# Trigonometry

Ted Sundstrom, Grand Valley State University

Steven Schlicker, Grand Valley State University

Copyright Year: 2016

Publisher: Grand Valley State University

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

While other texts may include more topics, my experience is that there's never enough time in the semester to get to them all. This text covers circular and right-triangle trigonometry, analytic trigonometry (identities and trigonometric... read more

While other texts may include more topics, my experience is that there's never enough time in the semester to get to them all. This text covers circular and right-triangle trigonometry, analytic trigonometry (identities and trigonometric equations), and applications, and spends just enough time on vectors, complex numbers, and polar coordinates to neatly round out the semester.

I have not encountered any errors during my review of the book.

Trigonometry is vital to mathematics and engineering, as well as to many of the natural sciences. I don't see that changing anytime soon. The applications in this book (of which there are no few) are relevant to these disciplines.

The tone of this book is very comfortable, almost conversational, yet it is not lacking in the necessary mathematical rigor. It presents ideas and techniques in an appropriate way without sounding aloof. New ideas are introduced by relating them to familiar ideas, and students are guided into using their own reasoning and thereby developing their own understanding of new concepts. Each section typically starts with an example or a problem for motivating the following discussion.

One of the major features of this book are its Progress Checks, which are essentially example problems that the students work out for themselves. These Progress Checks are usually constructed so as to guide the student's learning of a new topic. Occasionally, however, a Progress Check amounts to one of two things: either the repetition of a technique that was just demonstrated in the text, or a problem in which the student is expected to develop a new idea for themselves with too little guidance from the book. The teacher should be aware of where such issues occur and design class instruction of affected topics accommodatively.

I think this book is as modular as a trigonometry textbook can be. It's certainly well organized. I would caution against being too free with reorganizing the text. For one, topics in the text typically build off those that came before. This is by necessity. For another, research indicates that students develop a more cohesive understanding of trigonometry if circular trigonometry is studied before right-triangle trigonometry, and the connection between the two made explicit, as this book does. So, regardless of how modular this book may or may not be, I believe it is in the teacher's best interest to present topics in the order laid out by the book, as much as possible.

The book begins by looking fairly in depth at angle measure, a topic which is often glossed over in trigonometry texts. The focus for the first several sections is on the sine and cosine functions, so that students can develop a thorough understanding of how those functions behave before turning attention to the other four trig functions, all of which can be viewed as derivative of the sine and cosine. Circular trigonometry is covered before right-triangle trig, which, as mentioned before, encourages a more holistic understanding of trig than the reverse approach. In many textbooks, vectors are withheld until later in the book, where they are lumped in with other "applications" of trigonometry. Here, they are presented immediately following the traditional topics of right-triangle trig, which I like. The remaining topics--trig identities, trig equations, and complex numbers--are covered in the standard order. A comprehensive overview of the algebra of complex numbers is presented prior to the trigonometry of complex numbers.

The book was easily legible, and all charts and diagrams were clear and easy to read. The book has many links to external applets, all of which (that I checked) worked.

There were no grammatical errors that I noticed.

This book is appropriate for any student who might use it, regardless of background.

In many respects, this book conforms to current understanding of trigonometry education. A significant amount of time is spent early developing students' understanding of angles and angle measure. Students learn not only the process of measuring angles, but also the nature of angles themselves. Both radian and degree measure are presented in terms of subtended arcs, so that students can move between the two measures fluidly. Later, transformations of trig functions (typically a difficult subject for students) is studied by focusing on the arguments of functions and by invoking function composition. This example is typical of the book's healthy tendency to incorporate old concepts into new, but it also avoids the common trap of presenting transformations in purely geometric terms, whereby students come to understand a graphs as a "shape" and not the trace of a relationship. This book's approach instead encourages a "process view" of functions by attending to the relationship between input and output, and how that relationship changes when two functions are composed. This book is not perfect. I would like a greater emphasis to be placed on the covariation between input and output of trig functions, particularly when it comes to the graphs of the sine and cosine and how the shapes of those graphs are determined by the ways in which input and output change together. But overall, this book does a better job of presenting trigonometry to students than most books I've seen, whether open or retail.

This text was created for a three-credit trigonometry course (MATH123-Trigonometry) at Grand Valley State University. Other than conic sections, the text covers everything that is typically included in a first trigonometry course. However, the... read more

This text was created for a three-credit trigonometry course (MATH123-Trigonometry) at Grand Valley State University. Other than conic sections, the text covers everything that is typically included in a first trigonometry course. However, the comprehensiveness of this text exceeds other texts in many important ways. Trigonometric concepts are developed in a very thorough, patient and coherent manner that most certainly speaks to students, while developing their mathematical understanding and analytical thinking. It takes the time to engage the reader in the thought process using links to well developed and integrated diagrams, Geogrebra applets, worksheets, and YouTube screencast videos (developed by Grand Valley State University). The beginning review activities, focus questions, progress checks, and sections summaries provide direction and support active and deep learning. Answers and hints to selected exercise and geometric facts about circles and triangles are included in the Appendices. This text takes a very interesting reverse chronological approach to trigonometry. Trigonometric functions are introduced as circular functions and later as trigonometric functions. I am now convinced that this provides a more holistic perspective of trigonometry and better clarifies many concepts such as the dimensionless nature of radians.

I did not note any errors or bias in the content of the book.

The textbook begins with the most relevant applications of trigonometry, but is not written in a manner that will require necessary updates. If needed, it could be easily updated. The author's invite feedback, especially from students using the text.

The text makes even the most complex of the trigonometric concepts clear and accessible to students. It develops concepts and the definitions of terminology so that students can both understand and remember them. It includes progress checks for students to check their understanding. Section summaries list important definitions and results proven in the section. The disciplined and motivated student could learn trigonometry from this text on their own. Some students may need further clarification of the reading.

The text is internally consistent in terms of terminology and framework. Each of the beginning activities reviews prior mathematical work necessary for the new section. The answers to these beginning activities are developed in the material later in the section. Each section has progress checks with answers to these checks in Appendix A. Each section is summarized listing important definitions and results proven in the section.

This text is best read from the beginning to the end to fully understand how ideas are developed. However, each chapter can stand on its own. Each reading section is fairly small and the text within each section is broken up into small chunks to keep the reader engaged, active, and interested.

This text is very organized, well structured, and flows very well. It is particularly strong in its clear and logical development of topics. There is the right amount of rhythm and variety to keep the reader engaged. It uses language that is more conversational and approachable than most mathematical texts.

There were a few links to figures that did not seems to take me exactly to the figure, but at least close to them. Otherwise, I did not note any interface issues, navigation problems, distortion of images, or any other distracting or confusing display features.

I noted only a couple of grammatical errors in Progress Check 1.2 and on page 168 where a word or two seemed to be missing.

This text is not culturally insensitive in any way. Rather the variety of resources and beginning activities engage students with different learning styles and mathematical backgrounds.

I think this text is excellent. It uses a guided approach and has a conversational, interesting, and engaging tone that promises to effectively reach all learners from the serious math student to those with math anxiety. I will definitely make this text available to my students to get their feedback and hope to use it as the main text in the future. It could greatly facilitate teaching and learning both in online and traditional courses.

## Table of Contents

- Note to Students
- Preface
- The Trigonometric Functions
- Graphs of the Trigonometric Functions
- Triangles and Vectors
- Trigonometric Identities and Equations
- Complex Numbers and Polar Coordinates
- Appendix A: Answers for the Progress Checks
- Appendix B: Answers and Hints for Selected Exercises
- Appendix C: Some Geometric Facts about Triangles and Parallelograms
- Index

## About the Book

This trigonometry textbook is different than other trigonometry books in that it is free to download, and the reader is expected to do more than read the book and is expected to study the material in the book by working out examples rather than just reading about them. So this book is not just about mathematical content but is also about the process of learning and doing mathematics. That is, this book is designed not to be just casually read but rather to be engaged.

Since this can be a difficult task, there are several features of the book designed to assist students in this endeavor. In particular, most sections of the book start with a beginning activity that review prior mathematical work that is necessary for the new section or introduce new concepts and definitions that will be used later in that section. Each section also contains several progress checks that are short exercises or activities designed to help readers determine if they are understanding the material. In addition, the text contains links to several interactive Geogebra applets or worksheets. These applets are usually part of a beginning activity or a progress check and are intended to be used as part of the textbook.

## About the Contributors

### Authors

**Ted Sundstrom **is a mathematics professor at Grand Valley State University in Allendale, MI.

**Steven Schlicker** is a mathematics professor at Grand Valley State University in Allendale, MI.