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Chapter 12: Trigonometry Stages 5.1/5.2 – Year 9 Unit length: 4 weeks Strand: Measurement and Geometry Substrand: Rationale: Trigonometry is a branch of geometry that is very important in fields such as navigation, surveying, engineering, astronomy and architecture. Students will use basic trigonometry to find unknown sides and angles in right-angled triangles. Teacher: Outcomes: Review of earlier work from Stage 4 and selections from: Right-Angled Triangles (Trigonometry) [Stages 5.1, 5.2◊] Dates taught: A student: Content statements: • selects and uses appropriate strategies to solve problems (MA5.1-2WM) • provides reasoning to support conclusions that are appropriate to the context (MA5.1-3WM) • selects appropriate notations and conventions to communicate mathematical ideas and solutions (MA5.2-1WM) • applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression (MA5.1-10MG) • applies trigonometry to solve problems, including problems involving bearings (MA5.2-13MG) • Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles (ACMMG223) • • Apply trigonometry to solve right-angled triangle problems (ACMMG224) Solve right-angled triangle problems including those involving direction and angles of elevation and depression (ACMMG245) Right-Angled Triangles (Trigonometry) ________ to ________ Teacher reflection Strengths Weaknesses Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12 Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1 1 Resources: Literacy: • • Foundation worksheets • Drag-and-drop activities GeoGebra activities adjacent bearing opposite side tangent ratio angle of depression cosine ratio similar triangles trigonometric ratios hypotenuse sine ratio trigonometry Other angle of elevation Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12 Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1 2 Student book / eBook 12:01 Right-angled triangles 12:02 Similar right-angled triangles: The ratio of sides 12:03 Tangent ratio Content dot points • identify the hypotenuse, adjacent sides and opposite sides with respect to a given angle in a right-angled triangle in any orientation • label sides of right-angled triangles in different orientations in relation to a given angle • label the side lengths of a right-angled triangle in relation to a given angle • define the sine, cosine and tangent ratios for angles in right-angled triangles • use similar triangles to investigate the constancy of the sine, cosine and tangent ratios for a given angle in rightangled triangles • • Register Technology Drag-and-drop Maths terms 12 Resources and suggestions Foundation worksheet 12:01 – Sides of right-angled triangles Review labelling sides according to the angle given. Use triangles in different orientations. GeoGebra Ratios of the sides of similar rightangled tringles use trigonometric notation find the size in degrees and minutes of unknown angles in right-angled triangles Use Questions 1 and 2 of Exercise 12:02 to discover the relationship between the sides of a triangle for a 30 angle and 50 angle. Use this to generalise the pattern. Preview the GeoGebra activity on p. 360. Read through pp. 361–362 to understand why tan = o/a. Show students how to draw the commonly used Greek letters and liken them to other pronumerals previously used. Foundation worksheet 12:03 – The tangent ratio Investigation 12:03 – Tangent ratio (p. 365) Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12 Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1 3 12:04 Finding an unknown side • use a calculator to find the values of the trigonometric ratios, given angles measured in degrees and minutes GeoGebra Finding an unknown side using the tangent ratio Foundation worksheet 12:04 – Finding unknown sides using the tangent ratio Help students draw diagrams representing the problems described. Preview the GeoGebra activity on p. 368. 12:05 Finding an unknown angle • • • 12:06 Sine and cosine ratios • 12:07 Finding unknown sides with sine and cosine A Finding a short side • use a calculator to find approximations of the trigonometric ratios for a given angle measured in degrees use a calculator to find an angle correct to the nearest degree, given one of the trigonometric ratios for the angle use a calculator to find the size in degrees and minutes of an angle, given a trigonometric ratio for the angle Drag-and-drop find the lengths of unknown sides in right-angled triangles where the given angle is measured in degrees and minutes Drag-and-drop select and use appropriate trigonometric ratios in right-angled triangles to find unknown sides, including the hypotenuse B Finding the hypotenuse 12:08 Using sine and cosine to find an unknown angle Finding angles Ensure students know how to use their calculators when finding unknown angles. Foundation worksheet 12:05 – Finding unknown angles using the tangent ratio The trigonometric ratios Use memory strategies to remember the relationship between the trigonometric ratios, e.g. Soh Cah Toa or ‘Some Old Hare Came A Hopping Through Our Area’, p. 373. Drag-and-drop Finding sides Foundation worksheet 12:07A – Finding unknown sides using sine and cosine GeoGebra Finding an unknown side using the sine and cosine ratios Preview the GeoGebra activity on p. 378. Foundation worksheet 12:07B – Finding the hypotenuse Preview the GeoGebra activity on p. 381. Finding the hypotenuse using the sine and cosine ratios • select and use appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest degree GeoGebra Using the trig ratios to find an angle Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12 Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1 Preview the GeoGebra activity on p. 384. 4 12:09 Miscellaneous exercises A Angles of elevation and depression B Compass bearings C Other topics • solve a variety of practical problems involving angles of elevation and depression, including problems for which a diagram is not provided • draw diagrams to assist in solving practical problems involving angles of elevation and depression • interpret three-figure bearings and compass bearings • interpret directions given as bearings and represent them in diagrammatic form • solve a variety of practical problems involving bearings, including problems for which a diagram is not provided • draw diagrams to assist in solving practical problems involving bearings • check the reasonableness of solutions to problems involving bearings Drag-and-drop Bearings 1 Bearings 2 Review Foundation worksheet 12:09B – Compass bearings Use compasses as a class to navigate around the school. Come back to the classroom and draw the route taken including the relevant bearings. Make use of Maths terms 12. (p. 390) Diagnostic test 12 – Use the right-hand column to assist in remediation when errors occur. (p. 391) Assignment 12A – Exam-style questions for revision (p. 392) Assignment 12B – Working mathematically problems (p. 393) Assignment 12C – Use this cumulative revision to review previous topics (p. 394) Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2 Teaching Program — Chapter 12 Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0053 1 5