 # Trigonometry

## PrefacePreface

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.
There are many reasons why students might find trigonometry difficult, among them:
• The subject inherently involves a great deal of technical detail, which can be allowed to obscure the main ideas.
• The subject is often taught with the analytical rigor appropriate to a precalculus course -- before students have acquired the necessary facility with functions.
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.)
Nor do we begin with a chapter about angles, including coterminal and reference angles, converting degrees to minutes and seconds, radians, arc length, and angular velocity, before the trig ratios are even mentioned. We have tried to address these issues as follows:
• Chapter 1 reviews only the most basic facts about triangles and circles that students will need to begin their study of trigonometry, and may be omitted or assigned as homework. Other facts about functions and angles are introduced when they are needed. For example, minutes and seconds are discussed in the context of parallax in the section on Law of Sines in Chapter 3. Nautical bearings occur in Section 4.1, Angles and Rotation.
• Chapter 2 introduces the three (not six) basic trig ratios, and considers angles in the first quadrant only. We believe this initial simplicity allows students to focus on the fundamental concepts without simultaneously trying to master a welter of peripheral detail.
• In Chapter 3 we introduce reference angles for the second quadrant in order to study obtuse triangles and the Laws of Sines and Cosines. Reference angles are covered again in more generality in Chapter 4.
• Chapter 4 considers angles as rotations in preparation for the graphs of sine and cosine. Note that the applications of periodic functions in this chapter are functions of degrees only, to fit with our approach: radians come later, after students have some experience with sinusoidal graphs.
• Chapter 5 begins with a section on algebraic manipulations with trig ratios, a skill that is often neglected but can engender endless confusion for students. This chapter treats only simple equations and identities; more equations and identities appear in Chapters 7 and 8. We solve equations both graphically and analytically, and we use graphs as well as algebra to verify trigonometric identities.
• Chapter 6 introduces radians and the circular functions of real numbers. Most of this chapter and Chapter 7 revisit basic skills such as analyzing graphs and solving equations, but working now in radians rather than degrees.
• Chapter 8 studies identities and their use in more detail, including the sum and difference formulas and the double angle identities. Inverse trig functions are included here, and are the three reciprocal trig functions.
• Chapters 9 and 10 cover ancillary topics; typical trigonometry courses may include one or more of these topics: vectors, polar coordinates, and complex numbers.
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.
Katherine Yoshiwara