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(BQ) Part 1 book College algebra trigonometry has contents: Equations and inequalities, graphs and functions, polynomial and rational functions, inverse, exponential, and logarithmic functions, trigonometric functions.
www.downloadslide.com Global edition College Algebra & Trigonometry For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition sixth edition Lial Hornsby Schneider Daniels College Algebra & Trigonometry Sixth edition Margaret L Lial • John Hornsby • David I Schneider • Callie J Daniels G LOBa l edition This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author Pearson Global Edition Lial_06_1292151951_Final.indd 23/09/16 11:45 AM www.downloadslide.com Get the most out of MyMathLab® MyMathLab, Pearson’s online learning management system, creates personalized experiences for students and provides powerful tools for instructors With a wealth of tested and proven resources, each course can be tailored to fit your specific needs Talk to your Pearson Representative about ways to integrate MyMathLab into your course for the best results Data-Driven Reporting for Instructors MyMathLab’s comprehensive online gradebook automatically tracks students’ results to tests, quizzes, homework, and work in the study plan • The Reporting Dashboard, found under More Gradebook Tools, makes it easier than ever to identify topics where students are struggling, or specific students who may need extra help • Learning in Any Environment Because classroom formats and student needs continually change and evolve, MyMathLab has built-in flexibility to accommodate various course designs and formats • With a new, streamlined, mobile-friendly design, students and instructors can access courses from most mobile devices to work on exercises and review completed assignments • www.mymathlab.com A00_LIAL1953_06_GE_FEP.indd 07/09/16 4:35 pm www.downloadslide.com CHAPTER 6 Analytic Geometry Available in MyMathLab® for Your Course Achieve Your Potential Success in math can make a difference in your life MyMathLab is a learning experience with resources to help you achieve your potential in this course and beyond MyMathLab will help you learn the new skills required, and also help you learn the concepts and make connections for future courses and careers Visualization and Conceptual Understanding These MyMathLab resources will help you think visually and connect the concepts NEW! Guided Visualizations These engaging interactive figures bring mathematical concepts to life, helping students visualize the concepts through directed explorations and purposeful manipulation Guided Visualizations are assignable in MyMathLab and encourage active learning, critical thinking, and conceptual learning Video Assessment Exercises Assignable in MyMathLab, Example Solution Videos present the detailed solution process for every example in the text Additional Quick Reviews cover definitions and procedures for each section Assessment exercises check conceptual understanding of the mathematics www.mymathlab.com A00_LIAL1953_06_GE_FEP.indd 07/09/16 4:35 pm www.downloadslide.com Preparedness and Study Skills MyMathLab® gives access to many learning resources which refresh knowledge of topics previously learned Getting Ready material and Skills for Success Modules are some of the tools available Getting Ready Students refresh prerequisite topics through skill review quizzes and personalized homework integrated in MyMathLab With Getting Ready content in MyMathLab students get just the help they need to be prepared to learn the new material Skills for Success Modules Skills for Success Modules help foster success in collegiate courses and prepare students for future professions Topics such as “Time Management,” “Stress Management” and “Financial Literacy” are available within the MyMathLab course www.mymathlab.com A00_LIAL1953_06_GE_FEP.indd 07/09/16 4:35 pm www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com College Algebra & Trigonometry SIXTH EDITION GLOBAL EDITION Margaret L Lial American River College John Hornsby University of New Orleans David I Schneider University of Maryland Callie J Daniels St Charles Community College Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am www.downloadslide.com Editorial Director: Chris Hoag MathXL Content Manager: Kristina Evans Editor in Chief: Anne Kelly Marketing Manager: Claire Kozar Editorial Assistant: Ashley Gordon Marketing Assistant: Fiona Murray Program Manager: Danielle Simbajon Senior Author Support/Technology Specialist: Joe Vetere Project Manager: Christine O’Brien Rights and Permissions Project Manager: Gina Cheselka Program Management Team Lead: Karen Wernholm Procurement Specialist: Carol Melville Project Management Team Lead: Peter Silvia Associate Director of Design: Andrea Nix Assistant Acquisitions Editor, Global Edition: Murchana Borthakur Program Design Lead: Beth Paquin Senior Project Editor, Global Edition: Amrita Naskar Text Design: Cenveo® Publisher Services Manager, Media Production, Global Edition: Vikram Kumar Illustrations: Cenveo® Publisher Services Senior Manufacturing Controller, Global Edition: Jerry Kataria Cover Design: Lumina Datamatics Ltd Media Producer: Jonathan Wooding Cover Image: gusenych/Shutterstock TestGen Content Manager: John Flanagan Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2017 The rights of Margaret L Lial, John Hornsby, David I Schneider, and Callie J Daniels to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled College Algebra & Trigonometry, 6th edition, ISBN 978-0-13-411252-7, by Margaret L Lial, John Hornsby, David I Schneider, and Callie J Daniels, published by Pearson Education © 2017 Acknowledgments of third-party content appear on page 1181, which constitutes an extension of this copyright page All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC 1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners TI-84 Plus C screenshots from Texas Instruments Courtesy of Texas Instruments MYMATHLAB, MYMATHLAB PLUS, MATHXL, LEARNING CATALYTICS, and TESTGEN are exclusive trademarks owned by Pearson Education, Inc or its affiliates in the U.S and/or other countries Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos, or other trade dress are for demonstrative or descriptive purposes only Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc or its affiliates, authors, licensees, or distributors ISBN 10: 1-292-15195-1 ISBN 13: 978-1-292-15195-3 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 Typeset by Cenveo® Publisher Services Printed and bound by Vivar in Malaysia A01_LIAL1953_06_GE_FM.indd 22/09/16 5:34 pm www.downloadslide.com To Rhonda, Sandy, and Betty Johnny To my MS & T professors, Gus Garver, Troy Hicks, and Jagdish Patel C.J.D A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am Contents www.downloadslide.com Preface 17 Resources for Success 22 R Review of Basic Conceptsâ•… 27 R.1 Setsâ•… 28 Basic Definitions╇ ■╇ Operations on Sets R.2 Real Numbers and Their Propertiesâ•… 35 Sets of Numbers and the Number Line╇ ■╇Exponents╇■╇ Order of Operations╇ ■ Properties of Real Numbers╇ ■╇ Order on the Number Line╇ ■╇ Absolute Value R.3 Polynomialsâ•… 50 Rules for Exponents╇ ■╇Polynomials╇■╇ Addition and Subtraction╇ ■╇ Multiplication╇ ■╇Division R.4 Factoring Polynomialsâ•… 62 Factoring Out the Greatest Common Factor╇ ■╇ Factoring by Grouping╇ ■╇ Factoring Trinomials╇ ■╇ Factoring Binomials╇ ■╇ Factoring by Substitution R.5 Rational Expressionsâ•… 72 Rational Expressions╇ ■╇ Lowest Terms of a Rational Expression╇ ■╇ Multiplication and Division╇ ■╇ Addition and Subtraction╇ ■╇ Complex Fractions R.6 Rational Exponentsâ•… 81 Negative Exponents and the Quotient Rule╇ ■╇ Rational Exponents╇ ■╇ Complex Fractions Revisited R.7 Radical Expressionsâ•… 92 Radical Notation╇ ■╇ Simplified Radicals╇ ■╇ Operations with Radicals╇ ■╇ Rationalizing Denominators Test Prepâ•… 103╇ ■╇ Review Exercisesâ•… 107╇ ■╇Testâ•… 111 1.1 Equations and Inequalitiesâ•… 113 Linear Equationsâ•… 114 Basic Terminology of Equations╇ ■╇ Linear Equations╇ ■╇ Identities, Conditional Equations, and Contradictions╇ ■╇ Solving for a Specified Variable (Literal Equations) 1.2 Applications and Modeling with Linear Equationsâ•… 120 Solving Applied Problems╇ ■╇ Geometry Problems╇ ■╇ Motion Problems╇ ■╇ Mixture Problems╇ ■╇ Modeling with Linear Equations 1.3 Complex Numbersâ•… 131 Basic Concepts of Complex Numbers╇ ■╇ Operations on Complex Numbers A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am www.downloadslide.com CONTENTS 1.4 Quadratic Equations 139 The Zero-Factor Property ■ The Square Root Property ■ Completing the Square ■ The Quadratic Formula ■ Solving for a Specified Variable ■ The Discriminant Chapter Quiz (Sections 1.1–1.4) 149 1.5 Applications and Modeling with Quadratic Equations 150 Geometry Problems ■ The Pythagorean Theorem ■ Height of a Projected Object ■ Modeling with Quadratic Equations 1.6 Other Types of Equations and Applications 162 Rational Equations ■ Work Rate Problems ■ Equations with Radicals ■ Equations with Rational Exponents ■ Equations Quadratic in Form Summary Exercises on Solving Equations 175 1.7 Inequalities 176 Linear Inequalities ■ Three-Part Inequalities ■ Quadratic Inequalities ■ Rational Inequalities 1.8 Absolute Value Equations and Inequalities 188 Basic Concepts ■ Absolute Value Equations ■ Absolute Value Inequalities ■ Special Cases ■ Absolute Value Models for Distance and Tolerance Test Prep 196 ■ Review Exercises 201 ■ Test 207 Graphs and Functions 209 2.1 Rectangular Coordinates and Graphs 210 Ordered Pairs ■ The Rectangular Coordinate System ■ The Distance Formula ■ The Midpoint Formula ■ Equations in Two Variables 2.2 Circles 221 Center-Radius Form ■ General Form ■ An Application 2.3 Functions 229 Relations and Functions ■ Domain and Range ■ Determining Whether Relations Are Functions ■ Function Notation ■ Increasing, Decreasing, and Constant Functions 2.4 Linear Functions 245 Basic Concepts of Linear Functions ■ Standard Form Ax + By = C ■ Slope ■ Average Rate of Change ■ Linear Models Chapter Quiz (Sections 2.1–2.4) 259 2.5 Equations of Lines and Linear Models 260 Point-Slope Form ■ Slope-Intercept Form ■ Vertical and Horizontal Lines ■ Parallel and Perpendicular Lines ■ Modeling Data ■ Graphical Solution of Linear Equations in One Variable Summary Exercises on Graphs, Circles, Functions, and Equations 273 A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am www.downloadslide.com 576 CHAPTER 5 Trigonometric Functions 54 Cloud Ceiling The U.S Weather Bureau defines a cloud ceiling as the altitude of the lowest clouds that cover more than half the sky To determine a cloud ceiling, a powerful searchlight projects a circle of light vertically on the bottom of the cloud An observer sights the circle of light in the crosshairs of a tube called a clinometer A pendant hanging vertically from the tube and resting on a protractor gives the angle of elevation Find the cloud ceiling if the searchlight is located 1000 ft from the observer and the angle of elevation is 30.0° as measured with a clinometer at eye-height ft (Assume three significant digits.) Cloud Searchlight Observer ft 30.0° 1000 ft 55 Height of Mt Everest The highest mounmi 34 tain peak in the world is Mt Everest, 27 located in the Himalayas The height of u this enormous mountain was determined in 1856 by surveyors using trigonometry 14,545 ft long before it was first climbed in 1953 This difficult measurement had to be done from a great distance At an altitude of 14,545 ft on a different mountain, the straight-line distance to the peak of Mt Everest is 27.0134 mi and its angle of elevation is u = 5.82° (Source: Dunham, W., The Mathematical Universe, John Wiley and Sons.) (a) Approximate the height (in feet) of Mt Everest (b) In the actual measurement, Mt Everest was over 100 mi away and the curvature of Earth had to be taken into account Would the curvature of Earth make the peak appear taller or shorter than it actually is? 56 Error in Measurement A degree may seem like a very small unit, but an error of one degree in measuring an angle may be very significant For example, suppose a laser beam directed toward the visible center of the moon misses its assigned target by 30.0″ How far is it (in miles) from its assigned target? Take the distance from the surface of Earth to that of the moon to be 234,000 mi (Source: A Sourcebook of Applications of School Mathematics by Donald Bushaw et al.) The two methods of expressing bearing can be interpreted using a rectangular coordinate system Suppose that an observer for a radar station is located at the origin of a coordinate system Find the bearing of an airplane located at each point Express the bearing using both methods 57 - 4, 02 61 - 5, 52 58 15, 02 62 - 3, - 32 Solve each problem See Examples and 59 10, 42 60 10, - 22 63 12, - 22 65 Distance Flown by a Plane A plane flies 1.3 hr at 110 mph on a bearing of 38° It then turns and flies 1.5 hr at the same speed on a bearing of 128° How far is the plane from its starting point? 64 12, 22 N 128° N 38° x M06_LIAL1953_06_GE_C05.indd 576 31/08/16 4:24 pm www.downloadslide.com 5.4 Solutions and Applications of Right Triangles 577 66 Distance Traveled by a Ship A ship travels 55 km on a bearing of 27° and then travels on a bearing of 117° for 140 km Find the distance from the starting point to the ending point N 117° N 27° 55 km 140 km x 67 Distance between Two Sailors Two sailors leave a port at the same time The first sailor sails on a bearing of 30° at 16 knots (nautical miles per hour) and the second on a bearing of 120° at 20 knots How far apart are they after 2.5 hr? 68 Distance between Two Boats Two boats leave a port at the same time The first boat sails on a bearing of 55° at 14 knots and the second on a bearing of 325° at 20 knots How far apart are they after 1.5 hr? 69 Distance between Two Docks Two docks are located on an east-west line 2587 ft apart From dock A, the bearing of a coral reef is 58° 22′ From dock B, the bearing of the coral reef is 328° 22′ Find the distance from dock A to the coral reef 70 Distance between Two Lighthouses Two lighthouses are located on a north-south line From lighthouse A, the bearing of a ship 3742 m away is 129° 43′ From lighthouse B, the bearing of the ship is 39° 43′ Find the distance between the lighthouses 71 Distance between Two Ships A ship leaves its home port and sails on a bearing of S 61° 50′ E Another ship leaves the same port at the same time and sails on a bearing of N 28° 10′ E If the first ship sails at 24.0 mph and the second ship sails at 28.0 mph, find the distance between the two ships after hr N 28° 10Ј x W E 61° 50Ј S 72 Distance between Transmitters Radio direction finders are set up at two points A and B, which are 2.50 mi apart on an east-west line From A, it is found that the bearing of a signal from a radio transmitter is N 36° 20′ E , and from B the bearing of the same signal is N 53° 40′ W Find the distance of the transmitter from B N Transmitter N 36° 20′ 53° 40′ 2.50 mi A B 73 Flying Distance The bearing from A to C is S 52° E The bearing from A to B is N 84° E The bearing from B to C is S 38° W A plane flying at 250 mph takes 2.4 hr to go from A to B Find the distance from A to C M06_LIAL1953_06_GE_C05.indd 577 31/08/16 4:24 pm www.downloadslide.com 578 CHAPTER 5 Trigonometric Functions 74 Flying Distance The bearing from A to C is N 64° W The bearing from A to B is S 82° W The bearing from B to C is N 26° E A plane flying at 350 mph takes 1.8 hr to go from A to B Find the distance from B to C 75 Distance between Two Cities The bearing from Winston-Salem, North Carolina, to Danville, Virginia, is N 42° E The bearing from Danville to Goldsboro, North Carolina, is S 48° E A car traveling at 65 mph takes 1.1 hr to go from Winston-Salem to Danville and 1.8 hr to go from Danville to Goldsboro Find the distance from Winston-Salem to Goldsboro 76 Distance between Two Cities The bearing from Atlanta to Macon is S 27° E, and the bearing from Macon to Augusta is N 63° E An automobile traveling at 62 mph needs hr to go from Atlanta to Macon and hr to go from Macon to Augusta Find the distance from Atlanta to Augusta Solve each problem See Examples and 77 Height of a Pyramid The angle of elevation from a point on the ground to the top of a pyramid is 35° 30′ The angle of elevation from a point 135 ft farther back to the top of the pyramid is 21° 10′ Find the height of the pyramid h 35° 30′ 21° 10′ 135 ft 78 Distance between a Whale and a Lighthouse A whale researcher is watching a whale approach directly toward a lighthouse as she observes from the top of this lighthouse When she first begins watching the whale, the angle of depression to the whale is 15° 50′ Just as the whale turns away from the lighthouse, the angle of depression is 35° 40′ If the height of the lighthouse is 68.7 m, find the distance traveled by the whale as it approached the lighthouse 15° 50′ 35° 40′ 68.7 m x 79 Height of an Antenna A scanner antenna is on top of the center of a house The angle of elevation from a point 28.0 m from the center of the house to the top of the antenna is 27° 10′, and the angle of elevation to the bottom of the antenna is 18° 10′ Find the height of the antenna 80 Height of Mt Whitney The angle of elevation from Lone Pine to the top of Mt Whitney is 10° 50′ A hiker, traveling 7.00 km from Lone Pine along a straight, level road toward Mt Whitney, finds the angle of elevation to be 22° 40′ Find the height of the top of Mt Whitney above the level of the road 81 Find h as indicated in the figure 82. Find h as indicated in the figure h h 29.5° 392 ft 49.2° 41.2° 52.5° 168 m M06_LIAL1953_06_GE_C05.indd 578 31/08/16 4:24 pm www.downloadslide.com 5.4 Solutions and Applications of Right Triangles 579 83 Distance of a Plant from a Fence In one area, the lowest angle of elevation of the sun in winter is 23° 20′ Find the minimum distance x that a plant needing full sun can be placed from a fence 4.65 ft high 84 Distance through a Tunnel A tunnel is to be built from A to B Both A and B are visible from C If AC is 1.4923 mi and BC is 1.0837 mi, and if C is 90°, find the measures of angles A and B 4.65 ft 23° 20Ј x Plant Tunnel A B 1.4923 mi 1.0837 mi C 85 Height of a Plane above Earth Find the minimum height h above the surface of Earth so that a pilot at point A in the figure can see an object on the horizon at C, 125 mi away Assume 4.00 * 10 mi as the radius of Earth 125 mi C A h B NOT TO SCALE 86 Length of a Side of a Piece of Land A piece of land has the shape shown in the figure Find the length x 198.4 m 52° 20Ј x 30° 50Ј 87 (Modeling) Distance between Two Points A variation of the subtense bar method that surveyors use to determine larger distances d between two points P and Q is shown in the figure The subtense bar with length b is placed between points P and Q so that the bar is centered on and perpendicular to the line of sight between P and Q Angles a and b are measured from points P and Q, respectively (Source: Mueller, I and K Ramsayer, Introduction to Surveying, Frederick Ungar Publishing Co.) P b/2 a b/2 b Q (a) Find a formula for d involving a , b , and b (b) Use the formula from part (a) to determine d if a = 37′ 48″, b = 42′ 03″, and b = 2.000 cm M06_LIAL1953_06_GE_C05.indd 579 31/08/16 4:24 pm www.downloadslide.com 580 CHAPTER 5 Trigonometric Functions 88 (Modeling) Distance of a Shot Put A shot-putter trying to improve performance may wonder whether there is an optimal angle to aim for, or whether the velocity (speed) at which the ball is thrown is more important The figure shows the path of a steel ball thrown by a shot-putter The distance D depends on initial velocity v, height h, and angle u when the ball is released U u h h D One model developed for this situation gives D as D= v sin u cos u + v cos u 21v sin u22 + 64h 32 Typical ranges for the variables are v: 33–46 ft per sec; h: 6–8 ft; and u: 40°9 45° (Source: Kreighbaum, E and K Barthels, Biomechanics, Allyn & Bacon.) (a) To see how angle u affects distance D, let v = 44 ft per sec and h = ft Calculate D, to the nearest hundredth, for u = 40°, 42°, and 45° How does distance D change as u increases? (b) To see how velocity v affects distance D, let h = and u = 42° Calculate D, to the nearest hundredth, for v = 43, 44, and 45 ft per sec How does distance D change as v increases? (c) Which affects distance D more, v or u? What should the shot-putter to improve performance? P Q d S N R M u u 2 R 89 (Modeling) Highway Curves A basic highway curve connecting two straight sections of road may be circular In the figure in the margin, the points P and S mark the beginning and end of the curve Let Q be the point of intersection where the two straight sections of highway leading into the curve would meet if extended The radius of the curve is R, and the central angle u denotes how many degrees the curve turns (Source: Mannering, F and W Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) (a) If R = 965 ft and u = 37°, find the distance d between P and Q C (b) Find an expression in terms of R and u for the distance between points M and N 90 (Modeling) Stopping Distance on a Curve Refer to Exercise 89 When an automobile travels along a circular curve, objects like trees and buildings situated on the inside of the curve can obstruct the driver’s vision In the figure, the minimum distance d that should be cleared on the inside of the highway is modeled by the equation d R u R NOT TO SCALE u d = R a1 - cos b (Source: Mannering, F and W Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) 57.3S (a) It can be shown that if u is measured in degrees, then u ≈ R , where S is the safe stopping distance for the given speed limit Compute d to the nearest foot for a 55 mph speed limit if S = 336 ft and R = 600 ft (b) Compute d to the nearest foot for a 65 mph speed limit given S = 485 ft and R = 600 ft (c) How does the speed limit affect the amount of land that should be cleared on the inside of the curve? M06_LIAL1953_06_GE_C05.indd 580 31/08/16 4:24 pm www.downloadslide.com CHAPTER 5 Test Prep 581 Chapter Test Prep Key Terms 5.1 line line segment (or segment) ray endpoint of a ray angle side of an angle vertex of an angle initial side terminal side positive angle negative angle degree acute angle right angle obtuse angle straight angle complementary angles (complements) supplementary angles (supplements) minute second angle in standard position quadrantal angle coterminal angles 5.2 sine (sin) cosine (cos) tangent (tan) cotangent (cot) secant (sec) cosecant (csc) degree mode reciprocal 5.3 side opposite side adjacent cofunctions reference angle 5.4 exact number significant digits angle of elevation angle of depression bearing New Symbols U right angle symbol (for a right triangle) Greek letter theta ∙ degree ∙ minute ∙ second m Quick Review Concepts 5.1 Examples Angles Types of Angles Two positive angles with a sum of 90° are complementary angles Two positive angles with a sum of 180° are supplementary angles degree = 60 minutes 1∙ ∙ 60∙2 minute = 60 seconds 1∙ ∙ 60∙2 Coterminal angles have measures that differ by a multiple of 360° Their terminal sides coincide when in standard position 70° and 90° - 70° = 20° are complementary 70° and 180° - 70° = 110° are supplementary 15° 30′ 45″ = 15° + 30 ° 45 ° + 60 3600 = 15.5125° 1° 30 # 60′ = 60° and 1° 45 45″ # 3600″ = 3600° 30′ Decimal degrees The acute angle u in the figure is in standard position If u measures 46°, find the measure of a positive and a negative coterminal angle y 46° + 360° = 406° 46° - 360° = - 314° u M06_LIAL1953_06_GE_C05.indd 581 x 31/08/16 4:24 pm www.downloadslide.com 582 CHAPTER 5 Trigonometric Functions Concepts Examples 5.2 Trigonometric Functions Trigonometric Functions Let 1x, y2 be a point other than the origin on the terminal side of an angle u in standard position The distance from the point to the origin is r = 2x + y The six trigonometric functions of u are defined as follows y sin U ∙ r csc U ∙ x cos U ∙ r tan U ∙ y 1x 02 x r r x y 02 sec U ∙ 1x 02 cot U ∙ y 02 y x y If the point - 2, 32 is on the terminal side of an angle u in standard position, find the values of the six trigonometric functions of u Here x = - and y = 3, so r = 21 - 222 + 32 = 24 + = 213 sin u = 3213 13 cos u = - 213 13 tan u = - csc u = 213 sec u = - 213 cot u = - See the summary table of trigonometric function values for quadrantal angles in this section If cot u = - , find tan u Reciprocal Identities sin U ∙ 1 cos U ∙ tan U ∙ csc U sec U cot U csc U ∙ 1 sec U ∙ cot U ∙ sin U cos U tan U tan u = 1 = = cot u -3 Find sin u and tan u, given that cos u = Pythagorean Identities sin2 U ∙ cos2 U ∙ 1 tan2 U ∙ ∙ sec2 U ∙ cot2 U ∙ csc2 U sin2 u + cos2 u = 1 =1 25 22 25 Quotient Identities tan u = 222 - sin u = cos u 23 Signs of the Trigonometric Functions 23 Square Subtract 25 = - 222 23 # 23 23 = - 266 Simplify the complex fraction, and rationalize the denominator x > 0, y > 0, r > I All functions positive II Sine and cosecant positive M06_LIAL1953_06_GE_C05.indd 582 23 To find tan u, use the values of sin u and cos u from sin u above and the quotient identity tan u = cos u sin U cos U ∙ tan U ∙ cot U cos U sin U III Tangent and cotangent positive Replace cos u with 222 Choose the negative root sin u = - x < 0, y < 0, r > Pythagorean identity 23 ≤ = 1 sin2 u = y and sin u sin2 u + ¢ sin2 u + x < 0, y > 0, r > 23 x x > 0, y < 0, r > Identify the quadrant(s) of any angle u that satisfies sin u 0, tan u Because sin u in quadrants III and IV, and tan u in quadrants I and III, both conditions are met only in quadrant III IV Cosine and secant positive 31/08/16 4:24 pm www.downloadslide.com CHAPTER 5 Test Prep 583 Concepts Examples 5.3 Trigonometric Function Values and Angle Measures Right-Triangle-Based Definitions of Trigonometric Functions Let A represent any acute angle in standard position y side opposite hypotenuse r csc A ∙ ∙ sin A ∙ ∙ r hypotenuse y side opposite cos A ∙ side adjacent hypotenuse x r ∙ sec A ∙ ∙ r hypotenuse x side adjacent tan A ∙ y side opposite side adjacent x ∙ cot A ∙ ∙ x side adjacent y side opposite B Hypotenuse 25 A Side opposite A C 24 Side adjacent to A sin A = 24 cos A = tan A = 25 25 24 csc A = 25 25 24 sec A = cot A = 24 Cofunction Identities For any acute angle A, cofunction values of complementary angles are equal sin A ∙ cos 90∙ ∙ A2 cos A ∙ sin 90∙ ∙ A2 sin 55° = cos190° - 55°2 = cos 35° tan A ∙ cot 90∙ ∙ A2 cot A ∙ tan 90∙ ∙ A2 tan 72° = cot190° - 72°2 = cot 18° sec A ∙ csc 90∙ ∙ A2 csc A ∙ sec 90∙ ∙ A2 sec 48° = csc190° - 48°2 = csc 42° Function Values of Special Angles U sin U cos U tan U cot U sec U csc U 30° 23 23 23 45° 22 22 23 23 2 22 22 60° 23 23 Reference Angle U∙ for U in 0∙, 360∙2 23 U in Quadrant I II III IV U∙ is u 180° - u u - 180° 360° - u Finding Trigonometric Function Values for Any Nonquadrantal Angle U Step 1 Add or subtract 360° as many times as needed to obtain an angle greater than 0° but less than 360° Step 2 Find the reference angle u′ Step 3 Find the trigonometric function values for u′ Step 4 Determine the correct signs for the values found in Step To approximate a trigonometric function value of an angle in degrees, make sure the calculator is in degree mode 60° 45° √2 30° 45° √3 30°9 60° right triangle 45°9 45° right triangle Quadrant I: For u = 25°, u′ = 25° Quadrant II: For u = 152°, u′ = 28° Quadrant III: For u = 200°, u′ = 20° Quadrant IV: For u = 320°, u′ = 40° Find sin 1050° Coterminal angle 1050° - 21360°2 = 330° in quadrant IV The reference angle for 330° is u′ = 30° sin 1050° = - sin 30° Sine is negative in quadrant IV = - sin 30° = Approximate each value cos 50° 15′ = cos 50.25° ≈ 0.63943900 csc 32.5° = M06_LIAL1953_06_GE_C05.indd 583 1 ≈ 1.86115900 sin 32.5° csc u = sin u 31/08/16 4:24 pm www.downloadslide.com 584 CHAPTER 5 Trigonometric Functions Concepts Examples To find the corresponding angle measure given a trigonometric function value, use an appropriate inverse function Find an angle u in the interval 30° , 90°2 that satisfies each condition in color cos u ≈ 0.73677482 u ≈ cos-110.736774822 u ≈ 42.542600° csc u ≈ 1.04766792 sin u ≈ 1.04766792 u ≈ sin-1 a u ≈ 72.65° 5.4 sin u = csc1 u b 1.04766792 Solutions and Applications of Right Triangles Solving an Applied Trigonometry Problem Step 1 Draw a sketch, and label it with the given information Label the quantity to be found with a variable Find the angle of elevation of the sun if a 48.6-ft flagpole casts a shadow 63.1 ft long Step 1 See the sketch We must find u Sun Flagpole 48.6 ft u Shadow 63.1 ft 48.6 63.1 Step 2 Use the sketch to write an equation relating the given quantities to the variable Step tan u = Step 3 Solve the equation, and check that the answer makes sense tan u ≈ 0.770206 Expressing Bearing Use one of the following methods Method 1 When a single angle is given, bearing is measured in a clockwise direction from due north Step u = tan - 0.770206 u ≈ 37.6° The angle of elevation rounded to three significant digits is 37.6°, or 37° 40′ Example: 220° Example: S 40° W N Method 2 Start with a north-south line and use an acute angle to show direction, either east or west, from this line 40° 220° S M06_LIAL1953_06_GE_C05.indd 584 31/08/16 4:24 pm www.downloadslide.com CHAPTER 5 Review Exercises Chapter 585 Review Exercises Give the measures of the complement and the supplement of an angle measuring 35° Find the angle of least positive measure that is coterminal with each angle -51° -174° 792° Work each problem Rotating Propeller The propeller of a speedboat rotates 650 times per Through how many degrees does a point on the edge of the propeller rotate in 2.4 sec? Rotating Pulley A pulley is rotating 320 times per Through how many degrees does a point on the edge of the pulley move in sec? Convert decimal degrees to degrees, minutes, seconds, and convert degrees, minutes, seconds to decimal degrees If applicable, round to the nearest second or the nearest thousandth of a degree 119° 08′ 03″ 47° 25′ 11″ 10 -61.5034° 275.1005° Find the six trigonometric function values for each angle Rationalize denominators when applicable 11 12 y y θ x θ 13 y θ x (–2, 0) (1, –√3) x (–3, –3) Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side Rationalize denominators when applicable 14 19, - 22 15 13, - 42 18 A 623, - B 19 A - 222, 222 B 16 11, - 52 17 - 8, 152 An equation of the terminal side of an angle u in standard position is given with a restriction on x Sketch the least positive such angle u, and find the values of the six trigonometric functions of u 20 y = - 5x, x … 21 5x - 3y = 0, x Ú 22 12x + 5y = 0, x Ú Complete the table with the appropriate function values of the given quadrantal angles If the value is undefined, say so M06_LIAL1953_06_GE_C05.indd 585 U sin U cos U tan U cot U sec U csc U 23 180° 24 - 90° 31/08/16 4:24 pm www.downloadslide.com 586 CHAPTER 5 Trigonometric Functions Give all six trigonometric function values for each angle u Rationalize denominators when applicable 23 , and cos u 25 cos u = - , and u is in quadrant III 26 sin u = 27 sec u = - 25, and u is in quadrant II 28 tan u = 2, and u is in quadrant III 29 sec u = , and u is in quadrant IV 30 sin u = - , and u is in quadrant III Find exact values of the six trigonometric functions for each angle A 31 A 32 61 11 58 40 60 A 42 Find exact values of the six trigonometric functions for each angle Do not use a calculator Rationalize denominators when applicable 33 1020° 34 120° 35 - 1470° 36 - 225° Find all values of u, if u is in the interval 30°, 360°2 and u has the given function value 37 cos u = - 38 sin u = - 39 sec u = - 223 40 cot u = - Find the sine, cosine, and tangent function values for each angle 41 y 42. y u 0 x x u (1, –Ë 3) (–3, –3) Use a calculator to approximate the value of each expression Give answers to six decimal places 43 sec 222° 30′ 44 sin 72° 30′ 45 csc 78° 21′ 46 cot 305.6° 47 tan 11.7689° 48 sec 58.9041° Use a calculator to find each value of u, where u is in the interval 30°, 90°2 Give answers in decimal degrees to six decimal places 49 sin u = 0.82584121 50 cot u = 1.1249386 51 cos u = 0.97540415 52 sec u = 1.2637891 53 tan u = 1.9633124 54 csc u = 9.5670466 Find two angles in the interval 30°, 360°2 that satisfy each of the following Round answers to the nearest degree 55 sin u = 0.73135370 M06_LIAL1953_06_GE_C05.indd 586 56 tan u = 1.3763819 31/08/16 4:24 pm www.downloadslide.com CHAPTER 5 Review Exercises 587 Solve each problem 57 A student wants to use a calculator to find the value of cot 25° However, instead of entering tan 25 , he enters tan-1 25 Assuming the calculator is in degree mode, will this produce the correct answer? Explain 58 Explain the process for using a calculator to find sec -1 10 Solve each right triangle In Exercise 60, give angles to the nearest minute In Exercises 61 and 62, label the triangle ABC as in Exercises 59 and 60 59 B 60 B c a c = 748 C 58° 30Ј A b 61 A = 39.72°, b = 38.97 m A b = 368.1 a = 129.7 C 62 B = 47° 53′, b = 298.6 m Solve each problem 63 Height of a Tower The angle of elevation from a point 93.2 ft from the base of a tower to the top of the tower is 38° 20′ Find the height of the tower 38° 20Ј 93.2 ft 64 Height of a Tower The angle of depression from a television tower to a point on the ground 36.0 m from the bottom of the tower is 29.5° Find the height of the tower 29.5° 36.0 m 65 Length of a Diagonal One side of a rectangle measures 15.24 cm The angle between the diagonal and that side is 35.65° Find the length of the diagonal 66 Length of Sides of an Isosceles Triangle The length of each of the two equal sides of an isosceles triangle is 30 m The angle between these sides is 120° Find the length of the third side 67 Distance between Two Points The bearing of point B from point C is 254° The bearing of point A from point C is 344° The bearing of point A from point B is 32° If the distance from A to C is 780 m, find the distance from A to B 68 Distance Traveled by an Airplane The bearing from point A to point B is S 65° E, and the bearing from point B to point C is N 25° E If an airplane travels from A to B, a distance of 90 km, and then from B to C, a distance of 84 km, how far is it from A to C? M06_LIAL1953_06_GE_C05.indd 587 31/08/16 4:24 pm www.downloadslide.com 588 CHAPTER 5 Trigonometric Functions 69 Distance between Two Cities╇ Two buses leave an intersection at the same time One heads due north at 45 mph The other travels due east After hr, they reach cities A in the north and B in the east, respectively The bearing of the bus headed toward city A from the bus headed toward city B is 324° How far apart are the cities A and B? 70 (Modeling) Height of a Satellite╇ Artificial satellites that orbit Earth often use VHF signals to communicate with the ground VHF signals travel in straight lines The height h of the satellite above Earth and the time T that the satellite can communicate with a fixed location on the ground are related by the model h=R¢ cos 180T P - 1≤â•›, where R = 3955 mi is the radius of Earth and P is the period for the satellite to orbit Earth (Source: Schlosser, W., T Schmidt-Kaler, and E Milone, Challenges of Astronomy, Springer-Verlag.) (a)╇Find h to the nearest mile when T = 25 and P = 140 (Evaluate the cosine function in degree mode.) (b)╇What is the value of h to the nearest mile if T is increased to 30 min? Chapter 5 Test Give the measures of the complement and the supplement of an angle measuring 67° Find the measure of each marked angle (7x + 19)° (2x – 1)° (–3x + 5)° (–8x + 30)° Perform each conversion 74° 18′ 36″ to decimal degrees 45.2025° to degrees, minutes, seconds Solve each problem Find the angle of least positive measure that is coterminal with each angle (a)╇390° (b)╇ - 80° (c)╇810° Rotating Tire╇ A tire rotates 450 times per Through how many degrees does a point on the edge of the tire move in sec? Sketch an angle u in standard position such that u has the least positive measure, and the given point is on the terminal side of u Then find the values of the six trigonometric functions for the angle If any of these are undefined, say so 12, - 72 Work each problem 10, - 22 10 Draw a sketch of an angle in standard position having the line with the equation 3x - 4y = 0, x … 0, as its terminal side Indicate the angle of least positive measure u, and find the values of the six trigonometric functions of u M06_LIAL1953_06_GE_C05.indd 588 19/09/16 12:01 pm www.downloadslide.com CHAPTER 5 Test 589 11 Complete the table with the appropriate function values of the given quadrantal angles If the value is undefined, say so sin U cos U tan U cot U sec U csc U 90° -360° 630° U 12 If the terminal side of a quadrantal angle lies along the negative x-axis, which two of its trigonometric function values are undefined? 13 Identify the possible quadrant(s) in which u must lie under the given conditions (a) cos u 0, tan u (b) sin u 0, csc u (c) cot u 0, cos u 14 Find the five remaining trigonometric function values of u if sin u = quadrant II and u is in Solve each problem 15 Find exact values of the six trigonometric functions for angle A in the right triangle A 13 12 16 Find the exact value of each variable in the figure z w 45° x 30° y Find exact values of the six trigonometric functions for each angle Rationalize denominators when applicable 18 -135° 17 240° 19 990° Find all values of u, if u is in the interval 30°, 360°2 and has the given function value 20 cos u = - 22 21 csc u = - 223 22 tan u = Solve each problem 23 How would we find cot u using a calculator, if tan u = 1.6778490? Evaluate cot u 24 Use a calculator to approximate the value of each expression Give answers to six decimal places (a) sin 78° 21′ (b) tan 117.689° (c) sec 58.9041° 25 Find the value of u in the interval 30°, 90°4 in decimal degrees, if sin u = 0.27843196 Give the answer to six decimal places 26 Solve the right triangle A c B M06_LIAL1953_06_GE_C05.indd 589 58° 30Ј b a = 748 C 31/08/16 4:24 pm www.downloadslide.com 590 CHAPTER 5 Trigonometric Functions 27 Antenna Mast Guy Wire A guy wire 77.4 m long is attached to the top of an antenna mast that is 71.3 m high Find the angle that the wire makes with the ground 28 Height of a Flagpole To measure the height of a flagpole, Jan Marie found that the angle of elevation from a point 24.7 ft from the base to the top is 32° 10′ What is the height of the flagpole? 29 Altitude of a Mountain The highest point in Texas is Guadalupe Peak The angle of depression from the top of this peak to a small miner’s cabin at an approximate elevation of 2000 ft is 26° The cabin is located 14,000 ft horizontally from a point directly under the top of the mountain Find the altitude of the top of the mountain to the nearest hundred feet 30 Distance between Two Points Two ships leave a port at the same time The first ship sails on a bearing of 32° at 16 knots (nautical miles per hour) and the second on a bearing of 122° at 24 knots How far apart are they after 2.5 hr? 31 Distance of a Ship from a Pier A ship leaves a pier on a bearing of S 62° E and travels for 75 km It then turns and continues on a bearing of N 28° E for 53 km How far is the ship from the pier? 32 Find h as indicated in the figure h 41.2° 52.5° 168 m M06_LIAL1953_06_GE_C05.indd 590 31/08/16 4:24 pm ... Geometric Definition of Conic Sections Test Prep 10 24 ■ Review Exercises 10 26 ■ Test 10 29 11 Further Topics in Algebra 10 31 11. 1 Sequences and Series 10 32 Sequences ■ Series and Summation Notation ... and Rules 11 .2 Arithmetic Sequences and Series 10 43 Arithmetic Sequences ■ Arithmetic Series A 01_ LIAL1953_06_GE_FM.indd 15 15 /09 /16 10 :40 am www.downloadslide.com 16 CONTENTS 11 .3 Geometric... Probability Test Prepâ•… 11 03╇ ■╇ Review Exercisesâ•… 11 07╇ ■╇Testâ•… 11 11 Appendices╅╇ 11 13 Appendix Aâ•… Polar Form of Conic Sectionsâ•… 11 13 Equations and Graphs╇ ■╇ Conversion from