Version 01 Trigonometry II Apprenticeship and Workplace Mathematics (Grade 10/Literacy Foundations Level 7) © 2012 by Open School BC http://mirrors.org/presskit/buttons/88x31/eps/by-nc.eps This work is licensed under the Creative Commons Attribution-NonCommercial 4. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/4.0/ Permissions beyond the scope of this license are identified in the materials by a copyright symbol and are outlined below. To request permission to use the exclusions to this Creative Commons license, contact the author/publisher of the third party materials: Third party copyright exclusions include: All photographs used under license from Shutterstock.com with exception of the following: • Photo "two food high kick" by Xander • Photo "funiculaire" by Funiculaire du Vieux Quebec. The Data Pages were reproduced with permission from the BC Ministry of Education.
Course History New, March 2012 Project Partners This course was developed in partnership with the Distributed Learning Resources Branch of Alberta Education and the following organizations: • Black Gold Regional Schools • Calgary Board of Education • Edmonton Public Schools • Peace Wapiti School Division No. 76 • Pembina Hills Regional Division No. 7 • Rocky View School Division No. 41 Project Management: Jennifer Riddel Content Revisions: Jennifer Riddel, Ester Moreno Edit: Leanne Baugh, Monique Brewer Math Edit: Learning Centre of the Greater Victoria School District Continuing Education Program: Nigel Cocking, Keith Myles, Bill Scott School District 47, Powell River: Tania Hobson OSBC: Christina Teskey Module Tests: Barb Lajeunesse, Michael Finnigan (SD 34) Copyright: Ilona Ugro Production Technicians: Sharon Barker, Beverly Carstensen, Dennis Evans, Brian Glover Art Coordination: Christine Ramkeesoon Media Coordination: Janet Bartz Art: Cal Jones Flash Programming: Sean Cunniam Narration Recording: MOH Productions and Neil Osborne Voice Talent: Felix LeBlanc, Kate Eldridge, Wendy Webb and MOH Productions Advisors: JD Caudle (Yukon Territory), Randy Decker (SD 40), Bev Fairful (Yukon Territory), Sonya Fern (SD 62), Sandra Garfinkel (SD 39), Richard Giroday (SD 58), Sharon Hann (SD 39), Tim Huttemann (SD 20), Dan Laidlaw (SD 73), Heather Lessard (SD 53), Gloria Lowe (SD 6), Jan Malcolm (SD 36), Christina Teskey (OSBC), Jennifer Waughtal (SD 57), Ray Wong (SD 91) Table of Contents Table of Contents Section Organization.
1 Lesson A: The Cosine Ratio. 3 Lesson B: Using the Cosines to Solve Problems. 33 Lesson C: General Problems. 107 © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | iii Viewing Your PDF Learning Package This PDF Learning Package is designed to be viewed in Acrobat.
If you are using the optional media resources, you should be able to link directly to the resource from the pdf viewed in Acrobat Reader. The links may not work as expected with other pdf viewers. Download Adobe Acrobat Reader: http://get.com/reader/ Section Organization Section Organization This section on Trigonometry is made up of several lessons. Lessons Lessons have a combination of reading and hands-on activities to give you a chance to process the material while being an active learner.
Each lesson is made up of the following parts: Essential Questions The essential questions included here are based on the main concepts in each lesson. These help you focus on what you will learn in the lesson. Focus This is a brief introduction to the lesson. Get Started This is a quick refresher of the key information and skills you will need to be successful in the lesson.
Activities Throughout the lesson you will see three types of activities: • Try This activities are hands-on, exploratory activities. • Self-Check activities provide practice with the skills and concepts recently taught. • Mastering Concepts activities extend and apply the skills you learned in the lesson. You will mark these activities using the solutions at the end of each section.
Explore Here you will explore new concepts, make predictions, and discover patterns. Bringing Ideas Together This is the main teaching part of the lesson. Here, you will build on the ideas from the Get Started and the Explore. You will expand your knowledge and practice your new skills.
Lesson Summary This is a brief summary of the lesson content as well as some instructions on what to do next. © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | v Section Organization At the end of each section you will find: Solutions This contains all of the solutions to the Activities. Appendix Here you will find the Data Pages along with other extra resources that you need to complete the section. You will be directed to these as needed.
Glossary This is a list of key terms and their definitions. Throughout the section, you will see the following features: Icons Throughout the section you will see a few icons used on the left-hand side of the page. These icons are used to signal a change in activity or to bring your attention to important instructions. AWM online resource (optional) This indicates a resource available on the internet.
If you do not have access, you may skip these sections. Solutions Calculator vi | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Section Organization My Notes The column on the outside edge of most pages is called “My Notes”. You can use this space to: • write questions about things you don’t understand. • note things that you want to look at again.
• draw pictures that help you understand the math. • identify words that you don’t understand. • connect what you are learning to what you already know. • make your own notes or comments.
Materials and Resources There is no textbook required for this course. You will be expected to have certain tools and materials at your disposal while working on the lessons. When you begin a lesson, have a look at the list of items you will need. You can find this list on the first page of the lesson, right under the lesson title.
In general, you should have the following things handy while you work on your lessons: • a scientific calculator • a ruler • a geometry set • Data Pages (found in the Appendix) © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | vii Trigonometry II Trigonometry II Photo by Sergei Bachlakov © 2010 Imagine the excitement of carrying the Olympic torch! More than 12 000 torch- bearers from all parts of Canada carried the flame during the 106-day relay that started in Victoria, British Columbia. The torch route went through all of the territories and provinces of Canada, and ended in Vancouver, the host city of the 2010 Winter Olympic Games. Wayne Gretzky had the honour of lighting the outdoor cauldron—what a spectacular event! Take a close look at that cauldron. Its design incorporates many of angles and triangles! The Winter Olympics required many new facilities, and these buildings and structures were designed by architects who applied mathematics.
One of their mathematical tools is trigonometry. In this section you will: • apply similarity to right triangles. • generalize patterns from similar right triangles. • apply the trigonometric ratios tangent, sine, and cosine to solve problems.
© OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 1 Trigonometry II—Lesson A: The Cosine Ratio Lesson A The Cosine Ratio To complete this lesson, you will need: In this lesson, you will complete: • a metric ruler • 7 activities • a protractor • a calculator Essential Questions • What is the cosine ratio? • How is the cosine ratio used to find unknown sides and angles in right triangles? © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 3 Trigonometry II—Lesson A: The Cosine Ratio My Notes Focus Photo by Dainis Derics © 2010 The 2010 Olympic bobsleigh and skeleton track at Whistler, British Columbia is 1450 m in length with a vertical drop of 152 m and an average slope of 10. In the women’s skeleton event, the combined mass of the competitor and sled cannot exceed 92 kg. Part of this combined weight pushes the runners against the track and part of the weight pushes the sled down the slope. On average, what percentage of the mass pushes the sled down the track? At the end of this lesson you will have the trigonometric skills to answer this question.
Get Started Do you remember what you learned in previous lessons about the relationship between the angles in a triangle? _________________________________________________________________ The measures of the angles in any triangle add up to 180º. In Activity 1 you will review the relationship between the two acute angles of a right triangle. 4 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Trigonometry II—Lesson A: The Cosine Ratio Activity 1 My Notes Try This You will need a protractor and a straightedge to complete this activity. Step 1: Use your protractor and straightedge to draw a right triangle PQR of any size.
An example is shown below. P q r Q p R Did You Know? The letters that are used to name triangles are usually placed alphabetically. But the letters could be placed in any order. For example, the previous triangle could have been named any of the following: ∆QRP, ∆PRQ, or ∆RQP.
Step 2: Measure ∠P and ∠R to the nearest degree. © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 5 Trigonometry II—Lesson A: The Cosine Ratio 2. Will the sum of the two acute angles of a right triangle always be My Notes the same as the answer you obtained in Question 1? Explain. What name is given to a pair of angles with this sum? ______________________________________________________________ ______________________________________________________________ 4.
In right ∆ABC, ∠A and ∠B are acute angles. If ∠B = 47°, what is the size of ∠A? Draw a diagram to support your answer. 6 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Trigonometry II—Lesson A: The Cosine Ratio 5. What is the complement of a 60° angle? My Notes Turn to the solutions at the end of the section and mark your work.
Explore In previous lessons, you explored the sine and tangent ratios. For a given acute angle in a right triangle, the tangent ratio was defined as opposite opposite , and the sine ratio was defined as. adjacent hypotenuse You discovered that for similar triangles, the values of the tangent ratios were the same regardless of how large or small the right triangles were. This was also true for the sine ratio.
So, if you knew the angle, you could determine the ratio. And, if you knew the ratio, you could determine the angle. Activity 2 Try This In this activity you will explore one other trigonometric ratio involving a different pair of sides within a right triangle. You will compare this ratio with ratios involving the same pair of sides from other right triangles.
For this comparison you will use your knowledge of similar right triangles. You will need your metric ruler, protractor, and calculator. © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 7 Trigonometry II—Lesson A: The Cosine Ratio Consider the right triangles in the following diagram. My Notes B3 B2 B1 A C1 C2 C3 Step 1: On a full piece of paper, draw a diagram similar to the one above.
Make an enlargement of it, so that the diagram takes up the entire page. Label the diagram using the same lettering. Step 2: For right triangle ΔAB1C1, measure to the nearest millimetre and record the length of sides AC1 and AB1 in the table below. For right triangle ΔAB2C2, measure to the nearest millimetre and record the length of sides AC2 and AB2 in the table below.
For right triangle ΔAB3C3, measure to the nearest millimetre and record the length of sides AC3 and AB3 in the table below. Step 3: Complete the last column of the table using your calculator.