Version 01 Trigonometry I 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 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, 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 Tangent Ratio.
3 Lesson B: Using Tangents to Solve Problems. 31 Lesson C: The Sine Ratio. 51 Lesson D: Using Sines to Solve Problems. 137 © 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 sefe 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 I Trigonometry I Photo by Roca © 2010 Hikers and skiers who are out on snowy slopes need to be careful to avoid areas that are high risk for avalanches. Many of these outdoor enthusiasts carry a device called a clinometer. A clinometer can be used to measure the angle of inclination of a slope. Hikers and skiers can measure the incline of the slopes around them and avoid risky areas.
Hikers and skiers aren’t the only ones who use clinometers. Surveyors, geologists, sailors, city planners, and engineers use clinometers to measure distances indirectly. How can measuring an angle of incline help one determine an unknown distance? Well, the angle measurements are used together with a mathematical concept called trigonometry. Trigonometry, as you will discover in this section, involves relationships arising from the sides and angles of right triangles.
© OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 1 Trigonometry I In this section you will: • apply similarity to right triangles. • generalize patterns from similar right triangles. • build and use a clinometer to measure distances indirectly. • apply the trigonometric ratios tangent and sine to solve problems.
2 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Trigonometry I—Lesson A: The Tangent Ratio Lesson A The Tangent Ratio To complete this lesson, you will need: In this lesson, you will complete: • a piece of cardboard (8 ½ by 11") • 9 activities • a drinking straw • tape • scissors • a paper clip or other small object to use as a weight • the clinometer template from the appendix • protractor • metric ruler • a calculator Essential Questions • What is the tangent ratio? • How is the tangent ratio used to find unknown sides and angles in right triangles? © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 3 Trigonometry I—Lesson A: The Tangent Ratio My Notes Focus Photo by Ewan Chesser © 2010 and photo by Joe Gough © 2010 Imagine building a tree house. Some measurements, such as the height of the tree, would be difficult to take—it’s not easy (or safe) to climb a tree with a tape measure in your hand! Luckily, we have trigonometry! In the construction industry, surveying allows people to use trigonometry to calculate unknown lengths so that direct measurements are not required. In this lesson you’ll create a device for measuring angles and you’ll begin your study of trigonometry. 4 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Trigonometry I—Lesson A: The Tangent Ratio Get Started My Notes Photo by Pitroviz © 2010 In the following activity you will make a clinometer.
A clinometer is a device that you put to your eye to see how high an object is off the ground. Clinometers are often used in forestry to measure tree heights or elevation changes. Activity 1 Try This Step 1: Get the “Clinometer Template” from the appendix, and cut it out. Step 2: Glue the template to a sheet of cardboard.
Then, cut out the semicircular protractor. © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 5 Trigonometry I—Lesson A: The Tangent Ratio Step 3: Tape a drinking straw to the base of the protractor as shown. My Notes 0 4: Poke a small hole through the crosshairs at the centre of the protractor’s base. Pass one end of your thread through the hole.
Knot and tape that end of the thread so it will not slip back through. Tie a weight (such as a paper clip) to the free end of the thread. 0 clip 6 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Trigonometry I—Lesson A: The Tangent Ratio Step 5: Let the weight swing free. When the straw is horizontal, the My Notes thread should lie along the zero degree line as shown in the diagram.
When you look through the straw at an object higher than you are, your partner should be able to read the angle of elevation. When the weight hangs down and is motionless, the string will lie along a particular angle. This is the angle of elevation. object horizon Read angle here.
When you look through the straw at an object lower than you are, your partner should be able to read the angle of depression from the scale when the weight hangs down motionlessly. object Read angle here. © OPEN SCHOOL BC APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT | 7 Trigonometry I—Lesson A: The Tangent Ratio Practise looking through the straw and measuring angles of elevation My Notes and depression. What is the greatest angle of elevation you can measure? ______________________________________________________________ 2.
What is the greatest angle of depression you can measure? ______________________________________________________________ Keep your clinometer: you will need it to complete the next lesson. Turn to the solutions at the end of the section and mark your work. Ratio Skills An important algebra skill you will use repeatedly throughout this section involves equal ratios. You will now review how to determine if ratios are equal.
Example 1 10 8 Are and equal ratios? 15 12 Solution Method 1: Reduce each fraction (ratio) to its simplest terms. 10 10 5 8 8 4 = ÷ = ÷ 15 15 5 12 12 4 2 2 = = 3 3 2 Since each ratio reduces to , the ratios are equal. 3 8 | APPRENTICESHIP AND WORKPLACE MATHEMATICS ETEXT © OPEN SCHOOL BC Trigonometry I—Lesson A: The Tangent Ratio Method 2: Compare the cross products. My Notes 10 8 = 15 12 10 × 12 = 120 15 × 8 = 120 Because the cross products are equal, the ratios are equal.
The Cross Product Principle The previous example illustrates the following principle in algebra. a c If = then ad = bc. b d and a c If ad = bc then =. b d This is the cross product principle.