Ebook Algebra & trigonometry: Enhanced with graphing utilities (Sixth edition) - Part 1

Ebook Algebra & trigonometry: Enhanced with graphing utilities (Sixth edition) - Part 1 presents the following content: Chapter 1 graphs, equations, and inequalities; chapter 2 graphs; chapter 3 functions and their graphs; chapter 4 linear and quadratic functions; chapter 5 polynomial and rational functions; chapter 6 exponential and logarithmic functions; chapter 7 trigonometric functions.

Innovative Technology to Help You Succeed MyMathLab can improve any learning environment—whether you are taking a lab-based, hybrid, fully online, or a traditional lecture-style course INTERACTIVE FIGURES Math comes alive with new Interactive Figures in MyMathLab! Your instructor may choose to assign assessment questions that are written to accompany each figure This interaction will lead you to fully understand key mathematical concepts in a hands-on, engaging way A HISTORY OF SUCCESS Results show that you can improve your grade by using the videos, animations, interactive figures, step-by-step examples, and personalized feedback in MyMathLab To see the growing list of case studies for yourself, visit www.mymathlab.com/success-stories www.mymathlab.com Prepare for Class “Read the Book” Feature Description Benefit Page Every chapter begins with… Chapter Opening Article & Project Each chapter begins with a current article and ends with a related project The Article describes a real situation The Project lets you apply what you learned to solve a related problem 398, 501 NEW! Internet-based Projects The projects allow for the integration of spreadsheet technology that students will need to be a productive member of the workforce The projects allow the opportunity for students to collaborate and use mathematics to deal with issues that come up in their lives 398, 501 Every section begins with… Learning Objectives Each section begins with a list of objectives Objectives also appear in the text where the objective is covered These focus your studying by emphasizing what’s most important and where to find it 419 Most sections contain… Most sections begin with a list of key concepts to review with page numbers Ever forget what you’ve learned? This feature highlights previously learned material to be used in this section Review it, and you’ll always be prepared to move forward 419 Now Work the ‘Are You Prepared?’ Problems Problems that assess whether you have the prerequisite knowledge for the upcoming section Not sure you need the Preparing for This Section review? Work the ‘Are You Prepared?’ problems If you get one wrong, you’ll know exactly what you need to review and where to review it! 419, 430 “Now Work ” These follow most examples and direct you to a related exercise We learn best by doing You’ll solidify your understanding of examples if you try a similar problem right away, to be sure you understand what you’ve just read 428 PROBLEMS WARNING Warnings are provided in the text These point out common mistakes and help you to avoid them 453 Explorations and Seeing the Concept These represent graphing utility activities to foreshadow a concept or solidify a concept just presented You will obtain a deeper and more intuitive understanding of theorems and definitions 252, 425 These provide alternative descriptions of select definitions and theorems Does math ever look foreign to you? This feature translates math into plain English 421 Calculus Icon These appear next to information essential for the study of calculus Pay attention if you spend extra time now, you’ll better later! 365 Showcase EXAMPLES These examples provide “how-to” instruction by offering a guided, stepby-step approach to solving a problem With each step presented on the left and the mathematics displayed on the right, students can immediately see how each step is employed 337–338 Model It! Marked with These are examples and problems that require you to build a mathematical model from either a verbal description or data The homework Model It! problems are marked by purple numbers It is rare for a problem to come in the form, “Solve the following equation” Rather, the equation must be developed based on an explanation of the problem These problems require you to develop models that will allow you to describe the problem mathematically and suggest a solution to the problem 444, 473 PREPARING FOR THIS SECTION In Words Examples and Problems Practice “Work the Problems” Feature Description Benefit Page “Assess Your Understanding” contains a variety of problems at the end of each section ‘Are You Prepared?’ Problems These assess your retention of the prerequisite material you’ll need Answers are given at the end of the section exercises This feature is related to the Preparing for This Section feature Do you always remember what you’ve learned? Working these problems is the best way to find out If you get one wrong, you’ll know exactly what you need to review and where to review it! 419, 430 Concepts and Vocabulary These short-answer questions, mainly Fill-in-the-Blank and True/False items, assess your understanding of key definitions and concepts in the current section It is difficult to learn math without knowing the language of mathematics These problems test your understanding of the formulas and vocabulary 431 Skill Building Correlated to section examples, these problems provide straightforward practice It’s important to dig in and develop your skills These problems provide you with ample practice to so 431–433 Mixed Practice These problems offer comprehensive assessment of the skills learned in the section by asking problems that relate to more than one concept or objective These problems may also require you to utilize skills learned in previous sections Learning mathematics is a building process Many concepts are interrelated These problems help you see how mathematics builds on itself and also see how the concepts tie together 433 Applications and Extensions These problems allow you to apply your skills to real-world problems These problems also allow you to extend concepts leamed in the section You will see that the material learned within the section has many uses in everyday life 433–435 Explaining Concepts: Discussion and Writing “Discussion and Writing” problems are colored red These support class discussion, verbalization of mathematical ideas, and writing and research projects To verbalize an idea, or to describe it clearly in writing, shows real understanding These problems nurture that understanding Many are challenging but you’ll get out what you put in 436 NEW! Interactive Exercises In selected exercise sets, applets are provided to give a “hands-on” experience “Now Work ” Many examples refer you to a related homework problem These related problems are marked by a pencil and yellow numbers If you get stuck while working problems, look for the closest Now Work problem and refer back to the related example to see if it helps 429 Every chapter concludes with a comprehensive list of exercises to pratice Use the list of objectives to determine the objective and examples that correspond to the problems Work these problems to verify you understand all the skills and concepts of the chapter Think of it as a comprehensive review of the chapter 495–499 PROBLEMS Chapter Review Problems The applets allow students to interact with mathematics in an active learning environment By exploring a variety of scenarios, the student is able to visualize the mathematics and develop a deeper conceptual understanding of the material 345–346 Review “Study for Quizzes and Tests” Feature Description Benefit Page Chapter Reviews at the end of each chapter contain… “Things to Know” A detailed list of important theorems, formulas, and definitions from the chapter Review these and you’ll know the most important material in the chapter! 494–495 “You should be able to…” Contains a complete list of objectives by section, examples that illustrate the objective, and practice exercises that test your understanding of the objective Do the recommended exercises and you’ll have mastery over the key material If you get something wrong, review the suggested examples and page numbers and try again 495–496 Review Exercises These provide comprehensive review and practice of key skills, matched to the Learning Objectives for each section Practice makes perfect These problems combine exercises from all sections, giving you a comprehensive review in one place 496–499 CHAPTER TEST About 15–20 problems that can be taken as a Chapter Test Be sure to take the Chapter Test under test conditions—no notes! Be prepared Take the sample practice test under test conditions This will get you ready for your instructor’s test If you get a problem wrong, watch the Chapter Test Prep video 499–500 CUMULATIVE REVIEW These problem sets appear at the end of each chapter, beginning with Chapter They combine problems from previous chapters, providing an ongoing cumulative review These are really important They will ensure that you are not forgetting anything as you go These will go a long way toward keeping you constantly primed for the final exam 500 CHAPTER PROJECTS The Chapter Project applies what you’ve learned in the chapter Additional projects are available on the Instructor’s Resource Center (IRC) The Project gives you an opportunity to apply what you’ve learned in the chapter to solve a problem related to the opening article If your instructor allows, these make excellent opportunities to work in a group, which is often the best way of learning math 501 NEW! Internet-based Projects In selected chapters, a web-based project is given The projects allow the opportunity for students to collaborate and use mathematics to deal with issues that come up in their lives 501 This page intentionally left blank ALGEBRA & TRIGONOMETRY Enhanced with Graphing Utilities Sixth Edition Michael Sullivan Chicago State University Michael Sullivan, III Joliet Junior College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editor in Chief: Anne Kelly Sponsoring Editor: Dawn Murrin Assistant Editor: Joseph Colella Executive Marketing Manager: Roxanne McCarley Marketing Manager: Peggy Sue Lucas Marketing Assistant: Justine Goulart Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Peggy McMahon Procurement Manager/Boston: Evelyn Beaton Procurement Specialist: Debbie Rossi Procurement Media Specialist: Ginny Michaud Senior Author Support/Technology Specialist: Joe Vetere Associate Director of Design, USHE North and West: Andrea Nix Senior Design Specialist: Heather Scott Interior and Cover Design: Tamara Newnam Cover Image (background): iStockphoto/Simfo Image Manager:/Image Management Services: Rachel Youdelman Photo Research: PreMedia Global Permissions Project Manager: Michael Joyce Media Producer: Christina Maestri Software Development: Kristina Evans, Mary Durnwald, and Marty Wright Full-Service Project Management: Cenveo Publisher Services/Nesbitt Graphics, Inc Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this text appear on page xxviii of the book Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book, and Pearson was aware of a trademark claim, the designations have been printed in initial caps or all caps Microsoft® and Windows® are registered trademarks of the Microsoft Corporation in the U.S.A and other countries Screen shots and icons reprinted with permission from the Microsoft Corporation This book is not sponsored or endorsed by or affiliated with the Microsoft Corporation Library of Congress Cataloging-in-Publication Data Sullivan, Michael, 1942Algebra & trigonometry : enhanced with graphing utilities/Michael Sullivan, Michael Sullivan, III –6th ed p cm Includes bibliographical references and index ISBN 978-0-321-78483-4 (alk paper) Algebra–Textbooks Trigonometry–Textbooks Algebra–Graphic methods Trigonometry–Graphic methods I Sullivan, Michael, 1967 July 2- II Title III Title: Algebra and trigonometry QA154.3.S75 2013 512’.13–dc23 2011024234 Copyright ©2013, 2009, 2006, 2003, 2000 Pearson Education, Inc All rights reserved Manufactured in the United States of America This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 201-236-3290 10—CRK—15 14 13 12 11 ISBN-10: 0-321-78483-9 ISBN-13: 978-0-321-78483-4 For the Family Katy (Murphy) and Pat Mike and Yola Dan and Sheila Colleen (O’Hara) and Bill Shannon, Patrick, Ryan Michael, Kevin, Marissa Maeve, Sean, Nolan Kaleigh, Billy, Timmy This page intentionally left blank 584 CHAPTER Trigonometric Functions STEP 2: The graph of y = sin 12x - p2 lies between -3 and on the y-axis One f p f 2p p 3p cycle begins at x = +p= To find = and ends at x = + = v v v 2 p 3p the five key points, divide the interval c , d into four subintervals, each 2 p of length p , = , by finding the following values of x: NOTE We can also find the interval defining one cycle by solving the inequality … 2x - p … 2p Then p … 2x … 3p p 3p … x … 2 ᭿ p p p 3p + = 4 1st x-coordinate 2nd x-coordinate 3p p + =p 4 p+ p 5p = 4 5p p 3p + = 4 3rd x-coordinate 4th x-coordinate 5th x-coordinate STEP 3: Use these values of x to determine the five key points on the graph: p a , 0b a 3p , 3b 1p, 02 a 5p , -3b 3p , 0b a STEP 4: Plot these five points and fill in the graph of the sine function as shown in Figure 110(a) Extending the graph in each direction, we obtain Figure 110(b) Figure 110 y ␲, 3) ( 3––– ␲– , 3) (Ϫ –– Ϫ1 y ␲, 3) ( 3––– ␲, 3) ( 7––– ␲– , 0) ( –– ␲ ––– ␲ ––– ( ␲, 0) 3––– (␲, 0) ␲ ␲ ––– 5––– ␲– , 0) (Ϫ –– ␲– , 0) ( –– ␲– Ϫ –– Ϫ1 x Ϫ2 ␲ ––– ␲ ––– Ϫ2 Ϫ3 Ϫ3 ␲, Ϫ3) (5––– ␲ ␲ 3––– ␲, 0) ( 5––– (2π, 0) (π, 0) ␲ 7––– ␲, Ϫ3) (5––– (b) ␲– , Ϫ3) ( –– (a) x 9––– ␲, 0) (3––– ␲, Ϫ3) (9––– p The graph of y = sin 12x - p2 = sin c 2a x - b d may also be obtained using transformations See Figure 111 Figure 111 y y ␲– , 1) ( –– ␲ Ϫ1 Ϫ3 Multiply by 3; Vertical stretch by a factor of Figure 112 Ϫ3 ␲ x ␲, Ϫ3) (3––– ␲ ␲ Ϫ3 x ␲– , Ϫ3) ( –– Replace x by 2x ; Replace x by x Ϫ ␲2 ; Horizontal compression Shift right ␲ by a factor of 12 units ␲– (b) y ϭ sin x (c) y ϭ sin (2x ) (d) y ϭ sin 2(x Ϫ –– 2) ] ϭ sin (2x Ϫ ␲) 3␲ ␲ Ϫ3 ␲, 3) ( 3––– ␲ 2␲ x ␲, Ϫ3) (3––– y ␲– , 3) ( –– [ (a) y ϭ sin x Ϫ2 ␲ 2␲ x ␲, Ϫ1) (3––– y ␲– , 3) ( –– Check: Figure 112 shows the graph of Y1 = sin 12x - p2 using a graphing utility To graph a sinusoidal function of the form y = A sin 1vx - f2 + B, first graph the function y = A sin 1vx - f2 and then apply a vertical shift SECTION 7.8 Phase Shift; Sinusoidal Curve Fitting 585 Finding the Amplitude, Period, and Phase Shift of a Sinusoidal Function and Graphing It EXAM PL E Find the amplitude, period, and phase shift of y = cos 14x + 3p2 + and graph the function STEP 1: Begin by graphing y = cos 14x + 3p2 Comparing Solution y = cos 14x + 3p2 = cos c 4a x + 3p bd y = A cos 1vx - f2 = A cos c va x - f bd v to we see that A = 2, v = 4, and f = -3p The graph is a cosine curve with f 2p 2p p 3p = , and phase shift = = = amplitude A = 2, period T = v v 4 STEP 2: The graph of y = cos 14x + 3p2 lies between -2 and on the y-axis One NOTE We can also find the interval defining one cycle by solving the inequality f f 2p 3p 3p p p and ends at x = + = - = + = v v v 4 3p p To find the five key points, divide the interval c , - d into four 4 p p subintervals, each of the length , = , by finding the following values cycle begins at x = … 4x + 3p … 2p Then - 3p … 4x … - p - 3p p … x … 4 - ᭿ 3p - 3p p 5p + = 8 1st x-coordinate - 5p p p + = 8 2nd x-coordinate p p 3p + = 8 - 3rd x-coordinate 4th x-coordinate - 3p p p + = 8 5th x-coordinate STEP 3: The five key points on the graph of y = cos 14x + 3p2 are a- 3p , 2b a- 5p , 0b a- p , -2b a- 3p , 0b a- p , 2b STEP 4: Plot these five points and fill in the graph of the cosine function as shown in Figure 113(a) Extending the graph in each direction, we obtain Figure 113(b), the graph of y = cos 14x + 3p2 STEP 5: A vertical shift up unit gives the final graph See Figure 113(c) Figure 113 ␲, 2) (Ϫ 3––– ␲– , 2) (Ϫ –– y ␲– , 2) (Ϫ –– y ␲– , 3) (Ϫ –– ␲– , 2) ( –– y ␲– , 3) ( –– 4 ␲– , 1) ( –– ␲ Ϫ␲ ␲ Ϫ5––– ␲ Ϫ␲ 3␲ –– Ϫ––– Ϫ–– –– Ϫ3––– 8 ␲ –– Ϫ2 ␲– , Ϫ2) (Ϫ –– (a) x ␲ Ϫ␲ ␲ Ϫ␲ 3␲ –– Ϫ––– Ϫ–– –– Ϫ5––– 8 ␲– , Ϫ2) (Ϫ –– ␲ –– Ϫ2 (b) y ϭ cos (4x ϩ 3␲) ␲ –– x ␲ Ϫ␲ ␲ Ϫ␲ ␲ –– Ϫ3 ––– Ϫ–– –– Ϫ5––– 8 ␲ (Ϫ ––– , Ϫ1) Add 1; Vertical shift up unit ␲ –– ␲ –– x Ϫ2 (c) y ϭ cos (4x ϩ 3␲) ϩ 3p b d + may also be obtained using transformations See Figure 114 on the next page The graph of y = cos 14x + 3p2 + = cos c 4a x + 586 CHAPTER Trigonometric Functions Figure 114 y y (2␲, 2) ␲ Ϫ2 (␲, Ϫ2) (a) y ϭ cos x ␲– , 2) ( –– 2 2␲ x ␲ Ϫ2 Replace x by 4x; Horizontal compression by a factor of 14 ␲ ␲, 2) (Ϫ 3––– ␲ Ϫ5––– ␲ Ϫ3––– x ␲ –– Ϫ␲ Ϫ3––– –– Ϫ␲ ␲– , 2) ( –– ␲ –– –– Ϫ␲ 8 ␲ –– x Ϫ2 ␲– , Ϫ2) (Ϫ –– ␲– , Ϫ2) ( –– (b) y ϭ cos (4x ) y ␲– , 2) (Ϫ –– 3␲ (c) y ϭ cos [4(x ϩ )] ϭ cos (4x ϩ 3␲) Replace x by x ϩ 3␲ 4; Shift left 3␲ units Add 1; Vertical shift up unit ␲, 3) (Ϫ 3––– ␲, 1) (Ϫ5––– ␲– , 3) (Ϫ –– ␲, 1) (Ϫ3––– y ␲– , 3) ( –– ␲– , 1) (Ϫ –– ␲ Ϫ␲ ␲ Ϫ5––– ␲ Ϫ␲ ␲ –– Ϫ3 ––– Ϫ–– –– Ϫ3––– 8 ␲– , Ϫ1) (Ϫ –– ␲– , 1) ( –– ␲ –– ␲ –– x Ϫ2 (d) y ϭ cos (4x ϩ 3␲) ϩ Now Work PROBLEM SUMMARY Steps for Graphing Sinusoidal Functions STEP 1: STEP 2: STEP 3: STEP 4: STEP 5: y = A sin1vx - f2 + B or y = A cos1vx - f2 + B f 2p Determine the amplitude A , period T = , and phase shift v v f Determine the starting point of one cycle of the graph, Determine the ending point of one cycle of the v f 2p f f 2p 2p graph, + Divide the interval c , + d into four subintervals, each of length , v v v v v v Use the endpoints of the subintervals to find the five key points on the graph Plot the five key points and connect them with a sinusoidal graph to obtain one cycle of the graph Extend the graph in each direction to make it complete If B ϶ 0, apply a vertical shift Build Sinusoidal Models from Data Scatter diagrams of data sometimes take the form of a sinusoidal function Let’s look at an example The data given in Table 12 represent the average monthly temperatures in Denver, Colorado Since the data represent average monthly temperatures collected over many years, the data will not vary much from year to year and so will essentially repeat each year In other words, the data are periodic Figure 115 shows the scatter diagram of these data repeated over years, where x = represents January, x = represents February, and so on Notice that the scatter diagram looks like the graph of a sinusoidal function We choose to fit the data to a sine function of the form y = A sin 1vx - f2 + B where A, B, v, and f are constants SECTION 7.8 Table 12 Phase Shift; Sinusoidal Curve Fitting 587 Figure 115 y Month, x Average Monthly Temperature, °F January, 29.7 February, 33.4 March, 39.0 April, 48.2 May, 57.2 June, 66.9 July, 73.5 August, 71.4 September, 62.3 October, 10 51.4 November, 11 39.0 December, 12 31.0 75 30 25 x Source: U.S National Oceanic and Atmospheric Administration EXAM PL E Finding a Sinusoidal Function from Temperature Data Fit a sine function to the data in Table 12 Solution Begin with a scatter diagram of the data for one year See Figure 116 The data are fitted to a sine function of the form Figure 116 y = A sin 1vx - f2 + B 75 STEP 1: To find the amplitude A, we compute largest data value - smallest data value 73.5 - 29.7 = = 21.9 Amplitude = 13 25 To see the remaining steps in this process, superimpose the graph of the function y = 21.9 sin x, where x represents months, on the scatter diagram Figure 117 shows the two graphs To fit the data, the graph needs to be shifted vertically, shifted horizontally, and stretched horizontally STEP 2: Determine the vertical shift by finding the average of the highest and lowest data values Figure 117 75 13 Vertical shift = Ϫ25 Figure 118 75 13 25 73.5 + 29.7 = 51.6 Now superimpose the graph of y = 21.9 sin x + 51.6 on the scatter diagram See Figure 118 We see that the graph needs to be shifted horizontally and stretched horizontally STEP 3: It is easier to find the horizontal stretch factor first Since the temperatures repeat every 12 months, the period of the function is T = 12 Since 2p T= = 12, we find v 2p p = v= 12 588 CHAPTER Trigonometric Functions p xb + 51.6 on the scatter diagram See Figure 119 We see that the graph still needs to be shifted horizontally STEP 4: To determine the horizontal shift, use the period T = 12 and divide the interval 0, 12 into four subintervals of length 12 , = 3: Figure 119 Now superimpose the graph of y = 21.9 sin a 75 0, , 3, , 6, , 9, 12 13 25 The sine curve is increasing on the interval 10, 32 and is decreasing on the interval 13, 92, so a local maximum occurs at x = The data indicate that a maximum occurs at x = (corresponding to July’s temperature), so we must shift the graph of the function units to the right by replacing x by x - Doing this, we obtain p y = 21.9 sin a 1x - 42 b + 51.6 Multiplying out, we find that a sine y = A sin 1vx - f2 + B that fits the data is y = 21.9 sin a Figure 120 function of the form p 2p xb + 51.6 75 p 2p xb + 51.6 and the scatter diagram of the data are shown in Figure 120 The graph of y = 21.9 sin a The steps to fit a sine function 13 y = A sin 1vx - f2 + B 25 to sinusoidal data follow: Steps for Fitting a Sine Function y ‫ ؍‬A sin Vx ؊ F2 ؉ B to Data STEP 1: Determine A, the amplitude of the function Amplitude = largest data value - smallest data value STEP 2: Determine B, the vertical shift of the function Vertical shift = largest data value + smallest data value STEP 3: Determine v Since the period T, the time it takes for the data to 2p repeat, is T = , we have v 2p v= T STEP 4: Determine the horizontal shift of the function by using the period of the data Divide the period into four subintervals of equal length Determine the x-coordinate for the maximum of the sine function and the x-coordinate for the maximum value of the data Use this f information to determine the value of the phase shift, v Now Work PROBLEMS 29(a)–(c) SECTION 7.8 Phase Shift; Sinusoidal Curve Fitting 589 Let’s look at another example Since the number of hours of sunlight in a day cycles annually, the number of hours of sunlight in a day for a given location can be modeled by a sinusoidal function The longest day of the year (in terms of hours of sunlight) occurs on the day of the summer solstice For locations in the northern hemisphere, the summer solstice is the time when the Sun is farthest north In 2011, the summer solstice occurred on June 21 (the 171st day of the year) at 1:16 pm EDT The shortest day of the year occurs on the day of the winter solstice The winter solstice is the time when the Sun is farthest south (again, for locations in the northern hemisphere) In 2011, the winter solstice occurred on December 22 (the 356th day of the year) at 12:30 am (EST) EXAM PL E Finding a Sinusoidal Function for Hours of Daylight According to the Old Farmer’s Almanac, the number of hours of sunlight in Boston on the summer solstice is 15.30 and the number of hours of sunlight on the winter solstice is 9.08 (a) Find a sinusoidal function of the form y = A sin 1vx - f2 + B that fits the data (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year (c) Draw a graph of the function found in part (a) (d) Look up the number of hours of sunlight for April in the Old Farmer’s Almanac and compare it to the results found in part (b) Source: The Old Farmer’s Almanac, www.almanac.com/rise Solution largest data value - smallest data value 15.30 - 9.08 = = 3.11 (a) STEP 1: Amplitude = STEP 2: Vertical shift = = largest data value + smallest data value 15.30 + 9.08 = 12.19 STEP 3: The data repeat every 365 days Since T = v= So far, we have y = 3.11 sin a 2p = 365, we find v 2p 365 2p x - f b + 12.19 365 STEP 4: To determine the horizontal shift, we use the period T = 365 and divide the interval 0, 365 into four subintervals of length 365 , = 91.25: 0, 91.25 , 91.25, 182.5 , 182.5, 273.75 , 273.75, 365 The sine curve is increasing on the interval 10, 91.252 and is decreasing on the interval 191.25, 273.752, so a local maximum occurs at x = 91.25 Since the maximum occurs on the summer solstice at x = 171, we must shift the graph of the function 171 - 91.25 = 79.75 units to the right by replacing x by x - 79.75 Doing this, we obtain y = 3.11 sin a 2p 1x - 79.752 b + 12.19 365 Multiplying out, we find that a sine function of the form y = A sin 1vx - f2 + B that fits the data is y = 3.11 sin a 319p 2p xb + 12.19 365 730 590 CHAPTER Trigonometric Functions (b) To predict the number of hours of daylight on April 1, we let x = 91 in the function found in part (a) and obtain Figure 121 y = 3.11 sin a 16 2p 365 # 91 - 319 pb 730 + 12.19 Ϸ 12.79 400 So we predict that there will be about 12.79 hours = 12 hours, 47 minutes of sunlight on April in Boston (c) The graph of the function found in part (a) is given in Figure 121 (d) According to the Old Farmer’s Almanac, there will be 12 hours 45 minutes of sunlight on April in Boston Now Work PROBLEM 35 Certain graphing utilities (such as a TI-83, TI-84 Plus, and TI-86) have the capability of finding the sine function of best fit for sinusoidal data At least four data points are required for this process EX AM PL E Finding the Sine Function of Best Fit Use a graphing utility to find the sine function of best fit for the data in Table 12 Graph this function with the scatter diagram of the data Solution Enter the data from Table 12 and execute the SINe REGression program The result is shown in Figure 122 The output that the utility provides shows the equation y = a sin 1bx + c2 + d The sinusoidal function of best fit is y = 21.15 sin 10.55x - 2.352 + 51.19 where x represents the month and y represents the average temperature Figure 123 shows the graph of the sinusoidal function of best fit on the scatter diagram Figure 122 Figure 123 75 13 25 Now Work PROBLEMS 29(d) AND (e) 7.8 Assess Your Understanding Concepts and Vocabulary f For the graph of y = A sin1vx - f2 , the number is called v the True or False Only two data points are required by a graphing utility to find the sine function of best fit SECTION 7.8 Phase Shift; Sinusoidal Curve Fitting 591 Skill Building In Problems 3–14, find the amplitude, period, and phase shift of each function Graph each function Be sure to label key points Show at least two periods p y = sin12x - p2 y = sin13x - p2 y = cos a3x + b p p y = -2 cos a2x - b y = cos 12x + p2 y = -3 sina2x + b 2 y = sin 1px + 22 - 10 y = cos 12px + 42 + 12 y = cos 12px - 42 - 13 y = - sina - 2x + 11 y = cos 1px - 22 + p b 14 y = -3 cos a -2x + p b In Problems 15–18, write the equation of a sine function that has the given characteristics 16 Amplitude: p Period: Phase shift: 15 Amplitude: Period: p Phase shift: 17 Amplitude: Period: 3p Phase shift: - 18 Amplitude: Period: p Phase shift: - Mixed Practice In Problems 19–26, apply the methods of this and the previous section to graph each function Be sure to label key points and show at least two periods 19 y = tan14x - p2 23 y = - cot a2x + p b cot 12x - p2 p 24 y = - tana3x + b 20 y = 21 y = csc a2x - p b 25 y = -sec 12px + p2 sec 13x - p2 p 26 y = - csc a - px + b 22 y = Applications and Extensions 27 Alternating Current (ac) Circuits The current I, in amperes, flowing through an ac (alternating current) circuit at time t, in seconds, is p I 1t2 = 120 sina30pt - b tÚ Decade, x Major Hurricanes, H 1921–1930, 17 1931–1940, 16 1941–1950, 29 What is the period? What is the amplitude? What is the phase shift? Graph this function over two periods 1951–1960, 33 1961–1970, 27 28 Alternating Current (ac) Circuits The current I, in amperes, flowing through an ac (alternating current) circuit at time t, in seconds, is 1971–1980, 16 1981–1990, 16 1991–2000, 27 2001–2010, 33 I 1t2 = 220 sina60pt - p b tÚ What is the period? What is the amplitude? What is the phase shift? Graph this function over two periods 29 Hurricanes Hurricanes are categorized using the SaffirSimpson Hurricane Scale, with winds 111–130 miles per hour (mph) corresponding to a category hurricane, winds 131–155 mph corresponding to a category hurricane, and winds in excess of 155 mph corresponding to a category hurricane The following data represent the number of major hurricanes in the Atlantic Basin (category 3, 4, or 5) each decade from 1921 to 2010 (a) Draw a scatter diagram of the data for one period (b) Find a sinusoidal function of the form y = A sin 1vx - f2 + B that models the data (c) Draw the sinusoidal function found in part (b) on the scatter diagram Source: U.S National Oceanic and Atmospheric Administration (d) Use a graphing utility to find the sinusoidal function of best fit (e) Draw the sinusoidal function of best fit on a scatter diagram of the data 30 Monthly Temperature The data on the next page represent the average monthly temperatures for Washington, D.C (a) Draw a scatter diagram of the data for one period (b) Find a sinusoidal function of the form y = A sin1vx - f2 + B that models the data (c) Draw the sinusoidal function found in part (b) on the scatter diagram (d) Use a graphing utility to find the sinusoidal function of best fit 592 CHAPTER Trigonometric Functions (e) Graph the sinusoidal function of best fit on a scatter diagram of the data (c) Draw the sinusoidal function found in part (b) on the scatter diagram (d) Use a graphing utility to find the sinusoidal function of best fit (e) Graph the sinusoidal function of best fit on a scatter diagram of the data Month, x Average Monthly Temperature, ˚F January, 34.6 Month, x Average Monthly Temperature, ˚F February, 37.5 January, 31.8 March, 47.2 February, 34.8 April, 56.5 March, 44.1 May, 66.4 April, 53.4 June, 75.6 May, 63.4 July, 80.0 June, 72.5 August, 78.5 July, 77.0 September, 71.3 August, 75.6 October, 10 59.7 September, 68.5 November, 11 49.8 October, 10 56.6 December, 12 39.4 November, 11 46.8 December, 12 36.7 Source: U.S National Oceanic and Atmospheric Administration 31 Monthly Temperature The following data represent the average monthly temperatures for Indianapolis, Indiana Month, x Average Monthly Temperature, ˚F January, 25.5 February, 29.6 March, 41.4 April, 52.4 May, 62.8 June, 71.9 July, 75.4 August, 73.2 September, 66.6 October, 10 54.7 November, 11 43.0 December, 12 30.9 Source: U.S National Oceanic and Atmospheric Administration (a) Draw a scatter diagram of the data for one period (b) Find a sinusoidal function of the form y = A sin1vx - f2 + B that models the data (c) Draw the sinusoidal function found in part (b) on the scatter diagram (d) Use a graphing utility to find the sinusoidal function of best fit (e) Graph the sinusoidal function of best fit on a scatter diagram of the data 32 Monthly Temperature The following data represent the average monthly temperatures for Baltimore, Maryland (a) Draw a scatter diagram of the data for one period (b) Find a sinusoidal function of the form y = A sin 1vx - f2 + B that models the data Source: U.S National Oceanic and Atmospheric Administration 33 Tides The length of time between consecutive high tides is 12 hours and 25 minutes According to the National Oceanic and Atmospheric Administration, on Saturday, July 21, 2012, in Charleston, South Carolina, high tide occurred at 11:30 am (11.5 hours) and low tide occurred at 5:31 pm (17.5167 hours) Water heights are measured as the amounts above or below the mean lower low water The height of the water at high tide was 5.84 feet, and the height of the water at low tide was −0.37 foot (a) Approximately when will the next high tide occur? (b) Find a sinusoidal function of the form y = A sin1vx - f2 + B that models the data (c) Use the function found in part (b) to predict the height of the water at pm on July 21, 2012 34 Tides The length of time between consecutive high tides is 12 hours and 25 minutes According to the National Oceanic and Atmospheric Administration, on Saturday, July 21, 2012, in Sitka Sound, Alaska, high tide occurred at 2:37 am (2.6167 hours) and low tide occurred at 9:12 pm (9.2 hours) Water heights are measured as the amounts above or below the mean lower low water The height of the water at high tide was 11.09 feet, and the height of the water at low tide was −2.49 feet (a) Approximately when will the next high tide occur? (b) Find a sinusoidal function of the form y = A sin1vx - f2 + B that models the data (c) Use the function found in part (b) to predict the height of the water at pm 35 Hours of Daylight According to the Old Farmer’s Almanac, in Miami, Florida, the number of hours of sunlight on the summer solstice of 2011 was 13.75, and the number of hours of sunlight on the winter solstice was 10.55 (a) Find a sinusoidal function of the form y = A sin 1vx - f2 + B that models the data (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year (c) Draw a graph of the function found in part (a) Chapter Review (d) Look up the number of hours of sunlight for April in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b) 36 Hours of Daylight According to the Old Farmer’s Almanac, in Detroit, Michigan, the number of hours of sunlight on the summer solstice of 2011 was 15.30, and the number of hours of sunlight on the winter solstice was 9.10 (a) Find a sinusoidal function of the form y = A sin1vx - f2 + B that models the data (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year (c) Draw a graph of the function found in part (a) (d) Look up the number of hours of sunlight for April in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b) 37 Hours of Daylight According to the Old Farmer’s Almanac, in Anchorage, Alaska, the number of hours of sunlight on the summer solstice of 2011 was 19.42 and the number of hours of sunlight on the winter solstice was 5.48 593 (a) Find a sinusoidal function of the form y = A sin 1vx - f2 + B that models the data (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year (c) Draw a graph of the function found in part (a) (d) Look up the number of hours of sunlight for April in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b) 38 Hours of Daylight According to the Old Farmer’s Almanac, in Honolulu, Hawaii, the number of hours of sunlight on the summer solstice of 2011 was 13.43 and the number of hours of sunlight on the winter solstice was 10.85 (a) Find a sinusoidal function of the form y = A sin 1vx - f2 + B that models the data (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year (c) Draw a graph of the function found in part (a) (d) Look up the number of hours of sunlight for April in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b) Explaining Concepts: Discussion and Writing 39 Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis 40 Find an application in your major field that leads to a sinusoidal graph Write a paper about your findings CHAPTER REVIEW Things to Know Definitions Angle in standard position (p 503) Degree (1°) (p 504) Radian (p 506) Acute angle (p 516) Complementary angles (p 522) Cofunction (p 522) Trigonometric functions of any angle (p 539) Vertex is at the origin; initial side is along the positive x-axis revolution 360 The measure of a central angle of a circle whose rays subtend an arc whose length is equal to the radius of the circle p An angle u whose measure is 0Њ u 90Њ or u , u in radians p Two acute angles whose sum is 90° a radians b The following pairs of functions are cofunctions of each other: sine and cosine; tangent and cotangent; secant and cosecant 1Њ = P = (a, b) is a point, not the origin, on the terminal side of u a distance r from the origin: sin u = b r cos u = a r tan u = b , a a ϶0 r r a , b϶0 sec u = , a ϶ cot u = , b ϶ b a b The acute angle formed by the terminal side of u and either the positive or negative x-axis csc u = Reference angle of u (p 544) Periodic function (p 556) f 1u + p2 = f 1u2, for all u, p 0, where the smallest such p is the fundamental period Formulas counterclockwise revolution = 360Њ (p 504) = 2p radians (p 508) s = r u (p 507) u is measured in radians; s is the length of the arc subtended by the central angle u of the circle of radius r 594 A = CHAPTER Trigonometric Functions r u (p 510) A is the area of the sector of a circle of radius r formed by a central angle of u radians v is the linear speed along the circle of radius r; v is the angular speed (measured in radians per unit time) v = rv (p 511) Table of Values U (Radians) U (Degrees) sinU cos U tan U csc U sec U cot U 0° Not defined Not defined p 30° 23 23 223 23 p 45° 22 22 22 22 60° 23 2 23 223 23 90° Not defined Not defined 180° -1 Not defined -1 Not defined 270° -1 Not defined p p p 3p Not defined -1 Fundamental identities (p 519) sin u cos u tan u = cot u = cos u sin u 1 sec u = cot u = csc u = sin u cos u tan u sin2 u + cos2 u = tan2 u + = sec u cot u + = csc u y Properties of the trigonometric functions y = sin x (p 561) Domain: - ϱ x ϱ Range: - … y … Periodic: period = 2p (360Њ) Odd function y = cos x (p 563) y = tan x (pp 576–577) y = cot x (pp 578–579) y = csc x (p 579) Domain: - ϱ x ϱ Range: - … y … Periodic: period = 2p (360Њ) Even function Ϫ␲ Ϫ1 ␲ – ␲ ␲ – ␲ 2␲ 5–– ␲ x y Ϫ␲ Ϫ␲ – Ϫ1 p Domain: - ϱ x ϱ , except odd integer multiples of (90Њ) Range: - ϱ y ϱ Periodic: period = p (180Њ) Odd function p ␲ ␲ Ϫ 5–– Ϫ 3–– Vertical asymptotes at odd integer multiples of 2 Domain: - ϱ x ϱ , except integer multiples of p (180°) Range: - ϱ y ϱ Periodic: period = p (180Њ) Odd function 3␲ Ϫ ––– Vertical asymptotes at integer multiples of p Domain: - ϱ x ϱ , except integer multiples of p (180Њ) Range: y Ú y ϭ sin x Periodic: period = 2p (360Њ) Odd function 3␲ Ϫ––– Vertical asymptotes at integer multiples of p 2 3–– ␲ 2␲ 5–– ␲ x y ␲ Ϫ –– ␲ – Ϫ1 3–– ␲ 5–– ␲ x 5␲ ––– x y Ϫ␲ – ␲ – Ϫ1 3␲ ––– 2 y ϭ csc x y Ϫ1 ␲ – x Chapter Review p Domain: - ϱ x ϱ , except odd integer multiples of (90Њ) Range: y Ú y Periodic: period = 2p (360Њ) Even function p Ϫ ––– 3␲ ␲ Ϫ –– Vertical asymptotes at odd integer multiples of 2 Ϫ1 y = sec x (p 579) Sinusoidal graphs y = A sin(vx) + B, v y = A cos(vx) + B, v f bd + B v f y = A cos(vx - f) + B = A cos c v ax - b d + B v y = A sin(vx - f) + B = A sin c v ax - 2p (pp 556, 583) v Amplitude = A (pp 566, 583) f Phase shift = (p 583) v y ϭ cos x ␲ – 3–– ␲ 2 x y ϭ sec x Period = Objectives Section 7.1 You should be able to: 7.2 7.3 7.4 7.5 Convert between decimals and degrees, minutes, seconds measures for angles (p 505) Find the length of an arc of a circle (p 507) Convert from degrees to radians and from radians to degrees (p 507) Find the area of a sector of a circle (p 510) Find the linear speed of an object traveling in circular motion (p 511) Find the values of trigonometric functions of acute angles (p 516) Use the Fundamental Identities (p 518) Find the values of the remaining trigonometric functions, given the value of one of them (p 520) Use the Complementary Angle Theorem (p 522) Find the exact values of the trigonometric functions of p = 45Њ (p 528) Find the exact values of the trigonometric functions of p p = 30Њ and = 60Њ (p 529) Use a calculator to approximate the values of the trigonometric functions of acute angles (p 530) Model and solve applied problems involving right triangles (p 531) Find the exact values of the trigonometric functions for any angle (p 539) Use coterminal angles to find the exact value of a trigonometric function (p 541) Determine the signs of the trigonometric functions of an angle in a given quadrant (p 543) Find the reference angle of an angle (p 544) Use a reference angle to find the exact value of a trigonometric function (p 545) Find the exact values of the trigonometric functions of an angle, given information about the functions (p 546) Find the exact values of the trigonometric functions using the unit circle (p 550) Know the domain and range of the trigonometric functions (p 554) Use the periodic properties to find the exact values of the trigonometric functions (p 555) Use even–odd properties to find the exact values of the trigonometric functions (p 557) Example(s) Review Exercises 4– 48 49, 50 1–4 49 51, 52 2, 41 11, 12 4, 6, 16 13 –15 1, 5, 3, 5, 6–9 42 53–55 1, 46 10 43 44 7, 8, 10 7, 16–23 45 47 10 7, 8, 10, 14, 15 595 596 CHAPTER 7.6 Graph functions of the form y = A sin(vx) using transformations (p 561) Graph functions of the form y = A cos(vx) using transformations (p 563) Determine the amplitude and period of sinusoidal functions (p 564) Graph sinusoidal functions using key points (p 566) Find an equation for a sinusoidal graph (p 569) 7.7 Graph functions of the form y = A tan(vx) + B and y = A cot(vx) + B (p 577) Graph functions of the form y = A csc(vx) + B and y = A sec(vx) + B (p 579) 7.8 Trigonometric Functions 24 5–7 8, 25 33, 34 24, 25, 35 39, 40 1, 27 Graph sinusoidal functions of the form y = A sin(vx - f) + B (p 582) Build sinusoidal models from data (p 586) 1, 29, 30 1, 3–5 31, 36–38, 56 57 Review Exercises In Problems and 2, convert each angle in degrees to radians Express your answer as a multiple of p 135° 15° In Problems and 4, convert each angle in radians to degrees 2p 3 - 5p In Problems 5–15, find the exact value of each expression Do not use a calculator tan p p - sin sec a - p 5p b - cot a b 11 sin2 20Њ + 14 sin 45o - tan sec 20Њ p cos tan p + sin p 10 cos 540Њ - tan( -405Њ) 12 sec 50Њ cos 50Њ sin( -40Њ) p 3p + tana - b 13 sin 50Њ cos 40Њ 15 sin 400Њ sec( - 50Њ) cos 50Њ In Problems 16–23, find the exact value of each of the remaining trigonometric functions 16 sin u = , 12 , sin u 5 3p u in quadrant III 20 sin u = - , u 2p 13 u is acute 19 cos u = - , 22 csc u = -4, p u 18 sec u = - , tan u 21 tan u = , 180Њ u 270Њ 17 tan u = 3p 23 cot u = - 2, p u p In Problems 24–32, graph each function Each graph should contain at least two periods Use the graph to determine the domain and the range of each function 24 y = sin(4x) 25 y = - cos(2x) 26 y = tan(x + p) 27 y = - tan(3x) p 28 y = cot ax + b 29 y = sec (2x) 31 y = sin(2x + 4) - 32 y = cot a p 30 y = csc ax + b x p - b In Problems 33 and 34, determine the amplitude and period of each function without graphing 33 y = sin(2x) 34 y = -2 cos(3px) In Problems 35–38, find the amplitude, period, and phase shift of each function Graph each function Show at least two periods 35 y = sin(3x) p 36 y = -cos a x + b 2 37 y = sina x - p b 2 38 y = - cos(px - 6) Chapter Review In Problems 39 and 40, find a function whose graph is given 39 y Ϫ4␲ 4␲ 8␲ x 597 51 Angular Speed of a Race Car A race car is driven around a circular track at a constant speed of 180 miles per hour If the diameter of the track is mile, what is the angular speed of the car? Express your answer in revolutions per hour (which is equivalent to laps per hour) 52 Lighthouse Beacons The Montauk Point Lighthouse on Long Island has dual beams (two light sources opposite each other) Ships at sea observe a blinking light every seconds What rotation speed is required to this? 53 Measuring the Length of a Lake From a stationary hot-air balloon 500 feet above the ground, two sightings of a lake are made (see the figure) How long is the lake? Ϫ5 40 y 500 ft 65° 25° Ϫ4 Ϫ2 10 x 54 Finding the Width of a River Find the distance from A to C across the river illustrated in the figure Ϫ7 41 Find the value of each of the six trigonometric functions of the acute angle u in a right triangle where the length of the side opposite u is and the length of the hypotenuse is p 42 Use a calculator to approximate sin Use a calculator to approximate sec 10Њ Round answers to two decimal places A 43 Determine the signs of the six trigonometric functions of an angle u whose terminal side is in quadrant III 25° B 50 ft C 44 Find the reference angle of 750° 45 Find the exact values of the six trigonometric functions of 12 b is the point on the unit circle that t if P = a - , 3 corresponds to t 46 Find the exact values of sin t, cos t, and tan t if P = ( -2, 5) is the point on the circle that corresponds to t 47 What is the domain and the range of the secant function? What is the period? 55 Finding the Distance to Shore The Willis Tower in Chicago is 1454 feet tall and is situated about mile inland from the shore of Lake Michigan, as indicated in the figure An observer in a pleasure boat on the lake directly in front of the Willis Tower looks at the top of the tower and measures the angle of elevation as 5° How far offshore is the boat? 48 (a) Convert the angle 32Њ20Ј35Љ to a decimal in degrees Round the answer to two decimal places (b) Convert the angle 63.18° to DЊMЈSЉ form Express the answer to the nearest second 1454 ft 49 Find the length of the arc subtended by a central angle of 30° on a circle of radius feet What is the area of the sector? 50 The minute hand of a clock is inches long How far does the tip of the minute hand move in 30 minutes? How far does it move in 20 minutes? 5° mi Lake Michigan 598 CHAPTER Trigonometric Functions 56 Alternating Current The current I, in amperes, flowing through an ac (alternating current) circuit at time t is I = 220 sin a30pt + (a) (b) (c) (d) p b tÚ What is the period? What is the amplitude? What is the phase shift? Graph this function over two periods 57 Monthly Temperature The following data represent the average monthly temperatures for Phoenix, Arizona (a) Draw a scatter diagram of the data for one period (b) Find a sinusoidal function of the form y = A sin(vx - f) + B that models the data (c) Draw the sinusoidal function found in part (b) on the scatter diagram (d) Use a graphing utility to find the sinusoidal function of best fit (e) Graph the sinusoidal function of best fit on the scatter diagram In Problems 1–3, convert each angle in degrees to radians Express your answer as a multiple of p -400Њ 13° In Problems 4–6 convert each angle in radians to degrees p 9p 3p In Problems 7–12, find the exact value of each expression p cos a - cos( - 120Њ) 10 tan 330° sin 3p 5p b - cos 4 19p p 12 sin2 60Њ - cos 45Њ 11 sin - tan In Problems 13–16, use a calculator to evaluate each expression Round your answer to three decimal places 28p 2p 15 sec 229Њ 16 cot 17 Fill in each table entry with the sign of each function 13 sin 17Њ 14 cos sin U cos U Average Monthly Temperature, °F January, 51 February, 55 March, 63 April, 67 May, 77 June, 86 July, 90 August, 90 September, 84 October, 10 71 November, 11 59 December, 12 52 Source: U.S National Oceanic and Atmospheric Administration The Chapter Test Prep Videos are step-by-step test solutions available in the Video Resources DVD, in , or on this text’s Channel Flip back to the Student Resources page to see the exact web address for this text’s YouTube channel CHAPTER TEST 260° Month, x tan U sec U U in QI U in QII U in QIII U in QIV 18 If f 1x2 = sin x and f 1a2 = , find f - a2 csc U cot U In Problems 19–21, find the value of the remaining five trigonometric functions of u 3p 19 sin u = , u in quadrant II 20 cos u = , u 2p 12 p u p 21 tan u = - , In Problems 22–24, the point (x, y) is on the terminal side of angle u in standard position Find the exact value of the given trigonometric function 22 (2, 7), sin u 23 ( - 5, 11), cos u 24 (6, -3), tan u In Problems 25 and 26, graph the function 25 y = sina x p - b 26 y = tana -x + p b + 27 Write an equation for a sinusoidal graph with the following properties: 2p p A = -3 period = phase shift = 28 Logan has a garden in the shape of a sector of a circle; the outer rim of the garden is 25 feet long and the central angle of the sector is 50° She wants to add a 3-foot wide walk to the outer rim; how many square feet of paving blocks will she need to build the walk? 29 Hungarian Adrian Annus won the gold medal for the hammer throw at the 2004 Olympics in Athens with a winning distance of 83.19 meters.* The event consists of swinging a 16-pound weight attached to a wire 190 centimeters long in a circle and then releasing it Assuming his release is at a 45°angle to the ground, the hammer will travel a distance *Annus was stripped of his medal after refusing to cooperate with postmedal drug testing ... expression 57 - 3 # + # (3 - 2) 51 -6 + # 55 + 58 # - 3(4 + 2) - 59 # (3 - 5) + # - 60 - (4 # - + 2) 61 10 - - # + (8 - 3) # 62 - # - # (3 - 4) 49 - + 50 - + 53 + - 54 - - 63 (5 - 3) 64 (5 + 4) 52 - #... 65 + - 66 - - 67 # 10 21 68 5# 10 69 # 10 25 27 70 21 # 10 0 25 71 + 72 + 73 + 74 15 + SECTION R.2 75 + 18 12 76 + 15 77 30 18 79 20 15 80 35 14 18 81 11 27 83 1# 3 + 10 84 4 #1 + 85 # 78 Algebra. .. Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 20 1- 2 3 6-3 290 10 —CRK? ?15 14 13 12 11 ISBN -1 0 : 0-3 2 1- 7 848 3-9 ISBN -1 3 : 97 8-0 -3 2 1- 7 848 3-4 For