This guide covers the story of trigonometry. It is a swift overview, but it is complete in the context of the content discussed in beginning and advanced high-school courses. The purpose of these notes is to supplement and put into perspective the material of any course on the subject you may have taken or are currently taking. (These notes will be tough going for those encountering trigonometry for the very first time!)

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James Tanton (PhD, Princeton 1994, mathematics) is an education consultant and an ambassador for the Mathematical Association of America in Washington, D.C. He has taught mathematics both at university and high-school institutions. In 2004 James founded and directed the St. Mark's Institute of Mathematics, conducting mathematics outreach for students of all ages and designing and teaching graduate courses in mathematics for educators. James is also the author of the MAA books *Solve This!* and *Mathematics Galore!* and is writing and doing video work on problem-solving for the MAA's Curriculum Inspirations project. He serves as a writer or an advisor on a number of curriculum projects and regularly travels across the nation and overseas to work directly with educators. James is absolutely committed to sharing joyful and beautiful mathematical thinking and doing with all.

This is as much a guide to solving problems as it is to the study of trigonometry. Tanton opens each chapter with an explicit statement of the Common Core topics that are being addressed in that chapter. They are a reminder that in mathematics, a common core of learned topics has been a part of our agenda for a long time. Using text and diagrams, the topics of the chapter are explained.

This is followed by a featured problem for the chapter, where the author recapitulates the process that he followed solving it. These problems are not bunnies, they require some thought and insight when solving them. The last part of the chapter is a set of problems that have appeared in mathematical contests and solutions to all of them appear in the second section. They are multiple choice problems.

The real unique feature of this book is the set of ten problem solving strategies that Tanton employs when he solved the featured problems. They are:

Engage in successful flailing-this means simply writing down whatever it is you know about the problem.

Do something-just the act of writing down something related but not necessarily relevant can trigger a valuable thought.

Engage in wishful thinking-in this strategy if you need something, write it down. For example, if you need a +4 on one side of an equation put it on both sides and see what you have.

Draw a picture-always a sound strategy in mathematics, when possible.

Solve a smaller version of the same problem-or equivalently, solve a part of it if the problem can be segmented.

Eliminate incorrect choices-a standard tactic on a multiple choice question.

Perseverance is key- in other words, keep trying to examine the problem, look at it different ways.

Second-guess the author-look for clues in the statement, for example the mention that 131 is prime. Irrelevant or a clue?

Avoid hard work-specifically working out the precise value of large numbers. Is there a pattern at work?

Go to extremes-for example in a puzzle that deals with the ages of people, what if they were all the same age? What if the escalator was not moving?

Tanton references these strategies in his descriptions of solving the featured problems and manages to inject a bit of humor into the work.

Trigonometry is a topic that is core to the understanding of basic mathematics and in this book Tanton emphasizes that even experienced mathematicians will look at problems and have an initial reaction of "What?" If you teach trig, there are many interesting and effective pedagogical techniques in this book as well as problems that you can use. --Charles Ashbacher

This clearly written, engaging book combines a summary of high school trigonometry and a guide to problem solving. The book offers teachers ideas about how to present some topics in an interesting way along with a collection of challenging problems involving trigonometry. The first 18 chapters review all of high school trigonometry. Some chapters include an interesting historical vignette, such as the origin of the names of trigonometric functions, or an innovative approach to describing a concept, such as the sun moving through the sky for circle trigonometry.

Most chapters also contain math competition problems. The author describes how he solved a featured problem with one of ten problem-solving strategies. Readers can try additional math competition problems that are given at the end of the chapters. At the end of the book are two brief chapters reviewing polar coordinates and 55 pages of solutions to 100 math competition problems included here.

If you are looking for an excellent collection of challenging problems that use trigonometry or for a well-written review of trigonometry, this book would be a good choice. --Mathematics Teacher

The MAA has been publishing problem books for many years. It has also been the force behind the American Mathematics Competitions. So it's natural that they would publish problem books based on AMC problems. This is the first of a series of books collating AMC problems on specific themes.

There is a lot to like in the notion of books devoted to competitive mathematics, and the idea of collecting AMC problems thematically shows great promise of developing into a good resource for students preparing for contest mathematics and for those who coach them. While the series will draw largely on the AMC's contests, the material in this book, and potentially others in the series, will certainly find use at all competitive levels.

Those of us who are involved with the world of mathematics competitions--at any level, high school or college--know that many good contest problems can be solved with a careful use of results from trigonometry. This book collects the most meaningful results from trigonometry and, in addition to reviewing them for the reader, includes a robust collection of problems from previous contests, where their value to competitors can be easily demonstrated.

While this is not a textbook, a nice added feature is the connections to Common Core state standards. It's good to see that contest mathematics need not be separate from day-to-day classroom mathematics, and that there's more to contest uses of trigonometry than just clever tricks.

If this book is any indicator, these books will be an excellent addition to the MAA Problem Books series. --Mark Bollman, MAA Reviews

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Editorial:
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