating 5 out of 5]

A lovely book! Even I thought there were enough pictures. A bright, motivated teenager could work through just about everything in here.

[Alejandro · Rating 4 out of 5]

I was struggling to get a true understanding of the trigonometric functions in Calculus so I just devoured this book and it's pretty awesome. I got some really great insights on what trig functions really are and connected the geometric concepts with the algebraic counterparts.

The book isn't overly difficult, and I found it easy to study and complete the exercises concurrently with my advanced Calculus course. It significantly eased my workload in Calculus and clarified a lot of concepts.

[Arabella Sun · Rating 5 out of 5]

Good book. I used it only as a concept book, you'll obviously have to get a practice book. This book had like 4-5 questions per 'section'.

[Mykyta Kuzmenko · Rating 4 out of 5]

Неплохец. Без воды, со сложными задачами. Хорошо показана связь с колебательными процессами в физике.

[masnsen · Rating 5 out of 5]

Excellent

[Maxim Perepelitsyn · Rating 5 out of 5]

While I had failed to understand the subject at school, this book made everything clear. On par, quality wise, with other Gelfand's books.