Trigonometry

We have tried to make this edition of Trigonometry useful to students in a variety of programs. For example, students who have encountered elements of triangle trig in previous courses may be able to skip all or part of Chapters 1 through 3. Students preparing for technical courses may not need much of the material after Chapter 6 or 7. Chapters 9 and 10 cover vectors and polar coordinates, optional topics that occur in some trigonometry courses but are often reserved for precalculus.

In his beautiful book, Trigonometric Delights, Eli Maor enjoins us "Let’s not forget that trigonometry is, first and foremost, a practical discipline, born out of and deeply rooted in applications." After the New Math "[f]ormal definitions and legalistic verbosity—all in the name of mathematical rigor—replaced a real understanding of the subject." And formalism is "... certainly not the best way to motivate the beginning student."

The typical trigonometry student has just completed a second course in algebra. He or she has a nodding acquaintance with functions and is still very wary of irrational numbers. A statement such as "\(\sin \dfrac{5\pi}{3} = \dfrac{-\sqrt{3}}{2}\)" may well be greeted with panic and bewilderment. So we do not begin with a preliminary chapter covering all the mathematical topics needed for the rest of the course, including elements of analytic geometry and properties of functions such as domain and range, symmetry, transformations, composition, and inverse functions. (This material usually comprises most of a precalculus course, which is usually taught after trigonometry, where it is introduced using more familiar, hence easier, functions as examples.)

In addition to the Homework Problems, each Example in the book is followed by a similar Exercise for students to test their understanding. Each Section concludes with a Summary , a set of Study Questions, and a list of Skills to be addressed in the Homework. A Summary and a set of Review Problems follows each chapter. Chapters 1 through 8 include Activities for students to work through some of the main ideas. We have described the use of a graphing calculator, but other graphing utilities can easily be substituted.

Throughout we have been guided by the Rule of Four and use tables and graphical representation to illustrate concepts. We have taken care to include numerical examples and diagrams, both in Examples and in Homework Problems, to offer students some intuitive understanding for the more abstract ideas of trigonometry. Above all, we have tried to focus on the fundamental ideas of trigonometry by introducing them in their most basic form and returning later to look at them in greater detail.