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Ebook College algebra with trigonometry (9th edition): Part 1 includes the following content: Chapter R: Basic algebraic operations, chapter 1 equations and inequalities, chapter 2 graphs, chapter 3 functions, chapter 4 polynomial and rational functions, chapter 5 exponential and logarithmic functions, chapter 6 trigonometric functions. Please refer to the documentation for more details.
bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page i College Algebra with Trigonometry This page intentionally left blank bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page iii NINTH EDITION College Algebra with Trigonometry Raymond A Barnett Merritt College Michael R Ziegler Marquette University Karl E Byleen Marquette University Dave Sobecki Miami University Hamilton bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page iv COLLEGE ALGEBRA WITH TRIGONOMETRY, NINTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2011 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2008, 2001, and 1999 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper DOW/DOW ISBN 978–0–07–351950–0 MHID 0–07–351950–2 ISBN 978–0–07–729720–6 (Annotated Instructor’s Edition) MHID 0–07–729720–2 Vice President, Editor-in-Chief: Marty Lange Vice President, EDP: Kimberly Meriwether David Editorial Director: Stewart K Mattson Sponsoring Editor: John R Osgood Director of Development: Kristine Tibbetts Developmental Editor: Christina A Lane Marketing Manager: Kevin M Ernzen Lead Project Manager: Sheila M Frank Senior Production Supervisor: Kara Kudronowicz Senior Media Project Manager: Sandra M Schnee Designer: Tara McDermott Cover/Interior Designer: Ellen Pettergell (USE) Cover Image: © Comstock Images/Getty Images Senior Photo Research Coordinator: Lori Hancock Supplement Producer: Mary Jane Lampe Compositor: Aptara®, Inc Typeface: 10/12 Times Roman Printer: R R Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page Chapter R Opener: © Corbis RF; 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p 658: © Vol 48/Getty RF; p 662: © Getty RF Chapter 11 Opener: © Vol 6/Getty RF; p 733: © ThinkStock/PictureQuest RF; p 745: © Corbis RF Library of Congress Cataloging-in-Publication Data Barnett, Raymond A College algebra with trigonometry/Raymond A Barnett [et al.] — 9th ed p cm — (Barnett, Ziegler & Byleen’s precalculus series) Includes index ISBN 978-0-07-351950-0 — ISBN 0-07-351950-2 (hard copy : alk paper) Algebra–Textbooks Trigonometry–Textbooks I Title QA154.3.B368 2011 512.13–dc22 2009019473 www.mhhe.com bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page v The Barnett, Ziegler and Sobecki Precalculus Series College Algebra, Ninth Edition This book is the same as Precalculus without the three chapters on trigonometry ISBN 0-07-351949-9, ISBN 978-0-07-351-949-4 Precalculus, Seventh Edition This book is the same as College Algebra with three chapters of trigonometry added The trigonometric functions are introduced by a unit circle approach ISBN 0-07-351951-0, ISBN 978-0-07-351-951-7 College Algebra with Trigonometry, Ninth Edition This book differs from Precalculus in that College Algebra with Trigonometry uses right triangle trigonometry to introduce the trigonometric functions ISBN 0-07-735010-3, ISBN 978-0-07-735010-9 College Algebra: Graphs and Models, Third Edition This book is the same as Precalculus: Graphs and Models without the three chapters on trigonometry This text assumes the use of a graphing calculator ISBN 0-07-305195-0, ISBN 978-0-07-305195-6 Precalculus: Graphs and Models, Third Edition This book is the same as College Algebra: Graphs and Models with three additional chapters on trigonometry The trigonometric functions are introduced by a unit circle approach This text assumes the use of a graphing calculator ISBN 0-07-305196-9, ISBN 978-0-07-305-196-3 v This page intentionally left blank bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page vii About the Authors Raymond A Barnett, a native of and educated in California, received his B.A in mathematical statistics from the University of California at Berkeley and his M.A in mathematics from the University of Southern California He has been a member of the Merritt College Mathematics Department and was chairman of the department for four years Associated with four different publishers, Raymond Barnett has authored or co-authored 18 textbooks in mathematics, most of which are still in use In addition to international English editions, a number of the books have been translated into Spanish Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, Northern Kentucky University; Charles Burke, City College of San Francisco; John Fujii, Merritt College; Karl Byleen, Marquette University; and Dave Sobecki, Miami University Hamilton Michael R Ziegler received his B.S from Shippensburg State College and his M.S and Ph.D from the University of Delaware After completing postdoctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he held the rank of Professor in the Department of Mathematics, Statistics, and Computer Science Dr Ziegler published more than a dozen research articles in complex analysis and co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Karl Byleen before passing away unexpectedly in 2008 Karl E Byleen received his B.S., M.A., and Ph.D degrees in mathematics from the University of Nebraska He is currently an Associate Professor in the Department of Mathematics, Statistics, and Computer Science of Marquette University He has published a dozen research articles on the algebraic theory of semigroups and co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Michael Ziegler Dave Sobecki earned a B.A in math education from Bowling Green State University, then went on to earn an M.A and a Ph.D in mathematics from Bowling Green He is an associate professor in the Department of Mathematics at Miami University in Hamilton, Ohio He has written or co-authored five journal articles, eleven books and five interactive CD-ROMs Dave lives in Fairfield, Ohio with his wife (Cat) and dogs (Macleod and Tessa) His passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel, and home improvement projects vii This page intentionally left blank bar19502_fm_i-xxxii.qxd 12/10/09 11:47 PM Page ix Dedicated to the memory of Michael R Ziegler, trusted author, colleague, and friend bar19502_ch06_385-460.qxd 446 11/20/09 CHAPTER 6:21 PM Page 446 TRIGONOMETRIC FUNCTIONS Figure compares the graphs of the restricted cosine function and its inverse Notice that (0, 1), (͞2, 0), and (, Ϫ1) are on the restricted cosine graph Reversing the coordinates gives us three points on the graph of the inverse cosine function y y (Ϫ1, ) (0, 1) y ϭ cos x 2 , 0 Ϫ1 x y ϭ cosϪ1 x ϭ arccos x 0, 2 (, Ϫ1) (1, 0) Ϫ1 x Domain ϭ [Ϫ1, 1] Range ϭ [0, ] Inverse cosine function Domain ϭ [0, ] Range ϭ [Ϫ1, 1] Restricted cosine function (a) (b) Z Figure Inverse cosine function We complete the discussion by giving the cosine–inverse cosine identities: Z COSINE–INVERSE COSINE IDENTITIES cos (cosϪ1 x) ϭ x cosϪ1 (cos x) ϭ x ZZZ EXPLORE-DISCUSS EXAMPLE Ϫ1 Յ x Յ 0ՅxՅ f (f ؊1(x)) ؍x f ؊1(f(x)) ؍x Evaluate each of the following with a calculator Which illustrate a cosine–inverse cosine identity and which not? Discuss why (A) cos (cosϪ1 0.2) (B) cos [cosϪ1 (Ϫ2)] (C) cosϪ1 (cos 2) (D) cosϪ1 [cos (Ϫ3)] Exact Values Find exact values without using a calculator (A) arccos (Ϫ13ր2) SOLUTIONS (B) cos (cosϪ1 0.7) (C) sin [cosϪ1 (Ϫ13)] (A) y ϭ arccos (Ϫ13ր2) is equivalent to b Reference triangle associated with y 13 0ՅyՅ 5 13 yϭ ϭ arccos aϪ b cos y ϭ Ϫ Ϫ͙3 y a [Note: y Ϫ5͞6, even though cos (Ϫ5͞6) ϭ Ϫ͙3 ෆ͞2 because y must be between and , inclusive.] bar19502_ch06_385-460.qxd 11/20/09 6:21 PM Page 447 SECTION 6–6 (B) cos (cosϪ1 0.7) = 0.7 Ϫ1 Inverse Trigonometric Functions 447 Cosine–inverse cosine identity, because ؊1 Յ 0.7 Յ (Ϫ13); (C) Let y ϭ cos then cos y ϭ Ϫ13, Յ y Յ Draw a reference triangle associated with y Then sin y ϭ sin [cosϪ1 (Ϫ13)] can be determined directly from the triangle (after finding the third side) without actually finding y a2 ϩ b2 ϭ c2 b ϭ 232 Ϫ (Ϫ12) ϭ 18 ϭ 212 b a ϭ Ϫ1 cϭ3 c b Because b Ͼ in Quadrant II y a a Therefore, sin [cosϪ1 (Ϫ13)] ϭ sin y ϭ 212ր3 MATCHED PROBLEM Find exact values without using a calculator (A) arccos (12 ր2) (B) cosϪ1 (cos 3.05) (C) cot [cosϪ1 (Ϫ1ր 15)] EXAMPLE Calculator Values Find to four significant digits using a calculator (A) arccos 0.4325 (B) cosϪ1 2.137 (C) csc [cosϪ1 (Ϫ0.0349)] SOLUTIONS Set your calculator in radian mode (A) arccos 0.4325 ϭ 1.124 (B) cosϪ1 2.137 ϭ Error 2.137 is not in the domain of cos؊1 (C) csc [cosϪ1 (Ϫ0.0349)] ϭ 1.001 MATCHED PROBLEM Find to four significant digits using a calculator (A) cosϪ1 (0.6773) (B) arccos (Ϫ1.003) (C) cot [cosϪ1 (Ϫ0.5036)] Z Inverse Tangent Function To restrict the tangent function so that it becomes one-to-one, we choose the interval (Ϫ͞2, ͞2) Over this interval the restricted function passes the horizontal line test, and each range value is assumed exactly once as x moves across this restricted domain (Fig 7) We use this restricted tangent function to define the inverse tangent function bar19502_ch06_385-460.qxd 448 11/20/09 CHAPTER 6:21 PM Page 448 TRIGONOMETRIC FUNCTIONS y Z Figure y ϭ tan x is one-to-one over (Ϫ͞2, ͞2) y ϭ tan x Ϫ Ϫ2 Ϫ 3 Ϫ Ϫ1 2 x 3 Z DEFINITION Inverse Tangent Function The inverse tangent function, denoted by tanϪ1 or arctan, is defined as the inverse of the restricted tangent function y ϭ tan x, Ϫ͞2 Ͻ x Ͻ ͞2 So y ϭ tanϪ1 x y ϭ arctan x and are equivalent to tan y ϭ x where Ϫ͞2 Ͻ y Ͻ ͞2 and x is a real number In words, the inverse tangent of x, or the arctangent of x, is the number or angle y, Ϫ͞2 Ͻ y Ͻ ͞2, whose tangent is x Figure compares the graphs of the restricted tangent function and its inverse Notice that (Ϫ͞4, Ϫ1), (0, 0), and (͞4, 1) are on the restricted tangent graph Reversing the coordinates gives us three points on the graph of the inverse tangent function Also note that the vertical asymptotes become horizontal asymptotes for the inverse function y Z Figure Inverse tangent y y ϭ tan x function y ϭ tanϪ1 x ϭ arctan x Ϫ Ϫ1 , 1 x Ϫ , Ϫ1 1, 4 Ϫ1, Ϫ Ϫ1 Ϫ Domain ϭ Ϫ , Range ϭ (Ϫϱ, ϱ) Restricted tangent function Domain ϭ (Ϫϱ, ϱ) Range ϭ Ϫ , Inverse tangent function (a) (b) We now state the tangent–inverse tangent identities Z TANGENT–INVERSE TANGENT IDENTITIES tan (tanϪ1 x) ϭ x tanϪ1 (tan x) ϭ x Ϫϱ x ϱ Ϫր2 x ր2 f(f ؊1(x)) ؍x f ؊1(f(x)) ؍x x bar19502_ch06_385-460.qxd 11/20/09 6:21 PM Page 449 SECTION 6–6 EXAMPLE 449 Inverse Trigonometric Functions Exact Values Find exact values without using a calculator (A) tanϪ1 (Ϫ1ր 13) SOLUTIONS (B) tanϪ1 (tan 0.63) (A) y ϭ tanϪ1 (Ϫ1ր 13) is equivalent to Reference triangle associated with y /2 b Ϫ y 2 13 y ϭ Ϫ ϭ tanϪ1 aϪ b 13 tan y ϭ Ϫ ͙3 a y Ϫ1 Ϫ/2 [Note: y cannot be 11͞6 because y must be between Ϫ ր2 and ր2.] (B) tanϪ1 (tan 0.63) ϭ 0.63 Tangent–inverse tangent identity, because ؊͞2 Յ 0.63 Յ ͞2 Find exact values without using a calculator MATCHED PROBLEM (B) tan (tanϪ1 43) (A) arctan (Ϫ13) Z Summary We summarize the definitions and graphs of the inverse trigonometric functions discussed so far for convenient reference ؊1 ؊1 ؊1 Z SUMMARY OF sin , cos , AND tan y ϭ sinϪ1 x y ϭ cosϪ1 x y ϭ tanϪ1 x is equivalent to is equivalent to is equivalent to x ϭ sin y x ϭ cos y x ϭ tan y y where Ϫ1 Յ x Յ 1, Ϫ͞2 Յ y Յ ͞2 where Ϫ1 Յ x Յ 1, Յ y Յ where Ϫϱ Ͻ x Ͻ ϱ, Ϫ͞2 Ͻ y Ͻ ͞2 y y Ϫ1 x Ϫ1 Ϫ Ϫ2 y ϭ sinϪ1 x Domain ϭ [Ϫ1, 1] Range ϭ [Ϫ , ] Ϫ1 y ϭ cosϪ1 x Domain ϭ [Ϫ1, 1] Range ϭ [0, ] x y ϭ tanϪ1 x Domain ϭ (Ϫϱ, ϱ) Range ϭ Ϫ , x bar19502_ch06_385-460.qxd 450 11/23/09 CHAPTER 5:54 PM Page 450 TRIGONOMETRIC FUNCTIONS Z Inverse Cotangent, Secant, and Cosecant Functions (Optional) For completeness, we include the definitions and graphs of the inverse cotangent, secant, and cosecant functions Z DEFINITION Inverse Cotangent, Secant, and Cosecant Functions y ϭ cotϪ1 x y ϭ secϪ1 x y ϭ cscϪ1 x x ϭ cot y x ϭ sec y x ϭ csc y is equivalent to is equivalent to is equivalent to where y , Ϫϱ x ϱ where Յ y Յ , y ր2, |x| Ն where Ϫր2 Յ y Յ ր2, y 0, |x| Ն y y y yϭ Ϫ2 Ϫ1 x secϪ1 Domain: All real numbers Range: Ͻ y Ͻ Ϫ2 Ϫ1 Ϫ2 Ϫ1 y ϭ cscϪ1 x x y ϭ cotϪ1 x x Domain: x Յ Ϫ1 or x Ն Range: Յ y Յ , y /2 Ϫ x Domain: x Յ Ϫ1 or x Ն Range: Ϫ/2 Յ y Յ /2, y [Note: The domain restrictions used in defining secϪ1 and cscϪ1 are not universally agreed upon.] ANSWERS TO MATCHED PROBLEMS (A) ͞4 (A) ͞4 (A) Ϫ͞3 6-6 (B) Ϫ0.4 (B) 3.05 (B) 43 (C) Ϫ1͞2 (C) Ϫ1͞2 (B) Not defined (B) Not defined (C) Ϫ2.724 (C) Ϫ0.5829 Exercises Unless stated to the contrary, the inverse trigonometric functions are assumed to have real number ranges (use radian mode in calculator problems) A few problems involve ranges with angles in degree measure, and these are clearly indicated (use degree mode in calculator problems) Explain why the function f (x) ϭ sin x, for Յ x Յ , has no inverse Explain why the function f (x) ϭ cos x, for Ϫ/2 Յ x Յ /2, has no inverse Does tan (tan (A) 0.2945 (A) 0.8267 Ϫ1 x) ϭ x for all real x? Explain Does tan Ϫ1 (tan x) ϭ x for all real x? Explain If a function f has an inverse, how are the graphs of f and f Ϫ1 related? If f is increasing, is f Ϫ1 also increasing? Explain In Problems 7–18, find exact values without using a calculator cosϪ1 arcsin (13ր2) sinϪ1 10 arccos (13ր2) bar19502_ch06_385-460.qxd 11/20/09 6:21 PM Page 451 SECTION 6–6 tanϪ1 13 sinϪ1 (12ր2) cosϪ1 (12) 15 arccos 16 arctan (1ր 13) 17 18 tanϪ1 In Problems 59–64, determine whether the statement is true or false Explain In Problems 19–24, evaluate to four significant digits using a calculator 19 sinϪ1 0.9103 20 cosϪ1 0.4038 21 arctan 103.7 22 tanϪ1 43.09 23 arccos 3.051 24 arcsin 1.131 In Problems 25–30, find the exact value without using a calculator if the expression is defined 25 arctan (Ϫ13) 26 arccos (Ϫ13) 27 cosϪ1 (Ϫ 12ր2) 28 sinϪ1 (Ϫ13ր2) 29 arcsin (Ϫ2) 30 arctan (Ϫ1) In Problems 31–34, evaluate to four significant digits using a calculator Ϫ1 31 cot [cos (Ϫ0.7003)] 33 25 ϩ cos Ϫ1 (1 Ϫ 12) Ϫ1 32 sec [sin (Ϫ0.0399)] 34 12 ϩ tanϪ1 In Problems 35–46, find the exact value without using a calculator if the expression is defined 35 tan (tanϪ1 15) 36 tan [tanϪ1 (Ϫ10)] 37 sin [sinϪ1 (Ϫ1ր 13)] 38 sin (sinϪ1 15) 39 cos [cosϪ1 (Ϫ 12)] 40 cos [cosϪ1 (Ϫ1)] 41 sinϪ1 (sin 1.5) 42 sinϪ1 [sin (3ր2)] 43 cos Ϫ1 [ cos (Ϫ)] 44 cosϪ1 [cos (Ϫ ր5)] 45 tan Ϫ1 [tan ( ր2) ] 46 tan Ϫ1 [tan (5ր4)] 59 None of the six trigonometric functions is one-to-one 60 Each of the six inverse trigonometric functions is one-to-one 61 Each of the six inverse trigonometric functions is periodic 62 Each of the six inverse trigonometric functions is bounded 63 The function y ϭ sinϪ1 x is odd 64 The function y ϭ cosϪ1 x is even In Problems 65–72, graph each function over the indicated interval 65 y ϭ sinϪ1 x, Ϫ1 Յ x Յ 66 y ϭ cosϪ1 x, Ϫ1 Յ x Յ 67 y ϭ cosϪ1 (xր3), Ϫ3 Յ x Յ 68 y ϭ sinϪ1 (x͞2), Ϫ2 Յ x Յ 69 y ϭ sinϪ1 (x Ϫ 2), Յ x Յ 70 y ϭ cosϪ1 (x ϩ 1), Ϫ2 Յ x Յ 71 y ϭ tanϪ1 (2x Ϫ 4), Ϫ2 Յ x Յ 72 y ϭ tanϪ1 (2x ϩ 3), Ϫ5 Յ x Յ 73 The identity cos (cosϪ1 x) ϭ x is valid for Ϫ1 Յ x Յ (A) Graph y ϭ cos (cosϪ1 x) for Ϫ1 Յ x Յ (B) What happens if you graph y ϭ cos (cosϪ1 x) over a larger interval, say Ϫ2 Յ x Յ 2? Explain 74 The identity sin (sinϪ1 x) ϭ x is valid for Ϫ1 Յ x Յ (A) Graph y ϭ sin (sinϪ1 x) for Ϫ1 Յ x Յ (B) What happens if you graph y ϭ sin (sinϪ1 x) over a larger interval, say Ϫ2 Յ x Յ 2? Explain In Problems 47–52, find the exact degree measure without using a calculator if the expression is defined In Problems 75–78, write each expression as an algebraic expression in x free of trigonometric or inverse trigonometric functions 47 sinϪ1 (Ϫ12) 75 cos (sinϪ1 x) 76 sin (cosϪ1 x) 77 cos (arctan x) 78 tan (arcsin x) Ϫ1 49 tan (Ϫ1) 51 arccos (Ϫ13ր2) 48 cosϪ1 (Ϫ1) Ϫ1 50 sin (Ϫ 12) 52 arctan (Ϫ13) In Problems 53–56, find the degree measure of each to two decimal places using a calculator set in degree mode Ϫ1 53 cos 0.7253 55 arcsin (Ϫ0.3662) 451 58 Evaluate cosϪ1 [cos (Ϫ0.5)] with a calculator set in radian mode, and explain why this does or does not illustrate the inverse cosine–cosine identity 11 arctan 13 sinϪ1 (12) Inverse Trigonometric Functions Ϫ1 54 tan 12.4304 56 arccos (Ϫ0.9206) 57 Evaluate sinϪ1 (sin 2) with a calculator set in radian mode, and explain why this does or does not illustrate the inverse sine–sine identity In Problems 79 and 80, find f Ϫ1 (x) How must x be restricted in f Ϫ1(x)? 79 f (x) = ϩ cos (x Ϫ 3), Յ x Յ (3 ϩ ) 80 f (x) ϭ ϩ sin (x Ϫ 1), (1 Ϫ ր2) Յ x Յ (1 ϩ ր2) 81 The identity cosϪ1 (cos x) ϭ x is valid for Յ x Յ (A) Graph y ϭ cosϪ1 (cos x) for Յ x Յ (B) What happens if you graph y ϭ cosϪ1 (cos x) over a larger interval, say Ϫ2 Յ x Յ 2? Explain bar19502_ch06_385-460.qxd 452 12/11/09 CHAPTER 2:35 PM Page 452 TRIGONOMETRIC FUNCTIONS 82 The identity sinϪ1 (sin x) ϭ x is valid for Ϫ ր2 Յ x Յ ր2 (A) Graph y ϭ sinϪ1 (sin x) for Ϫ ր2 Յ x Յ ր2 (B) What happens if you graph y ϭ sinϪ1 (sin x) over a larger interval, say Ϫ2 Յ x Յ 2? Explain Verify these formulas, and find the length of the belt to two decimal places if D ϭ inches, d ϭ inches, and C ϭ inches C APPLICATIONS 83 PHOTOGRAPHY The viewing angle changes with the focal length of a camera lens A 28-millimeter wide-angle lens has a wide viewing angle and a 300-millimeter telephoto lens has a narrow viewing angle For a 35-millimeter format camera the viewing angle , in degrees, is given by ϭ tanϪ1 21.634 x where x is the focal length of the lens being used What is the viewing angle (in decimal degrees to two decimal places) of a 28-millimeter lens? Of a 100-millimeter lens? d DϾd 88 ENGINEERING For Problem 87, find the length of the belt if D ϭ inches, d ϭ inches, and C ϭ 10 inches 89 ENGINEERING The function y1 ϭ 4 Ϫ cosϪ1 D 1 ϩ 2x sin acosϪ1 b x x represents the length of the belt around the two pulleys in Problem 87 when the centers of the pulleys are x inches apart (A) Graph y1 in a graphing calculator (in radian mode), with the graph covering pulleys with their centers from to 10 inches apart (B) How far, to two decimal places, should the centers of the two pulleys be placed to use a belt 24 inches long? Solve by graphing y1 and y2 ϭ 24 in the same viewing window and finding the point of intersection using the INTERSECT command 90 ENGINEERING The function 84 PHOTOGRAPHY Referring to Problem 83, what is the viewing angle (in decimal degrees to two decimal places) of a 17-millimeter lens? Of a 70-millimeter lens? 85 (A) Graph the function in Problem 83 in a graphing calculator using degree mode The graph should cover lenses with focal lengths from 10 millimeters to 100 millimeters (B) What focal-length lens, to two decimal places, would have a viewing angle of 40°? Solve by graphing ϭ 40 and ϭ tanϪ1 (21.634͞x) in the same viewing window and finding the point of intersection using the INTERSECT command 86 (A) Graph the function in Problem 83 in a graphing calculator, in degree mode, with the graph covering lenses with focal lengths from 100 millimeters to 1,000 millimeters (B) What focal length lens, to two decimal places, would have a viewing angle of 10°? Solve by graphing ϭ 10 and ϭ tanϪ1 (21.634͞x) in the same viewing window and finding the point of intersection using the INTERSECT command y1 ϭ 6 Ϫ cosϪ1 represents the length of the belt around the two pulleys in Problem 88 when the centers of the pulleys are x inches apart (A) Graph y1 in a graphing calculator (in radian mode), with the graph covering pulleys with their centers from to 20 inches apart (B) How far, to two decimal places, should the centers of the two pulleys be placed to use a belt 36 inches long? Solve by graphing y1 and y2 = 36 in the same viewing window and finding the point of intersection using the INTERSECT command 91 MOTION The figure represents a circular courtyard surrounded by a high stone wall A floodlight located at E shines into the courtyard r 87 ENGINEERING The length of the belt around the two pulleys in the figure is given by C r L ϭ D ϩ (d Ϫ D) ϩ 2C sin where (in radians) is given by DϪd ϭ cosϪ1 2C 1 ϩ 2x sin acosϪ1 b x x E x Shadow d A D bar19502_ch06_385-460.qxd 11/20/09 6:21 PM Page 453 Review (A) If a person walks x feet away from the center along DC, show that the person’s shadow will move a distance given by Ϫ1 d ϭ 2r ϭ 2r tan x r 453 (B) Find d to two decimal places if r ϭ 100 feet and x ϭ 40 feet 92 MOTION In Problem 91, find d for r ϭ 50 feet and x ϭ 25 feet where is in radians [Hint: Draw a line from A to C.] CHAPTER 6-1 Review Trigonometric Ratios Angles and Their Measure An angle is formed by rotating (in a plane) a ray m, called the initial side of the angle, around its endpoint until it coincides with a ray n, called the terminal side of the angle The common endpoint of m and n is called the vertex If the rotation is counterclockwise, the angle is positive; if clockwise, negative Two angles are coterminal if they have the same initial and terminal sides An angle is in standard position in a rectangular coordinate system if its vertex is at the origin and its initial side is along the positive x axis Quadrantal angles have their terminal sides on a coordinate axis An angle of degree is 360 of a complete rotation Two positive angles are complementary if their sum is 90°; they are supplementary if their sum is 180° An angle of radian is a central angle of a circle subtended by an arc having the same length as the radius Radian measure: ϭ s r deg rad Radian–degree conversion: ϭ 180° radians If a point P moves through an angle and arc length s, in time t, on the circumference of a circle of radius r, then the (average) linear speed of P is vϭ s t and the (average) angular speed is ϭ t Because s ϭ r it follows that v ϭ r 6-2 Right Triangle Trigonometry A right triangle is a triangle with one 90° angle To solve a right triangle is to find all unknown angles and sides, given the measures of two sides or the measures of one side and an acute angle sin ϭ Opp Hyp csc ϭ Hyp Opp cos ϭ Adj Hyp sec ϭ Hyp Adj tan ϭ Opp Adj cot ϭ Adj Opp Hyp Opp Adj Computational Accuracy Angle to Nearest Significant Digits for Side Measure 1° 10Ј or 0.1° 1Ј or 0.01° 10Љ or 0.001° 6-3 Trigonometric Functions: A Unit Circle Approach If is a positive angle in standard position, and P is the point of intersection of the terminal side of with the unit circle, then the radian measure of equals the length x of the arc opposite ; and if is negative, the radian measure of equals the negative of the length of the intercepted arc The function W that associates with each real number x the point W(x) ϭ P is called the wrapping function, and the point P is called a circular point The function W(x) can be visualized as a wrapping of the real number line, with origin at (1, 0), around the unit circle—the positive real axis is wrapped counterclockwise and the negative real axis is wrapped clockwise—so that each real number is paired with a unique circular point The function W(x) is not one-to-one: for example, each of the real numbers 2k, k any integer, corresponds to the circular point (1, 0) bar19502_ch06_385-460.qxd 454 11/20/09 CHAPTER 6:21 PM Page 454 TRIGONOMETRIC FUNCTIONS v v x cos x ϭ a x P tan x ϭ (1, 0) (1, 0) u u Ϫ1 Ϫ2 v x v 1 (1, 0) Ϫ3 Ϫ1 u Ϫ1 Ϫ1 Ϫ2 a cot x ϭ a b b The trigonometric functions of any multiple of ր6 or ր4 can be determined exactly from the coordinates of the circular point A graphing calculator can be used to graph the trigonometric functions and approximate their values for arbitrary inputs 6-4 (1, 0) u a a x b a sec x ϭ Ϫ2 Properties of Trigonometric Functions The definition of the trigonometric functions implies that the following basic identities hold true for all replacements of x by real numbers for which both sides of an equation are defined: Reciprocal identities Ϫ2 csc x ϭ v sin x sec x ϭ cos x cot x ϭ tan x (a, b) Quotient identities x units arc length W(x) x rad (1, 0) tan x ϭ u sin x cos x cot x ϭ cos x sin x Identities for negatives sin (Ϫx) ϭ Ϫsin x cos (Ϫx) ϭ cos x tan (Ϫx) ϭ Ϫtan x The coordinates of key circular points in the first quadrant can be found using simple geometric facts; the coordinates of the circular point associated with any multiple of ր6 or ր4 can then be determined using symmetry properties (0, 1) sin2 x ϩ cos2 x ϭ A function f is periodic if there exists a positive real number p such that Coordinates of Key Circular Points v Pythagorean identity ( 12 , ͙32 ) ( ͙21 , ͙21 ) ( ͙32 , 12 ) 4 f(x ϩ p) ϭ f (x) for all x in the domain of f The smallest such positive p, if it exists, is called the fundamental period of f, or often just the period of f All the trigonometric functions are periodic u (1, 0) Graph of y ϭ sin x: y The six trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—are defined in terms of the coordinates (a, b) of the circular point W(x) that lies on the terminal side of the angle with radian measure x: sin x ϭ b cos x ϭ b b Ϫ2 Ϫ Ϫ1 Period: 2 Domain: All real numbers Range: [Ϫ1, 1] 2 3 4 x bar19502_ch06_385-460.qxd 11/20/09 6:21 PM Page 455 455 Review Graph of y ϭ csc x: Graph of y = cos x: y y ϭ csc x ϭ y sin x y ϭ sin x Ϫ Ϫ2 2 3 4 x Ϫ1 Ϫ Ϫ2 Ϫ 3 3 Ϫ Ϫ1 2 x Period: 2 Domain: All real numbers Range: [Ϫ1, 1] Period: 2 Domain: All real numbers except k, k an integer Range: All real numbers y such that y Յ Ϫ1 or y Ն Graph of y ϭ tan x: y Graph of y ϭ sec x: y Ϫ2 Ϫ 5 Ϫ 3 Ϫ Ϫ Ϫ1 2 3 5 y ϭ sec x ϭ x cos x y ϭ cos x Ϫ2 Ϫ 3 Ϫ Ϫ Ϫ1 3 2 x Period: Domain: All real numbers except ր2 ϩ k, k an integer Range: All real numbers Period: 2 Domain: All real numbers except ր2 ϩ k, k an integer Range: All real numbers y such that y Յ Ϫ1 or y Ն Graph of y = cot x: y Ϫ2 Ϫ 3 Ϫ Ϫ Ϫ1 3 2 x Associated with each angle that does not terminate on a coordinate axis is a reference triangle for The reference triangle is formed by drawing a perpendicular from point P ϭ (a, b) on the terminal side of to the horizontal axis The reference angle ␣ is the acute angle, always taken positive, between the terminal side of and the horizontal axis as indicated in the following figure b a a ␣ Period: Domain: All real numbers except k, k an integer Range: All real numbers b P ϭ (a, b) Reference Triangle (a, b) (0, 0) ␣ is always positive bar19502_ch06_385-460.qxd 456 6-5 11/20/09 CHAPTER 6:21 PM Page 456 TRIGONOMETRIC FUNCTIONS y ϭ cos Ϫ1 x ϭ arccos x if and only if cos y ϭ x, Յ y Յ and Ϫ1 Յ x Յ More General Trigonometric Functions and Models Let A, B, C be constants such that A and B Ͼ If y ϭ A sin (Bx ϩ C) or y ϭ A cos (Bx ϩ C): Period ϭ Amplitude ϭ |A| 2 B Phase shift ϭ Ϫ y (Ϫ1, ) y ϭ cosϪ1 x ϭ arccos x If y ϭ A sec (Bx ϩ C) or y ϭ csc (Bx ϩ C ): Period ϭ 2 B Phase shift ϭ Ϫ B 0, 2 C B (1, 0) If y ϭ A tan (Bx ϩ C ) or y ϭ A cot (Bx ϩ C ): Period ϭ C B Ϫ1 Phase shift ϭ Ϫ C B (Amplitude is not defined for the secant, cosecant, tangent, and cotangent functions, all of which are unbounded.) Sinusoidal regression is used to find the function of the form y ϭ A sin (Bx ϩ C ) ϩ k that best fits a set of data points x Domain ϭ [Ϫ1, 1] Range ϭ [0, ] Inverse cosine function y ϭ tanϪ1 x ϭ arctan x if and only if tan y ϭ x, Ϫ͞2 Ͻ y Ͻ ͞2 and x is any real number y 6-6 Inverse Trigonometric Functions y ϭ tanϪ1 x ϭ arctan x Ϫ1 y ϭ sin x ϭ arcsin x if and only if sin y ϭ x, Ϫ͞2 Յ y Յ ͞2 and Ϫ1 Յ x Յ 1, 4 y 1, (0, 0) Ϫ1 Ϫ1, Ϫ2 y ϭ sinϪ1 x ϭ arcsin x x Domain ϭ [Ϫ1, 1] Range ϭ [Ϫ , ] Ϫ1, Ϫ Ϫ1 Ϫ x Domain ϭ (Ϫϱ, ϱ) Range ϭ Ϫ , Inverse tangent function Inverse sine function CHAPTER Review Exercises Work through all the problems in this chapter review and check answers in the back of the book Answers to all review problems are there, and following each answer is a number in italics indicating the section in which that type of problem is discussed Where weaknesses show up, review appropriate sections in the text Find the radian measure of a central angle opposite an arc 15 centimeters long on a circle of radius centimeters In a circle of radius centimeters, find the length of an arc opposite an angle of 2.5 radians bar19502_ch06_385-460.qxd 11/20/09 6:21 PM Page 457 Review Exercises Solve the triangle: 20.2 feet ␣ 16 Indicate whether the angle is a Quadrant I, II, III, or IV angle or a quadrantal angle (A) Ϫ210° (B) 5͞2 (C) 4.2 radians b 35.2Њ 17 Which of the following angles are coterminal with 120°? (A) Ϫ240° (B) Ϫ7͞6 (C) 840° a 18 Which of the following have the same value as cos 3? (A) cos 3° (B) cos (3 radians) (C) cos (3 ϩ 2) Find the reference angle associated with each angle (A) ϭ ͞3 (B) ϭ Ϫ120° (C) ϭ Ϫ13͞6 (D) ϭ 210° 19 For which values of x, Յ x Յ 2, is each of the following not defined? (A) tan x (B) cot x (C) csc x In which quadrants is each negative? (A) sin (B) cos (C) tan If (4, Ϫ3) is on the terminal side of angle , find (A) sin (B) sec (C) cot Complete Table using exact values Do not use a calculator Table ° rad sin cos tan 0° csc sec ND* ր 12 ͞4 20 A circular point P ϭ (a, b) moves clockwise around the circumference of a unit circle starting at (1, 0) and stops after covering a distance of 8.305 units Explain how you would find the coordinates of point P at its final position and how you would determine which quadrant P is in Find the coordinates of P to three decimal places and the quadrant for the final position of P cot 30° 45° 457 In Problems 21–36, evaluate exactly without the use of a calculator 21 tan 22 sec 90° 23 cosϪ1 24 cos aϪ 60° 3 b 12 26 csc 300° 270° 27 arctan 13 28 sin 570° 360° 29 tanϪ1 (Ϫ1) 30 cot aϪ 31 arcsin aϪ b 32 cosϪ1 aϪ 33 cos (cosϪ1 0.33) 34 csc [tanϪ1 (Ϫ1)] 35 sin c arccos aϪ b d 36 tan asinϪ1 90° 25 sinϪ1 180° *ND ϭ Not defined What is the period of each of the following? (A) y ϭ cos x (B) y ϭ csc x (C) y ϭ tan x Indicate the domain and range of each (A) y ϭ sin x (B) y ϭ tan x 10 Sketch a graph of y ϭ sin x, Ϫ2 Յ x Յ 2 4 b 13 b Ϫ4 b 11 Sketch a graph of y ϭ cot x, Ϫ Ͻ x Ͻ Evaluate Problems 37–44 to four significant digits using a calculator 12 Verbally describe the meaning of a central angle in a circle with radian measure 0.5 37 cos 423.7° 38 tan 93°46Ј17Љ 13 Describe the smallest shift of the graph of y ϭ sin x that produces the graph of y ϭ cos x 39 sec (Ϫ2.073) 40 sinϪ1 (Ϫ0.8277) 41 arccos (Ϫ1.3281) 42 tanϪ1 75.14 14 Change 1.37 radians to decimal degrees to two decimal places 43 csc [cosϪ1 (Ϫ0.4081)] 44 sinϪ1 (tan 1.345) 15 Solve the triangle: 45 Find the exact degree measure of each without a calculator (A) ϭ sinϪ1(Ϫ12) (B) ϭ arccos (Ϫ12) c  15.7 cm ␣ 13.3 cm 46 Find the degree measure of each to two decimal places using a calculator (A) ϭ cosϪ1 (Ϫ0.8763) (B) ϭ arctan 7.3771 47 Evaluate cosϪ1 [cos (Ϫ2)] with a calculator set in radian mode, and explain why this does or does not illustrate the inverse cosine–cosine identity bar19502_ch06_385-460.qxd 458 11/20/09 6:21 PM CHAPTER Page 458 TRIGONOMETRIC FUNCTIONS 48 Sketch a graph of y ϭ Ϫ2 cos x, Ϫ1 Յ x Յ Indicate amplitude A and period P In Problems 60 and 61, determine whether the statement is true or false If true, explain why If false, give a counterexample 49 Sketch a graph of y ϭ Ϫ2 ϩ sin (x͞2), Ϫ4 Յ x Յ 4 60 If ␣ and  are the acute angles of a right triangle, then sin ␣ ϭ csc  50 Find the equation of the form y ϭ A cos Bx that has the graph shown here y 62 If in the figure the coordinates of A are (8, 0) and arc length s is 20 units, find: (A) The exact radian measure of (B) The coordinates of P to three significant digits Ϫ 61 If ␣ and  are the acute angles of a right triangle and ␣ ϭ , then all six trigonometric functions of ␣ are greater than 12 and less than 32 x s P ϭ (a, b) Ϫ6 A 51 Find the equation of the form y ϭ A sin Bx that has the graph shown here y 63 Find exactly the least positive real number for which (A) cos x ϭ Ϫ12 (B) csc x ϭ Ϫ 12 Ϫ1 x Ϫ1 52 Describe the smallest shift and/or reflection that transforms the graph of y ϭ tan x into the graph of y ϭ cot x 53 Simplify each of the following using appropriate basic identities: sin2 x (A) sin (Ϫx) cot (Ϫx) (B) Ϫ sin2 x 54 Sketch a graph of y ϭ sin [(x͞2) ϩ (͞2)] over the interval Ϫ4 Յ x Յ 4 55 Indicate the amplitude A, period P, and phase shift for the graph of y ϭ Ϫ2 cos [(͞2) x Ϫ (͞4)] Do not graph 56 Sketch a graph of y ϭ cosϪ1 x, and indicate the domain and range 57 Graph y ϭ 1͞(1 ϩ tan2 x) in a graphing calculator that displays at least two full periods of the graph Find an equation of the form y ϭ k ϩ A sin Bx or y ϭ k ϩ A cos Bx that has the same graph 58 Graph each equation in a graphing calculator and find an equation of the form y ϭ A tan Bx or y ϭ A cot Bx that has the same graph as the given equation Select the dimensions of the viewing window so that at least two periods are visible sin2 x cos2 x (A) y ϭ (B) y ϭ sin 2x sin 2x 59 Determine whether each function is even, odd, or neither 1 (A) f (x) ϭ (B) g(x) ϭ ϩ tan x ϩ tan2 x 64 Sketch a graph of y ϭ sec x, Ϫ͞2 Ͻ x Ͻ 3͞2 65 Sketch a graph of y ϭ tanϪ1 x, and indicate the domain and range 66 Indicate the period P and phase shift for the graph of y ϭ Ϫ5 tan (x ϩ ͞2) Do not graph 67 Indicate the period and phase shift for the graph of y ϭ csc (x͞2 Ϫ ͞4) Do not graph 68 Indicate whether each is symmetrical with respect to the x axis, y axis, or origin (A) Sine (B) Cosine (C) Tangent 69 Write as an algebraic expression in x free of trigonometric or inverse trigonometric functions: sec (sinϪ1 x) 70 Try to calculate each of the following on your calculator Explain the results (A) csc (Ϫ) (B) tan (Ϫ3͞2) (C) sinϪ1 71 The accompanying graph is a graph of an equation of the form y ϭ A sin (Bx ϩ C ), Ϫ1 Ͻ Ϫ C͞B Ͻ Find the equation y Ϫ Ϫ 4 Ϫ2 x bar19502_ch06_385-460.qxd 12/5/09 9:06 PM Page 459 459 Review Exercises 72 Graph y ϭ 1.2 sin 2x ϩ 1.6 cos 2x in a graphing calculator (Select the dimensions of the viewing window so that at least two periods are visible.) Find an equation of the form y ϭ A sin (Bx ϩ C) that has the same graph as the given equation Find A and B exactly and C to three decimal places Use the x intercept closest to the origin as the phase shift 73 A particular waveform is approximated by the first six terms of a Fourier series: yϭ sin 3x sin 5x sin 7x sin 9x sin 11x asin x ϩ ϩ ϩ ϩ ϩ b 11 (A) Express the length L of the line that touches the two outer sides of the canal and the inside corner in terms of (B) Complete Table 2, each to one decimal place, and estimate from the table the longest log to the nearest foot that can make it around the corner (The longest log is the shortest distance L.) Table (radians) 0.4 L (feet) 42.0 0.5 0.6 0.7 0.8 0.9 1.0 (A) Graph this equation in a graphing calculator for Ϫ3 Յ x Յ 3 and Ϫ2 Յ y Յ (B) The graph in part A approximates a waveform that is made up entirely of straight line segments Sketch by hand the waveform that the Fourier series approximates (C) Graph the function in part A in a graphing calculator and use the MINIMUM command to find the shortest distance L to one decimal place; the length of the longest log that can make it around the corner (D) Explain what happens to the length L as approaches or ͞2 This waveform is called a pulse wave or a square wave, and is used, for example, to test distortion and to synchronize operations in computers 79 MODELING SEASONAL BUSINESS CYCLES A soft drink company has revenues from sales over a 2-year period as shown by the accompanying graph, where R(t) is revenue (in millions of dollars) for a month of sales t months after February (A) Find an equation of the form R(t) ϭ k ϩ A cos Bt that produces this graph, and check the result by graphing (B) Verbally interpret the graph APPLICATIONS 74 ASTRONOMY A line from the sun to the Earth sweeps out an angle of how many radians in 73 days? Express the answer in terms of R(t ) 75 GEOMETRY Find the perimeter of a square inscribed in a circle of radius 5.00 centimeters 76 ANGULAR SPEED A wind turbine of rotor diameter 40 feet makes 80 revolutions per minute Find the angular speed (in radians per second) and the linear speed (in feet per second) of the rotor tip 77 ALTERNATING CURRENT The current I in alternating electrical current has an amplitude of 30 amperes and a period of 601 second If I ϭ 30 amperes when t ϭ 0, find an equation of the form I ϭ A cos Bt that gives the current at any time t Ն 78 RESTRICTED ACCESS A 10-foot-wide canal makes a right turn into a 15-foot-wide canal Long narrow logs are to be floated through the canal around the right angle turn (see the figure) We are interested in finding the longest log that will go around the corner, ignoring the log’s diameter 12 24 t 80 MODELING TEMPERATURE VARIATION The 30-year average monthly temperature, °F, for each month of the year for Los Angeles is given in Table (World Almanac) (A) Using month as the basic unit of time, enter the data for a 2-year period in your graphing calculator and produce a scatter plot in the viewing window Choose 40 Յ y Յ 90 for the viewing window (B) It appears that a sine curve of the form y ϭ k ϩ A sin (Bx ϩ C) will closely model these data The constants k, A, and B are easily determined from Table To estimate C, visually estimate to one decimal place the smallest positive phase shift from the plot in part A After determining A, B, k, and C, write the resulting equation (Your value of C may differ slightly from the answer at the back of the book.) (C) Plot the results of parts A and B in the same viewing window (An improved fit may result by adjusting your value of C slightly.) (D) If your graphing calculator has a sinusoidal regression feature, check your results from parts B and C by finding and plotting the regression equation 15 ft L Canal 10 ft Table x (months) 10 11 12 y (temperature) 58 60 61 63 66 70 74 75 74 70 63 58 bar19502_ch06_385-460.qxd 460 11/20/09 CHAPTER CHAPTER ZZZ 6:21 PM Page 460 TRIGONOMETRIC FUNCTIONS GROUP ACTIVITY A Predator–Prey Analysis Involving Mountain Lions and Deer Use sinusoidal regression to find a function of the form y ϭ k ϩ A sin (Bx ϩ C) that models the data, and plot the function and the data Write an analysis of the fluctuations and cycles of the deer population (B) Mountain Lion Population Analysis Repeat 1, 2, and of part (A) for the mountain lion data (C) Interrelationship of the Two Populations Discuss the dynamics of the two interdependent populations What causes the two populations to rise and fall, and why are they out of phase with each other? In some western state wilderness areas, deer and mountain lion populations are interrelated, because the mountain lions rely on the deer as a food source The population of each species goes up and down in cycles, but out of phase with each other A wildlife management research team estimated the respective populations in a particular region every years over a 16-year period, with the results shown in Table (A) Deer Population Analysis Enter the deer population data for the time interval [0, 16] in a graphing calculator and produce a scatter plot of the data Table Mountain Lion/Deer Populations Years 10 12 14 16 Deer 1,272 1,523 1,152 891 1,284 1,543 1,128 917 1,185 Mountain Lions 39 47 63 54 37 48 60 46 40 ... irrational number 12 Ϸ 1. 414 213 562 is approximated by the rational numbers 14 10 14 1 10 0 1, 414 1, 000 14 ,14 2 10 ,000 14 1,4 21 100,000 ϭ 1. 4 ϭ 1. 41 ϭ 1. 414 ϭ 1. 414 2 ϭ 1. 414 21 Using the idea... integers 1 ϩ 1 ϩ 3 Ϫ 4 Ϫ 5 ؒ aϪ bؒ 10 11 Ϭ 10 0 Ϭ 11 aϪ b aϪ b 13 17 ؒ ? ?1 15 a b ϩ 2? ?1 8 Ϭ 10 Ϭ 12 Ϭ a3 Ϫ b 2 14 aϪ b aϪ b 16 Ϫ(4? ?1 ϩ 3) bar19502_chR_0 01- 042.qxd 10 10 /12 /09 4:39 PM CHAPTER R Page 10 ... (B) 11 21 ϭ 11 (D) (? ?16 )1? ?4 is undefined (not a real number) (F) 13 2 ϭ Evaluate each expression: (A) 81? ?3 (B) 1? ?4 (C) 11 0,000 (D) (? ?1) 1ր5 (E) 1? ?27 (F) 01? ?8 bar19502_chR_0 01- 042.qxd 16 10 /12 /09