Trigonometry Workbook: Teacher Guide

Trigonometry Workbook

Mel Friedman

Research & Education Association

61 Ethel Road West

Piscataway, New Jersey 08854

E-mail: [email protected]

REA’s Ready, Set, Go!™

Trigonometry Workbook

Teacher Guide

Copyright © 2009 by Research & Education Association, Inc.

All rights reserved. No part of this book may be reproduced

in any form without the permission of the publisher.

Printed in the United States of America

9780738669861

LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY. Publication of this work is for the purpose of test preparation and related use and subjects as set forth herein. While every effort has been made to achieve a work of high quality, neither Research & Education Association, Inc., nor the authors and other contributors of this work guarantee the accuracy or completeness of or assume any liability in connection with the information and opinions contained herein. REA and the authors and other contributors shall in no event be liable for any personal injury, property or other damages of any nature whatsoever, whether special, indirect, consequential or compensatory, directly or indirectly resulting from the publication, use or reliance upon this work.

Ready, Set, Go!™ and REA® are trademarks of Research & Education Association, Inc.

Table of Contents

Title Page

Copyright Page

Introduction

About Research & Education Association

About the Author

Acknowledgments

Lesson One - Graphs of Points and Angles

Lesson Two - Basics of Trigonometric Ratios—Part 1

Lesson Three - Basics of Trigonometric Ratios—Part 2

Lesson Four - Basics of Trigonometric Ratios—Part 3

Lesson Five - Basics of Trigonometric Ratios—Part 4

QUIZ ONE - LESSONS 1-5

Lesson Six - Special Angles and Trigonometric Ratio Values

Lesson Seven - Inverse Trigonometric Values

Lesson Eight - Solving Right Triangles—Part 1

Lesson Nine - Solving Right Triangles—Part 2

QUIZ TWO - LESSONS 6 – 9

Lesson Ten - Solving Oblique Triangles—Part I

Lesson Eleven - Solving Oblique Triangles—Part 2

Lesson Twelve - Solving Four-sidled Figures

QUIZ THREE - LESSONS 10 – 12

Lesson Thirteen - Solving Trigonometric Equations—Part 1

Lesson Fourteen - Solving Trigonometric Equations—Part 2

Lesson Fifteen - Solving Trigonometric Equations—Part 3

QUIZ FOUR - LESSONS 13-15

CUMULATIVE EXAM

Classroom Activities

Classroom Activity-Lesson 6 - special Angles and Trigonometric Ratio Values—Activity I

Classroom Activity – Lessons 13, 14 & 15 - Who Wants to Be an A Student? Game Sheet

Answer Key

Notes

SCORECARD - Trigonometry

Introduction

Welcome

to the Ready, Set, Go!

Trigonometry Workbook!

About This Book

This book will help high school math students at all learning levels understand basic trigonometry. Students will develop the skills, confidence, and knowledge they need to succeed on high school math exams with emphasis on passing high school graduation exams.

Fifteen easy-to-follow lessons break down the material into the basics. In-depth, step-by-step examples and solutions reinforce student learning, while the Math Flash feature provides useful tips and strategies, including advice on common mistakes to avoid.

Students can take drills and quizzes to test themselves on the subject matter, then review any areas in which they need improvement or additional reinforcement. The book concludes with a final exam, designed to comprehensively test what students have learned.

The Ready, Set, Go! Trigonometry Workbook will help students master the basics of trigonometry—and help them face their next math test—with confidence!

Icons Explained

Icons make navigating through the book easier. The icons, explained below, highlight tips and strategies, where to review a topic, and the drills found at the end of each lesson.

Look for the Math Flash feature for helpful tips and strategies, including advice on how to avoid common mistakes.

When you see the Let’s Review icon, you know just where to look for more help with the topic on which you are currently working.

The Test Yourself! icon, found at the end of every lesson, signals a short drill that reviews the skills you have studied in that lesson.

The information shown with icons throughout will be helpful to you as you prepare to teach. Alert your students to these simple icons, which highlight:

♦ Additional tips and strategies on the topic, including advice to students on how to avoid common math mistakes.

♦ Links to previous lessons for those students who need more background for the current lesson.

♦ Short drills that give you timely feedback on the effectiveness of your lesson to help you gauge your students’ progress.

To the Teacher

As an educator, you understand how important it is for students to master the basic mathematical concepts in a variety of areas. These all-important skills are the building blocks for advanced education. However, all too often, through no fault of their own, some at risk students fall behind in their learning and development.

With the growing focus on the No Child Left Behind Act, REA designed this workbook based on the Curriculum Focal Points of the National Council of Teachers of Mathematics. This workbook was created to help students increase their math proficiency and succeed on crucial High Stakes Exams.

The workbook is organized so that students can learn each lesson at their own pace. When they feel confident that they’ve mastered the math skills for one area, they can move on to the next lesson. Easy-to-follow lessons focus exclusively on one math concept and offer numerous examples and detailed solutions to problems. The Math Flash feature includes problem-solving tips and strategies to help illustrate and reinforce various methods and ideas, while offering advice on how to avoid common mistakes.

This workbook can be utilized in conjunction with lesson plans, stand alone as a single teaching source, or be used in a group-learning environment. The practice quizzes and drills can be given in the classroom as part of the overall curriculum or used for independent study. A cumulative exam at the end of the workbook helps students (and their instructors) gauge their mastery of the subject matter.

We are confident that this workbook will help your students develop the necessary math skills, and build the confidence they need to succeed. As part of our Ready, Set, Go! series, this workbook makes a welcome addition to any remedial math program and can be incorporated into any learning environment.

Use of Calculators: We at REA recommend that, when possible, you permit students to use a calculator to do their work. Of course, you must be guided by your own state standard.

About Research & Education Association

Founded in 1959, Research & Education Association (REA) is dedicated to publishing the finest and most effective educational materials—including software, study guides, and test preps—for students in elementary school, middle school, high school, college, graduate school, and beyond.

Today, REA’s wide-ranging catalog is a leading resource for teachers, students, and professionals.

We invite you to visit us at www.rea.com to find out how REA is making the world smarter.

About the Author

Author Mel Friedman is a former classroom teacher and test-item writer for Educational Testing Service and ACT, Inc.

Acknowledgments

We would like to thank Larry Kling, Vice President, Editorial, for his editorial direction; Pam Weston, Vice President, Publishing, for setting the quality standards for production integrity and managing the publication to completion; Alice Leonard, Senior Editor, for project management and preflight editorial review; Diane Goldschmidt, Senior Editor, for post-production quality assurance; Rachel DiMatteo, Graphic Artist, for page design; Christine Saul, Senior Graphic Artist, for cover design; and Jeff LoBalbo, Senior Graphic Artist, for post-production file mapping.

We also gratefully acknowledge Heather Brashear for copyediting, and Kathy Caratozzolo of Caragraphics for typesetting.

A special thank you to Orsolina Cetta of Piscataway High School, Piscataway, NJ, and to Tracy Doherty and Mary Willi of Union Catholic High School, Scotch Plains, NJ, for their technical review of the lessons and tests in this book.

Finally, thank you to Laura Hoffman, math teacher, formerly of Needham High School, Needham, Massachusetts, for creating all of the activities that enhance the lessons.

Lesson One

Graphs of Points and Angles

Trigonometry is defined as the branch of mathematics that deals primarily with the relationship that exists between the sides and angles of a triangle. However, you will also discover other applications. In this lesson, you will be introduced to plotting points and the measures of angles on the xy-coordinate system. For this lesson, angle measures will not exceed 360°.

Your Goal: When you have completed this lesson, you should be able to determine the measure of an angle, as well as classify angles into categories. You will also be able to convert angles from one system of measurement to another.

The xy-coordinate system is also known as the rectangular coordinate system or the Cartesian coordinate system. We start with a flat surface, which is known as a plane. Then, we draw two number lines that are perpendicular to each other and that intersect at the number zero. The horizontal line is labeled the x-axis, and the vertical line is labeled the y-axis.

On the x-axis, all positive numbers lie to the right of the intersection point and all negative numbers lie to the left. On the y-axis, all positive numbers lie above the intersection point and all negative numbers lie below. For consistency, the numbers are evenly spaced. Figure 1.1 shows the xy-coordinate system with this information.

Figure 1.1

Even though the only numbers shown for each axis are – 4 through +4, be aware that each axis represents a real number line. This means that these lines include numbers that are less than – 4, greater than +4, as well as fractions, decimals, and square roots.

Notice that the x-axis and y-axis divide the plane into four regions, each of which is called a quadrant. These quadrants are numbered using Roman numerals I, II, III, and IV (instead of 1, 2, 3, and 4). The numbering begins in the upper right quadrant, continues in a counterclockwise direction, and ends in the lower right quadrant. Figure 1.2 illustrates this numbering system.

Figure 1.2

Each point on this plane is assigned a corresponding x value and a corresponding y value, which will be written in the form (x, y). The expression (x, y) is called the ordered pair for the corresponding point. Each of x and y is called a coordinate of the ordered pair. The point where the two axes intersect is labeled (0, 0) and is called the origin.

Suppose point A is located 2 units to the right and 3 units above (0, 0). Then (2, 3) is the ordered pair for point A. Any point in the first quadrant will always be assigned an ordered pair in which each of x and y is positive.

Suppose point B is located 1 unit to the left and 4 units above (0, 0). Then −1, 4) is the ordered pair for point B. Any point in the second quadrant will always be assigned an ordered pair in which x is negative and y is positive.

Suppose point C is located 3 units to the left and 5 units down from (0, 0). Then (3, – 5) is the ordered pair for point C. Any point in the third quadrant will always be assigned an ordered pair in which each of x and y is negative.

Suppose point D is located 4 units to the right and 2 units down from (0, 0). Then (4, – 2) is the ordered pair for point D. Any point in the fourth quadrant will always be assigned an ordered pair in which x is positive and y is negative. Figure 1.3 shows points A, B, C, and D.

Figure 1.3

Now consider point E, which is located 5 units directly to the right of (0, 0). Then (5, 0) is the ordered pair for point E. Notice that E lies on the positive portion of the x-axis. If point F is located 4 units above (0, 0), then its corresponding ordered pair is (0, 4). Notice that F lies on the positive portion of the y-axis.

Similarly, if point G lies 2 units to the left of (0, 0), then its