Pre-Calculus 11 Textbook PDF

McGraw-Hill Ryerson McGraw-Hill Ryerson Pre-Calculus Pre-Calculus Pre-Calculus 11 11 McAskill Watt Balzarini Bonifacio Carlson...

McGraw-Hill Ryerson McGraw-Hill Ryerson Pre-Calculus Pre-Calculus Pre-Calculus 11 11 McAskill Watt Balzarini Bonifacio Carlson 11 Johnson Kennedy Wardrop Explore our Web Site http://www.mcgrawhill.ca McGraw-Hill Ryerson Pre-Calculus Authors Assessment Consultant Advisors Bruce McAskill, B.Sc., B.Ed., M.Ed., Ph.D. Chris Zarski, B.Ed., M.Ed. John Agnew, School District 63 (Saanich), Mathematics Consultant, Victoria, British Wetaskiwin Regional Division No. 11, British Columbia Columbia Alberta Katharine Borgen, School District 39 Wayne Watt, B.Sc., B.Ed., M.Ed. (Vancouver) and University of British Pedagogical Consultant Mathematics Consultant, Winnipeg, Manitoba Columbia, British Columbia Terry Melnyk, B.Ed. Eric Balzarini, B.Sc., B.Ed., M.Ed. Barb Gajdos, Calgary Roman Catholic Edmonton Public Schools, Alberta School District 35 (Langley), British Columbia Separate School District No. 1, Alberta Len Bonifacio, B.Ed. Aboriginal Consultant Sandra Harazny, Regina Roman Catholic Edmonton Catholic Separate School District Chun Ong, B.A., B.Ed. Separate School Division No. 81, No. 7, Alberta Manitoba First Nations Education Resource Saskatchewan Centre, Manitoba Renée Jackson, University of Alberta, Alberta Scott Carlson, B.Ed., B.Sc. Golden Hills School Division No. 75, Alberta Gerald Krabbe, Calgary Board of Education, Differentiated Instruction Consultant Alberta Blaise Johnson, B.Sc., B.Ed. Heather Granger School District 45 (West Vancouver), British Prairie South School Division No. 210, Gail Poshtar, Calgary Catholic School Columbia Saskatchewan District, Alberta Ron Kennedy, B.Ed. Francophone Advisor Mathematics Consultant, Edmonton, Alberta Gifted and Career Consultant Mario Chaput, Pembina Trails School Rick Wunderlich Division Manitoba Harold Wardrop, B.Sc. School District 83 (North Okanagan/ Brentwood College School, Mill Bay Shuswap), British Columbia Luc Lerminiaux, Regina School Division (Independent), British Columbia No. 4, Saskatchewan Math Processes Consultant Contributing Author Inuit Advisor Reg Fogarty Stephanie Mackay Christine Purse, Mathematics Consultant, School District 83 (North Okanagan/ Edmonton Catholic Separate School District British Columbia Shuswap), British Columbia No. 7, Alberta Métis Advisor Technology Consultants Senior Program Consultants Greg King, Northern Lights School Division Ron Kennedy Bruce McAskill, B.Sc., B.Ed., M.Ed., Ph.D. No. 69, Alberta Mathematics Consultant, Edmonton, Alberta Mathematics Consultant, Victoria, British Ron Coleborn Technical Advisor Columbia School District 41 (Burnaby), British Darren Kuropatwa, Winnipeg School Wayne Watt, B.Sc., B.Ed., M.Ed. Columbia Division #1, Manitoba Mathematics Consultant, Winnipeg, Manitoba Toronto Montréal Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan New Delhi Santiago Seoul Singapore Sydney Taipei COPIES OF THIS BOOK McGraw-Hill Ryerson MAY BE OBTAINED BY Pre-Calculus 11 CONTACTING: Copyright © 2011, McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. McGraw-Hill Ryerson Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, or stored in a data base or retrieval system, without the prior written WEB SITE: permission of McGraw-Hill Ryerson Limited, or, in the case of photocopying or other http://www.mcgrawhill.ca reprographic copying, a licence from The Canadian Copyright Licensing Agency (Access Copyright). 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The Geometer’s Sketchpad®, Key Curriculum Press, 1150 65th Street, Emeryville, CA 94608, 1-800-995-MATH. VICE-PRESIDENT, EDITORIAL AND PUBLISHER: Beverley Buxton ASSOCIATE PUBLISHER AND CONTENT MANAGER: Jean Ford PROJECT MANAGER: Janice Dyer DEVELOPMENTAL EDITORS: Maggie Cheverie, Jackie Lacoursiere, Jodi Rauch MANAGER, EDITORIAL SERVICES: Crystal Shortt SUPERVISING EDITOR: Janie Deneau COPY EDITOR: Julie Cochrane PHOTO RESEARCH & PERMISSIONS: Linda Tanaka EDITORIAL ASSISTANT: Erin Hartley EDITORIAL COORDINATION: Jennifer Keay, Janie Reeson, Alexandra Savage-Ferr MANAGER, PRODUCTION SERVICES: Yolanda Pigden PRODUCTION COORDINATOR: Jennifer Hall INDEXER: Natalie Boon INTERIOR DESIGN: Pronk & Associates COVER DESIGN: Michelle Losier ART DIRECTION: Tom Dart, First Folio Resource Group Inc. ELECTRONIC PAGE MAKE-UP: Tom Dart, Kim Hutchinson, First Folio Resource Group Inc. COVER IMAGE: Courtesy of Ocean/Corbis Acknowledgements There are many students, teachers, and administrators who the publisher, authors, and consultants of Pre-Calculus 11 wish to thank for their thoughtful comments and creative suggestions about what would work best in their classrooms. Their input and assistance have been invaluable in making sure that the Student Resource and its related Teacher’s Resource meet the needs of students and teachers who work within the Western and Northern Canadian Protocol Common Curriculum Framework. Reviewers Stella Ablett Carol Funk Andrew Jones Kim Mucheson Mulgrave School, West Nanaimo/Ladysmith School St. George’s School (Private Comox Valley School Vancouver (Independent) District No. 68 School) District #71 British Columbia British Columbia British Columbia British Columbia Kristi Allen Howard Gamble Jenny Kim Yasuko Nitta Wetaskiwin Regional Public Horizon School Division #67 Concordia High School Richmond Christian School Schools Alberta (Private) (Private) Alberta Alberta British Columbia Jessika Girard Karen Bedard Conseil Scolaire Janine Klevgaard Vince Ogrodnick School District No. 22 Francophone No. 93 Clearview School Division Kelsey School Division (Vernon) British Columbia No. 71 Manitoba British Columbia Alberta Crystal Ozment Pauline Gleimius Gordon Bramfield B.C. Christian Academy Ana Lahnert Nipisihkopahk Education Grasslands Regional (Private) Surrey School District Authority Division No. 6 British Columbia No. 36 Alberta Alberta British Columbia Curtis Rey Marge Hallonquist Yvonne Chow Elk Island Catholic Schools Carey Lehner Hannover School Division Strathcona-Tweedsmur Alberta Saskatchewan Rivers School Manitoba School (Independent) Division No. 119 Oreste Rimaldi Jeni Halowski Alberta Saskatchewan School District No. 34 Lethbridge School District Lindsay Collins No. 51 Debbie Loo (Abbotsford) South East Cornerstone Alberta Burnaby School District #41 British Columbia School Division No. 209 British Columbia Wade Sambrook Jason Harbor Saskatchewan North East School Division Jay Lorenzen Western School Division Julie Cordova No. 200 Horizon School District #205 Manitoba St. Jamies-Assiniboia School Saskatchewan Sakatchewan James Schmidt Division Dale Hawken Teréza Malmstrom Pembina Trails School Manitoba St. Albert Protestant Calgary Board of Education Division Janis Crighton Separate School District Alberta Manitoba Lethbridge School District No. 6 Rodney Marseille Sonya Semail No. 51 Alberta School District No. 62 School District 39 Alberta Murray D. Henry British Columbia (Vancouver) Steven Daniel Prince Albert Catholic British Columbia Darren McDonald Department of Education, School Division #6 Parkland School Division Dixie Sillito Culture and Employment Saskatchewan No. 70 Prairie Rose School Division Northwest Territories Barbara Holzer Alberta Alberta Ashley Dupont Prairie South School Dick McDougall Clint Surry St. Maurice School Board Division Calgary Catholic School School District 63 (Saanich) Manitoba Saskatchewan District British Columbia Dee Elder Larry Irla Alberta Debbie Terceros Edmonton Public Schools Aspen View Regional Georgina Mercer Peace Wapiti School Alberta Division No. 19 Fort Nelson School District Division #76 Alberta No. 81 Alberta Janet Ferdorvich Alexis Board of Education Betty Johns British Columbia John Verhagen Alberta University of Manitoba Livingstone Range School (retired) Division Manitoba Alberta Contents A Tour of Your Textbook..................... vi Unit 2 Quadratics............................. 138 Unit 1 Patterns..................................... 2 Chapter 3 Quadratic Functions................... 140 3.1 Investigating Quadratic Functions in Chapter 1 Sequences and Series................... 4 Vertex Form............................................................ 142 1.1 Arithmetic Sequences..............................................6 3.2 Investigating Quadratic Functions in 1.2 Arithmetic Series..................................................... 22 Standard Form....................................................... 163 1.3 Geometric Sequences........................................... 32 3.3 Completing the Square...................................... 180 1.4 Geometric Series..................................................... 46 Chapter 3 Review......................................................... 198 1.5 Infinite Geometric Series..................................... 58 Chapter 3 Practice Test............................................. 201 Chapter 1 Review............................................................ 66 Chapter 4 Quadratic Equations.................. 204 Chapter 1 Practice Test................................................ 69 4.1 Graphical Solutions of Quadratic Unit 1 Project.................................................................... 71 Equations................................................................. 206 4.2 Factoring Quadratic Equations....................... 218 Chapter 2 Trigonometry................................ 72 4.3 Solving Quadratic Equations by 2.1 Angles in Standard Position............................... 74 Completing the Square...................................... 234 2.2 Trigonometric Ratios of Any Angle................. 88 4.4 The Quadratic Formula...................................... 244 2.3 The Sine Law......................................................... 100 Chapter 4 Review..................................................... 258 2.4 The Cosine Law.................................................... 114 Chapter 4 Practice Test............................................. 261 Chapter 2 Review......................................................... 126 Chapter 2 Practice Test............................................. 129 Unit 2 Project Wrap-Up.................... 263 Unit 1 Project................................................................. 131 Cumulative Review, Unit 1 Project Wrap-Up.................... 132 Chapters 3—4................................. 264 Unit 2 Test........................................ 266 Cumulative Review, Chapters 1—2................................. 133 Unit 3 Functions and Equations..... 268 Unit 1 Test........................................ 136 Chapter 5 Radical Expressions and Equations....................................................... 270 5.1 Working With Radicals....................................... 272 5.2 Multiplying and Dividing Radical Expressions............................................................ 282 5.3 Radical Equations................................................ 294 Chapter 5 Review......................................................... 304 Chapter 5 Practice Test............................................. 306 iv MHR Contents Chapter 6 Rational Expressions and Unit 4 Systems of Equations Equations....................................................... 308 and Inequalities............................... 420 6.1 Rational Expressions.......................................... 310 Chapter 8 Systems of Equations................ 422 6.2 Multiplying and Dividing Rational 8.1 Solving Systems of Equations Expressions............................................................ 322 Graphically.............................................................. 424 6.3 Adding and Subtracting Rational 8.2 Solving Systems of Equations Expressions............................................................ 331 Algebraically........................................................... 440 6.4 Rational Equations.............................................. 341 Chapter 8 Review......................................................... 457 Chapter 6 Review......................................................... 352 Chapter 8 Practice Test............................................. 459 Chapter 6 Practice Test............................................. 355 Unit 4 Project................................................................. 461 Chapter 7 Absolute Value and Reciprocal Chapter 9 Linear and Quadratic Functions....................................................... 356 Inequalities.................................................... 462 7.1 Absolute Value...................................................... 358 9.1 Linear Inequalities in Two Variables............ 464 7.2 Absolute Value Functions................................ 368 9.2 Quadratic Inequalities in 7.3 Absolute Value Equations................................ 380 One Variable........................................................... 476 7.4 Reciprocal Functions.......................................... 392 9.3 Quadratic Inequalities in Two Variables........................................................ 488 Chapter 7 Review......................................................... 410 Chapter 9 Review......................................................... 501 Chapter 7 Practice Test............................................. 413 Chapter 9 Practice Test............................................. 504 Unit 3 Project Wrap-Up.................... 415 Unit 4 Project................................................................. 506 Cumulative Review, Unit 4 Project Wrap-Up.................... 507 Chapters 5—7................................. 416 Cumulative Review, Unit 3 Test........................................ 418 Chapters 8—9................................. 508 Unit 4 Test........................................ 510 Answers............................................. 513 Glossary............................................. 586 Index.................................................. 592 Credits............................................... 596 Contents MHR v A Tour of Your Textbook Unit 1 Patterns Unit Opener Many problems are solved using patterns. Economic Each unit begins with a two-page and resource trends may be based on sequences and series. Seismic exploration identifies underground phenomena, such spread. The first page of the Unit as caves, oil pockets, and rock layers, by transmitting sound into the earth and timing the echo of the vibration. Surveyors Opener introduces what you will use triangulation and the laws of trigonometry to determine distances between inaccessible Unit 1 Project Canada’s Natural Resources learn in the unit. The Unit Project is points. All of these activities use patterns and aspects of the mathematics you will encounter Canada is a country rich with natural found in abundance in the Canadian resources. Petroleum, minerals, landscape. Canada is one of the exporters of minerals, mineral products, and forests are world’s leading and forest products. Resource developmen in this unit. has been a mainstay of Canada’s t introduced on the second page. Each economy for many years. In this project, you will explore one of Canada’s natural resources of petroleum, minerals, or forestry. from the categories You will collect and present data chosen resource to meet the following related to your Unit Project helps you connect the criteria: Looking Ahead Include a log of the journey leading to the discovery of your resource. In this unit, you will solve In Chapter 1, you will provide data on the production of your problems involving… you will apply your knowledge natural resource. Here of sequences and series to show arithmetic sequences increased or decreased over time, how production has math in the unit to real life using and series geometric sequences and series your chosen resource. In Chapter 2, you will use skills and the cosine law to explore the and make predictions about future developed with trigonometry, including development of the sine law infinite geometric series area where your resource was discovered. then explore the proposed site of You will experiences that may interest you. sine and cosine laws At the end of your project, you the development of your resource. your natural resource. will encourage potential investors to participate in Your final project may take many a written or visual presentation forms. It may be , a brochure, a video production, show. Or, you could use the interactive or a computer slide features of a whiteboard. In the Project Corner box at the end of most sections, you will find notes about Canada’s natural resources. information and You can use this information to data and facts about your chosen help gather resource. 2 MHR Unit 1 Patterns Unit 1 Patterns MHR 3 Project Corner boxes Unit 1 Project Unit 1 Project Wrap-Up throughout the chapters help Canada’s Natural Resources Canada’s Canada Cana Canada’ Natural Resources you gather information for Canada is the source of more than 60 mineral commodities, mmodities, including You need in investment capital to develop your resource. Prepare a eral fuels. metals, non-metals, structural materials, and mineral Unit 1 Project presentation to make to your investors to encourage them to invest in your project. project You can use a written or visual presentation, a brochure, a Quarrying and mining are among the oldest industries stries in Canada. In 1672, coal was discovered on Cape Breton Island. video production, produ a computer slide show presentation, or an interactive whiteboard presentation. your project. Some Project In the 1850s, gold discoveries in British Columbia, and increased production of Cape Breton coal marked in Canadian mineral history. Canada’s Natural Resources a, oil finds in Ontario, rked a turning point The emphasis of the Chapter 2 Task is the location of your resource. Your presentation presen Actual dat should include the following: data taken from Canadian sources on the production of your You will describe the route of discovery of the resource and the In 1896, gold was found in the Klondike District of what became YukonY chosen res resource. Use sequences and series to show how production Corner boxes include part of the North-West Territories that later became nds were evident in In the late 1800s, large deposits of coal and oil sands me Alberta. planned area of the resource. pectacular gold rushes. Territory, giving rise to one of the world’s most spectacular Chapter 2 Task We b Link has increased increa and sales. A fictitious or decreased over time, and to predict future production fictitiou account of a recent discovery of your resource, including a The Journey to Locate the Resource map of the area showing the accompanying distances. questions to help you to In the post-war era there were many major mineralal discoveries: deposits of nickel in Manitoba; zinc-lead, copper, and molybdenum Use the map provided. Include a brief log of the journey leading to in British Columbia; and base metals and asbestoss in Québec, Ontario,your discovery. The exploration map is the route that you followed to discover your chosen resource. To obtain a copy of an exploration map, go to www.mhrprecalc11.ca and follow the links. A proposal proposa for how the resource area will be developed over the next few years ritish Columbia. Manitoba, Newfoundland, Yukon Territory, and British begin thinking about and berta in 1947 was With your exploration map, determine the total distance of your The discovery of the famous Leduc oil field in Alberta um industry. followed by a great expansion of Canada’s petroleum In the late 1940s and early 1950s, uranium was discovered iscovered in route, to the nearest tenth of a kilometre. Begin your journey at point A and conclude at point J. Include the height of the Sawback Ridge and the width of Crow River in your calculations. Saskatchewan and Ontario. In fact, Canada is noww the world’s largest discussing your project. uranium producer. roduction in October Canada’s first diamond-mining operation began production 1998 at the Ekati mine in Lac de Gras, Northwest Territories, T Developing the Area of Your Planned Resource Your job as a resource development officer for the company is to followedpresent a possible area of development. You are restricted by land by the Diavik mine in 2002. boundaries to the triangular shape shown, with side AB of 3.9 km, side AC of 3.4 km, and ∠B = 60°. Chapter 1 Task Determine all measures of the triangular region that your company The Unit Projects in Units 1 Choose a natural resource that you would like to research. You may could develop. d in the Project Corner wish to look at some of the information presented boxes throughout Chapter 1 for ideas. Research your our chosen resource. Possible Proposed Development A e, including what it is, List interesting facts about your chosen resource, and 4 are designed for you to how it is produced, where it is exported, how much is exported, and so on. Look for data that would support using a sequence nce or series in complete in pieces, chapter discussing or describing your resource. List the terms for the sequence or series you include. 3.9 km 3.4 km h uence or series to Use the information you have gathered in a sequence n of the resource over a predict possible trends in the use or production by chapter, throughout the ten-year period. Describe any effects the production of the natural community. ral resource has on the 60° B unit. At the end of the unit, a C D Unit 1 Project MHR 71 132 MHR Unit 1 Project Wrap-Up Project Wrap-Up allows you Unit 1 Project MHR 131 to consolidate your work in a meaningful presentation. Unit 2 Project Wrap-Up Quadratic Functions in Everyday Life You can analyse quadratic functions and their related equations to solve problems and explore the nature of a quadratic. A quadratic can model the curve an object follows as it flies through the air. For example, consider the path of a softball, a tennis ball, a football, a baseball, a soccer ball, or a basketball. A quadratic function can also be used to design an object that has a specific curved shape needed for a project. Quadratic equations have many practical applications. Quadratic equations may be used in the design and sales of many products found in stores. They may be used to determine the safety and the life expectancy of a The Unit Projects in Units 2 and 3 provide an product. They may also be used to determine the best price to charge to maximize revenue. Complete one of the following two options. opportunity for you to choose a single Project Option 1 Quadratic Functions in Everyday Life Research a real-life situation that may be Option 2 Avalanche Control Research a ski area in Western Canada that modelled by a quadratic function. requires avalanche control Wrap-Up at the end of the unit. Search the Internet for two images or video clips, one related to objects in motion and one related to fixed objects. These items Determine the best location or locations to position avalanche cannons in your resort. Justify your thinking. should show shapes or relationships that are Determine three different quadratic functions parabolic. that can model the trajectories of avalanche Model each image or video clip with a control projectiles. quadratic function, and determine how Graph each function. Each graph should accurate the model is. illustrate the specific coordinates of where Research the situation in each image or video the projectile will land. clip to determine if there are reasons why it Write a one-page report to accompany should be quadratic in nature. your graphs. Your report should include Write a one-page report to accompany your the following: functions. Your report should include the  the location(s) of the avalanche control following: cannon(s)  where quadratic functions and equations  the intended path of the controlled are used in your situations avalanche(s)  when a quadratic function is a good model  the location of the landing point for to use in a given situation each projectile  limitations of using a quadratic function as a model in a given situation Unit 2 Project Wrap-Up MHR 263 vi MHR A Tour of Your Textbook Chapter Opener CHAPTER 1 Sequences and Series Each chapter begins with a two-page Many patterns and designs linked and the human body. Certain patterns to mathematics are found in nature occur more often than others. Logistic spirals, such as the Golden spread that introduces you to what Fibonacci number sequence. The Nature’s Numbers. Mean spiral, are based on the Fibonacci sequence is often called 13 8 you will learn in the chapter. 8 8 13 2 1 2 1 1 1 5 3 The opener includes information 3 The pattern of this logistic spiral 5 is found in the chambered nautilus, the inner ear, star clusters, cloud about a career that uses the skills growth, leaves on stems, petals rabbit reproduction also appear spiral pattern. patterns, and whirlpools. Seed on flowers, branch formations, and to be modelled after this logistic covered in the chapter. A Web Link There are many different kinds learn about sequences that can of sequences. In this chapter, you be described by mathematical rules. will Career Link Link allows you to learn more about We b Biomedical engineers combine To learn earn more about biology, ab the Fibonacci sequence, go to engineering, and mathematical www.mhrprecalc11.ca and follow the links. sciences to solve medical and health-relate d problems. Some research and develop artificial this career and how it involves the and replacement limbs. Others organs design MRI machines, laser systems, and microscopic machines used in surgery. Many biomedical engineers work in research and mathematics you are learning. Key Terms sequence arithmetic sequence geometric sequence common ratio Did You Know? In mathematics, the Fibonacci sequence is a sequence of natural numbers named after taken insulin or used an asthma you have benefited from the work development in health-related fields. If you have ever inhaler, of common difference geometric series Leonardo of Pisa, also known biomedical engineers. as Fibonacci. Each number is general term convergent series the sum of the two preceding arithmetic series divergent series numbers. We b Link Visuals on the chapter opener spread 1, 1, 2, 3, 5, 8, 13,... To learn earn more about ab biomedical engineering, go to www.mhrprecalc11.ca and follow the links. 4 MHR Chapter 1 show other ways the skills and Chapter 1 MHR 5 concepts from the chapter are used in daily life. 1.1 Arithmetic Sequences Numbered Sections Focus on... deriving a rule for determining the general term of an arithmetic sequence determining t1, d, n, or tn in a problem that involves an arithmetic sequence describing the relationship between an arithmetic The numbered sections in each chapter start with sequence and a linear function solving a problem that involves an arithmetic sequence Comets are made of frozen lumps of gas and rock and are a visual to connect the topic to a real setting. The often referred to as icy mudballs or dirty snowballs. In 1705, Edmond Halley predicted that the comet seen in 1531, 1607, and 1682 would be seen again in 1758. Halley’s prediction was accurate. This comet was later named in his purpose of this introduction is to help you make honour. The years in which Halley’s Comet has appeared approximately form terms of an arithmetic sequence. What makes this sequence arithmetic? connections between the math in the section and in Investigate Arithmetic Sequences the real world, or to make connections to what you Staircase Numbers A staircase number is the number of cubes needed to make a staircase that has at least two steps. already know or may be studying in other classes. Is there a pattern to the number of cubes in successive staircase numbers? How could you predict different staircase numbers? 1 2 3 4 5 6 7 8 9 10 Columns Part A: Two-Step Staircase Numbers To generate a two-step staircase number, add the numbers of cubes in two consecutive columns. The first staircase number is the sum of the number of cubes in column 1 and in column 2. 1 2 6 MHR Chapter 1 A Tour of Your Textbook MHR vii Three-Part Lesson Each section is organized in a 2.4 three-part lesson: Investigate, Link the The Cosine Law The cosine law relates the lengths cosine of one of its angles. 3. a) For ABC given in step of the sides of a given triangle to 1, determine the value of 2ab cos the C. Ideas, and Check Your Understanding. b) Determine the value of 2ab Focus on... cos C for ABC from step 2. c) Copy and complete a table sketching a diagram and solving like the one started below. Record a problem your using the cosine law results and collect data for the triangle drawn in step 2 recognizing when to use the cosine from at least three other people. law to solve a given problem Triangle Side Lengths (cm) explaining the steps in the given proof of c2 a2 + b2 2ab cos C the cosine law a = 3, b = 4, c = 5 Investigate The Canadarm2, also known as the Mobile Servicing System, a = , b = , c =  is a major part of the Canadian spacee 4. Consider the inequality you robotic system. It completed its found to be true in step 2, for the first official construction job relationship on the International Space Station between the values of c2 and a2 in July 2001. The robotic obotic arm can move + b2. Explain how equipment and assist astronauts your results from step 3 might be working in space. The robotic manipulator used to turn the inequality into The Investigate consists of short operated by controlling the angles is an equation. This relationship is of its joints. The final position known as the cosine law. can be calculated by using the trigonometric of the arm 5. Draw ABC in which ∠C ratios of those angles. is obtuse. Measure its side lengths. Determine whether or not your equation from step 4 holds. Reflect and Respond steps often accompanied by Investigate the Cosine Law 6. The cosine law relates the the cosine of lengths of the sides of a given triangle one of its angles. Under what conditions would you use