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1Introducing the Lesson(5 to 10 min)Have students work in small groups. Assign a primary trigonometric ratio toeach group, and ask students to discuss the ratio for a few minutes. To helpstudents get started, suggest topics such as: type of triangle for which a primary trigonometric ratio is used relationship represented by each primary trigonometric ratio use of a primary trigonometric ratio to determine a side length use of a primary trigonometric ratio to determine an angle measure connection between tangent and slope diagram to model a primary trigonometric ratio problem that involves a primary trigonometric ratio in the solutionThen ask a member of each group to describe the ratio in a few words, withor without a diagram.Pose these questions: How is an acute triangle different from a righttriangle? Can you use primary trigonometric ratios to solve acute triangles?Why not?2Teaching and Learning(35 to 45 min)Explore the MathHave students work through the exploration in pairs. Ensure that theyconstruct only acute triangles, except in part F.Copyright 2011 by Nelson Education Ltd.If students are using dynamic geometry software, the following ideas maybe helpful: Ensure that students can use the software to measure angles in a triangleand lengths of line segments, as noted in the Tech Support in the margin. Place more experienced users of the software with students who are lessproficient. Encourage students to seek assistance from their neighbourswhen necessary. Remind students to choose the precision for lengths and angle measures,as well as the units. Ensure that students know how to use the calculator in the dynamiclength of opposite side.geometry software to determine the ratiosin (angle) For part F, students could drag vertices to obtain an angle of almostexactly 90º, create a segment and rotate it, or create a right triangle byconstructing a perpendicular line. When they drag the right triangle, theangle must remain a right angle.8.1: Exploring the Sine Law 295

If students are using rulers and protractors, have them measure the sidelengths to tenths of a centimetre and the angles to the nearest degree. Theaccuracy will be less exact than the accuracy with dynamic geometrysoftware.Discuss why results can vary because of precision in measurements andbecause of rounding, when using the calculator in the dynamic geometrysoftware or when using a scientific calculator.Answers to Explore the MathA.–B. Answers may vary, e.g.,length of opposite sideAngleSideSine A 52.3ºa 2.6sin A 0.79a 3.3sin A B 55.5ºb 2.8sin B 0.82b 3.3sin B C 72.2ºc 3.2sin C 0.95c 3.3sin CC. The ratio is the same for all three angles in my triangle.D.296 Principles of Mathematics 10: Chapter 8: Acute Triangle TrigonometryCopyright 2011 by Nelson Education Ltd.sin (angle)

length of opposite sideAngleSideSine A 26.3ºa 2.6sin A 0.44a 6.0sin A B 65.2ºb 5.4sin B 0.91b 6.0sin B C 88.5ºc 6.0sin C 1.00c 6.0sin Csin (angle)length of opposite sideis the same for all three angles in thissin (angle)triangle, but it is different from the ratio for the angles in the firsttriangle.length of opposite sideE. The ratiois always the same for all three anglessin (angle)in a triangle.F.Copyright 2011 by Nelson Education Ltd.The ratiolength of opposite sideAngleSideSine A 48.2ºa 3.9sin A 0.75a 5.2sin A B 41.8ºb 3.5sin B 0.67b 5.2sin B C 90.0ºc 5.2sin C 1.00c 5.2sin Csin (angle)In a right triangle, the ratio is the same for all three angles.For some of the tables, if I calculated the answers using themeasurements in the other columns, some results would be slightlydifferent than my results using the software calculator. The differencesare because of the precision in the measurements and because ofrounding.8.1: Exploring the Sine Law 297

G. Replacing the sine ratio with the cosine or tangent ratio does not give thesame results. The cosine ratios are not the same for all three angles in atriangle. Similarly, the tangent ratios are not the same.H. i) In an acute triangle, the ratio of the length of a side to the sine of theopposite angle is equivalent for all three angle–side pairs.abcii) In an acute triangle ABC,. sin A sin B sin CAnswers to ReflectingI. Since the relationship involves the sine ratio, an appropriate name couldbe the sine law.J. Yes. Write the ratio of the length of the known side to the sine of theangle opposite the side. Create an equivalent ratio for the unknown sidelength and the sine of the opposite known angle, or for the length of theknown side and the sine of the unknown opposite angle. Equate the tworatios, and solve for the unknown. If you know two of the angles, youcan calculate the third angle and then use ratios to solve for the unknownside length.3Consolidation(10 to 15 min)Students should understand that the ratio of the length of a side to the sine ofthe opposite angle is equivalent for all three angle–side pairs in an acutetriangle. Students should also understand that this relationship can be usedto determine unknown side lengths and angle measures in an acute triangleif enough information is known.Copyright 2011 by Nelson Education Ltd.Students should be able to answer the Further Your Understanding questionsindependently.298 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry

8.2 APPLYING THE SINE LAWGOALUse the sine law to calculate unknownside lengths and angle measures inacute triangles.Lesson at a GlancePrerequisite Skills/ConceptsStudent Book Pages 428–434length of opposite sideis the same for all threesin (angle)angle–side pairs in an acute triangle. Apply the primary trigonometric ratios to determine side lengths andangle measures. Solve problems involving the properties of interior angles of a triangleand angles formed by parallel lines. Solve problems involving proportions. Understand that the ratioSpecific Expectations [Explore the development of the sine law within acute triangles (e.g., usedynamic geometry software to determine that the ratio of the side lengthsequals the ratio of the sines of the opposite angles;] follow the algebraicdevelopment of the sine law and identify the application of solvingsystems of equations [student reproduction of the development of theformula is not required]). Determine the measures of sides and angles in acute triangles, using thesine law [and the cosine law]. Solve problems involving the measures of sides and angles in acutetriangles.Preparation and PlanningPacing5 10 min Introduction30 40 min Teaching and Learning15 20 min ConsolidationMaterials rulerRecommended PracticeQuestions 3, 4, 6, 7, 8, 9, 11, 14Key Assessment QuestionQuestion 7Extra PracticeLesson 8.2 Extra PracticeNew Vocabulary/Symbolssine lawNelson Websitehttp://www.nelson.com/mathCopyright 2011 by Nelson Education Ltd.Mathematical Process Focus Problem Solving Selecting Tools and Computational Strategies RepresentingMATH BACKGROUND LESSON OVERVIEW In Lesson 8.1, students explored the relationship between each side in an acute triangle and the sine of its oppositeangle. They discovered that the ratiolength of opposite sidesin (angle)is equivalent for all three angle–side pairs. In Lesson 8.2, students learn the name of this relationship: the sine law. Students use the sine law to determine unknown side lengths and angle measures in acute triangles. They also usethe sine law to solve problems that involve acute triangles.8.2: Applying the Sine Law 299

1Introducing the Lesson(5 to 10 min)Initiate a discussion about the exploration for Lesson 8.1 by invitingstudents to explain their discoveries in their own words and to illustrate theirdiscoveries on the board, as appropriate. Alternatively, students could spenda few minutes discussing the exploration in groups. Ask them to decidewhat they think was their most important discovery in the exploration. Havea student from each group present their decision to the class.2Teaching and Learning(30 to 40 min)Learn About the MathExample 1 shows that the sine law is true for all acute triangles. It allowsstudents to discuss a proof of the sine law. Help students understand howthis proof extends their exploration for Lesson 8.1, in which they obtainedequivalent ratios. Using the prompts, lead students through the steps in theproof.A. Ben needed to create two right triangles. He used one right triangle todetermine an expression for sin B and the other right triangle todetermine an expression for sin C. Since AD was the opposite side in ABD and in ACD, he was able to relate the two triangles.B. If Ben drew a perpendicular line segment from vertex C to side AB, heabandare equal.would be able to show that the ratiossin Asin BC. The expressions c sin B and b sin C both describe AD and can be setequal to each other. Instead of dividing both sides of the equation bysin C and then by sin B, both sides can be divided by b and then by c.This will still result in an equation that describes only one side of thesin B sin Ctriangle on each side:. Similarly, dividing the second bcsin A sin C. Therefore,equation by a and then by c will result in acit makes sense that the sine law can also be written in the formsin A sin B sin C. abc300 Principles of Mathematics 10: Chapter 8: Acute Triangle TrigonometryCopyright 2011 by Nelson Education Ltd.Answers to Reflecting

3Consolidation(15 to 20 min)Apply the MathUsing the Solved ExamplesIn Example 2, three angle measures and the length of one side of an acutetriangle are given, so one angle–side pair and the measure of another anglecan be used to determine a side length. Students could talk about thesolution in pairs, before a class discussion. Encourage students to take anactive role in explaining the steps in the example in their own words. Posequestions such as these: How do you think Elizabeth decided which angle measures to use? If she had known only the measures of A and C, how could she havesolved the problem? Why does it make sense that side b is shorter than side a?In Example 3, two side lengths and one angle measure are given, so oneangle–side pair and the length of another side can be used to determine anangle measure. Discuss how using the form of the sine law with sine in thenumerator makes it easier to calculate the solution. Since the final step in thesolution uses the inverse of the sine ratio, a brief review of the inverse maybe helpful.Example 4 presents a problem that can be modelled with an acute triangle.Two angles and one side are given, but not an angle–side pair. Studentsshould understand how the measure of the third angle is determined byusing the sum of the angles in a triangle.Copyright 2011 by Nelson Education Ltd.Answer to the Key Assessment QuestionStudents should draw a diagram to help them visualize the parallelogram inthe multi-step problem for question 7. Students could use the informationabout parallelograms from Lesson 2.4, In Summary, as a reference. Afterstudents complete the solution, ask them to explain the steps.7. The long sides of the parallelogram measure 15.4 cm, to the nearesttenth of a centimetre.ClosingQuestion 14 gives students an opportunity to summarize a situation in whichthe sine law can be used to solve a problem. Students might complete thequestion on their own or with a partner. After students have completed thequestion, have them share their solutions: Invite a student to explain her or his solution to the class or a group. Thenask for a few different solutions. Emphasize that measurements can bedetermined in different ways. Ask: How are all the solutions the same?How are some solutions different from other solutions?8.2: Applying the Sine Law 301

Alternatively, have a student draw the triangle on the board. Then ask theclass: What could you calculate? How? What could you calculate next?How? When finished, provide time for students to compare their ownsolution with the class solution.Ensure that students realize how the sum of the interior angles of a trianglecan be used to determine R.Assessment and Differentiating InstructionWhat You Will See Students Doing When students understand If students misunderstand Students write the sine law correctly, with the value to bedetermined in the numerator.Students may be unable to identify opposite angles and sides.They may substitute incorrectly because of errors wheninterpreting measurements or drawing diagrams. They maynot write the sine law with the value to be determined in thenumerator.Students solve an equation correctly and state the results in aconcluding sentence.Students cannot solve proportions by isolating the value to bedetermined and then calculating and rounding. They may notremember how to use the inverse sine, or they may not statethe solution in a concluding sentence to answer the question,using the correct units.Students create and label a parallelogram correctly to modelthe situation.Students cannot create or label a parallelogram correctly tomodel the situation. They may label the given sidesincorrectly.Students realize that they need to determine the angleopposite a short side of the parallelogram. Then they use thesum of the interior angles of a triangle to determine the angleopposite a long side of the parallelogram.Students may have difficulty with multi-step problems. Theymay not realize that they need to determine the angleopposite a short side of the parallelogram before they candetermine the angle opposite a long side.Students use the sine law to determine the length of the longsides of the parallelogram.Students may correctly calculate the angle opposite a shortside but use this angle in the sine law to determine the lengthof a long side.Differentiating Instruction How You Can RespondEXTRA SUPPORT1. To help students visualize a triangle, guide them as they draw a diagram with both the given and unknown informationmarked. Suggest circling the information for each ratio in a different colour. Help them note whether they are determining aside length or angle measure, and remind them to place this value in the numerator of the sine law.2. Suggest strategies for checking accuracy. Students might divide the sine of each angle by the length of the opposite side tocheck that they are equal, divide the length of each side by the sine of the opposite angle to check that they are equal, oradd the angle measures to check that the sum is 180 .EXTRA CHALLENGE1. Students could create problems that can be solved using the sine law and then solve their problems.2. Have students investigate real-world situations in which the sine law would be useful. The contexts used for the Practisingquestions in Lesson 8.2 might help students think of situations to investigate.302 Principles of Mathematics 10: Chapter 8: Acute Triangle TrigonometryCopyright 2011 by Nelson Education Ltd.Key Assessment Question 7

MID-CHAPTER REVIEWBig Ideas Covered So Far The ratiolength of opposite sidesin (angle)is the same for all three angle–side pairs in an acute triangle. The sine law states that in any acute triangle, ABC,asin A bsin B csin C. The sine law can be used to solve a problem modelled by an acute triangle if you can determine two sides and the angleopposite one of these sides, or two angles and any side.Using the Frequently Asked QuestionsHave students keep their Student Books closed. Display the Frequently Asked Questions on a board. Have students discuss thequestions and use the discussion to draw out what the class thinks are good answers. Then have students compare the classanswers with the answers on Student Book page 435. Students can refer to the answers to the Frequently Asked Questions asthey work through the Practice Questions.Using the Mid-Chapter ReviewAsk students if they have any questions about any of the topics covered so far in the chapter. Review any topics that studentswould benefit from considering again. Assign Practice Questions for class work and for homework.To gain greater insight into students’ understanding of the material covered so far in the chapter, you may want to ask questionssuch as the following: How do you decide which sides and angles to use in the ratios for the sine law? How are solutions that use the sine law to determine a side length the same as solutions that use the sine law to determinean angle measure? How are these solutions different? The sine law can only be used if certain information about an acute triangle is given. Explain why. How would you explain the inverse sine ratio? What other way could you explain it?Copyright 2011 by Nelson Education Ltd. What have you learned about the sine law?Chapter 8 Mid-Chapter Review 303

Lesson at a GlanceGOALExplore the relationship between sidelengths and angle measures in atriangle using the cosines of angles.Prerequisite Skills/Concepts Solve problems involving proportions. Apply the Pythagorean theorem to determine side lengths.Specific Expectation Explore the development of the cosine law within acute triangles (e.g.,use dynamic geometry software to verify the cosine law[; follow thealgebraic development of the cosine law and identify its relationship tothe Pythagorean theorem and the cosine ratio [student reproduction of thedevelopment of the formula is not required]).]Mathematical Process Focus Reasoning and Proving ConnectingStudent Book Pages 437–439Preparation and PlanningPacing5 10 min Introduction35 45 min Teaching and Learning10 15 min ConsolidationMaterials dynamic geometry softwareRecommended PracticeQuestions 1, 2, 3, 4, 5New Vocabulary/Symbolscosine lawNelson Websitehttp://www.nelson.com/mathMATH BACKGROUND LESSON OVERVIEW Students should understand the Pythagorean theorem. In this lesson, students explore the cosine law for an acute triangle, ABC:a2 b2 c2 – 2 bc cos Ab2 a2 c2 – 2 ac cos Bc2 a2 b2 – 2 ab cos C The cosine law is an extension of the Pythagorean theorem to triangles that do not have a right angle.304 Principles of Mathematics 10: Chapter 8: Acute Triangle TrigonometryCopyright 2011 by Nelson Education Ltd.8.3 EXPLORING THE COSINE LAW

1Introducing the Lesson(5 to 10 min)Begin by reviewing the Pythagorean theorem: Draw and label an acute triangle, ABC. Ask students why the Pythagorean theorem cannot be used to determinethe measure of c. (The triangle does not have a right

Lesson 8.2 Extra Practice Lesson 8.3: Exploring the Cosine Law, pp. 437–439 Explore the relationship between side lengths and angle measures in a triangle using the cosines of angles. 1 day dynamic geometry software Lesson 8.4: Applying the Cosine Law, pp. 440–445 Use