Introduction
Introduction
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The subject inherently involves a great deal of technical detail, which can be allowed to obscure the main ideas.
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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 “[latex]\sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}[/latex]” 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.)
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Chapters 9 and 10 cover ancillary topics; typical trigonometry courses may include one or more of these topics: vectors, polar coordinates, and complex numbers.
Katherine Yoshiwara
Atascadero, CA 2018
This Introduction section was adapted from: “Front Matter.” Trigonometry, Yoshiwara Books, https://yoshiwarabooks.org/trig/frontmatter-3.html. Accessed 16 Dec. 2024.
About This Book
This textbook was created through Connecting the Pipeline: Libraries, OER, and Dual Enrollment from Secondary to Postsecondary, a $1.3 million project funded by LOUIS: The Louisiana Library Network and the Institute of Library and Museum Services. This project supports the extension of access to high-quality post-secondary opportunities to high school students across Louisiana and beyond by creating materials that can be adopted for dual enrollment environments. Dual enrollment is the opportunity for a student to be enrolled in high school and college at the same time.
The cohort-developed OER course materials are released under a license that permits their free use, reuse, modification, and sharing with others. This includes a corresponding course available in Moodle and Canvas that can be imported to other platforms. For access/questions, contact Affordable Learning Louisiana.
If you are adopting this textbook, we would be glad to know of your use via this brief survey.
License
Trigonometry, copyright © 2024 by LOUIS: The Louisiana Library Network, is licensed under a GNU Free Documentation except where otherwise noted. This is an adaptation of Trigonometry by Katherine Yoshiwara, licensed under a GNU Free Documentation License. That adapted text provides permission to copy, distribute, and/or modify the document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
Cover Image
The cover image is “For 2021: Cycles” by Kevin Dooley and licensed under a Creative Common Attribution 2.0 Generic license.
