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1 WORK PROGRAM MQ 9 NSW 5.2 Pathway Chapter 14 Right-angled trigonometry Strand: Measurement, Space and geometry, Number, Patterns and algebra Substrands and outcomes: Perimeter and area MS4.1 Uses formulae and Pythagoras’ theorem in calculating perimeter and area of circles and figures composed of rectangles and triangles Trigonometry MS5.1.2 Applies trigonometry to solve problems (diagram given) including those involving angles of elevation and depression Trigonometry MS5.2.3 Applies trigonometry to solve problems including those involving bearings Two-dimensional space SGS3.2b Measures, constructs and classifies angles Position SGS3.3 Uses a variety of mapping skills Properties of geometrical figures SGS4.3 Classifies, constructs, and determines the properties of triangles and quadrilaterals Fractions, decimals and percentages NS4.3 Operates with fractions, decimals, percentages, ratios and rates Algebraic techniques PAS4.4 Uses algebraic techniques to solve linear equations and simple inequalities Section Are you ready? (page 484) GC tips, Investigations, History of mathematics, Maths Quest challenge, 10 Quick Questions, Code puzzles SkillSHEETS, WorkSHEETS, Interactive games, Test yourself, Topic tests (CD–ROM) SkillSHEETS (page 484) 14.1: Measuring angle with a protractor 14.3: Rounding to a given number of decimal places 14.4: Solving equations of x the type a = to find x b 14.5: Solving equations of b the type a = to find x x Technology applications (CD–ROM) Learning outcomes SGS3.2b using a protractor to measure angles SGS 3.3 locating a place on a map which is a given direction from a landmark SGS4.3 recognising and classifying types of triangles on the basis 2 14.10: Using Pythagoras’ theorem 14.11 Identifying and using properties of a triangle 14.12: Understanding direction What is trigonometry? (page 485) Naming the sides of a right-angled triangle (page 485) WE 1, 2, 3a-c Ex 14A Naming the sides of a right-angled triangle (page 490) Investigation: The cosine and tangent ratios (page 488) Maths Quest challenge: Q1 (page 493) SkillSHEET 14.1: Measuring angles with a protractor (page 491) Game time 001 (page 493) Cabri geometry: Investigating the sine ratio (page 486) Cabri geometry: Investigating the cosine ratio (page 489) Cabri geometry: Investigating the tangent ratio (page 489) of their properties (acute-angled, rightangled, obtuse-angled, scalene, isosceles and equilateral triangles) MS4.1 using Pythagoras’ theorem to find the length of sides in rightangled triangles NS4.3 rounding decimals to a given number of places PAS4.4 solving equations using algebraic methods that involve up to and including three steps in the solution process and have solutions that are not necessarily whole numbers MS5.1.2 identifying the hypotenuse, adjacent and opposite sides with respect to a given angle in a right-angled triangle in any orientation recognising that the 3 Cabri geometry: Sine, cosine, tangent (page 491) Trigonometric ratios (page 494) WE 4, 5a-b Ex 14B Trigonometric ratios (page 496) Maths Quest challenge: Q1 (page 498) SkillSHEET 14.2: Labelling sides of a triangle (page 496) SkillSHEET 14.3: Rounding to a given number of decimal places (page 497) WorkSHEET 14.1 (page 498) Mathcad: Trigonometric ratios (page 496) Excel: Introducing the trig ratios (page 497) ratio of matching sides in similar right-angled triangles is constant for equal angles defining the sine, cosine and tangent ratios for angles in right-angled triangles labelling sides of rightangled triangles in different orientations in relation to a given angle (Applying strategies, Communicating) explaining why the ratio of matching sides in similar right-angled triangles is constant for equal angles (Communicating, Reasoning) MS5.1.2 defining the sine, cosine and tangent ratios for angles in right-angled triangles using trigonometric notation e.g. sin A labelling sides of rightangled triangles in different orientations 4 Finding trigonometric ratios using a calculator (page 499) WE 6a-c Ex 14C Finding trigonometric ratios using a calculator (page 500) Finding side lengths (page 500) WE 7, 8 Ex 14D Finding side lengths (page 502) GC tip – Casio: Finding sin, cos or tan of an angle (page 499) Angles and the calculator (page 505) WE 9a-b, 10, 11a-b Ex 14E Angles and the GC tip – Casio: Finding DMS or ‘ “ on a graphics calculator Maths Quest challenge: Q1 (page 504) SkillSHEET 14.4: Solving equations of the type x a = to find x b (page 502) SkillSHEET 14.5: Solving equations of the type b a = to find x x (page 503) SkillSHEET 14.6: Rearranging formulas (page 503) Game time 002 (page 504) WorkSHEET 14.2 (page 504) SkillSHEET 14.7: Rounding angles to the nearest degree (page 508) GC tip – TI: Finding sin, cos or tan of an angle (page 499) in relation to a given angle (Applying strategies, Communicating) MS5.1.2 using a calculator to find approximations of the trigonometric ratios of a given angle measured in degrees Excel: Using sine (page 502) Excel: Using cosine (page 502) Excel: Using cosine (DIY) (page 503) Excel: Using tangent (page 503) Excel: Using tangent (DIY) (page 503) Excel: Universal trigonometric calculator (page 503) Mathcad: Finding side lengths (page 503) MS5.1.2 using trigonometric notation e.g. sin A selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown sides, including the hypotenuse solving problems in practical situations involving right-angled triangles (Applying strategies) GC program – TI: Entering angles expressed in degrees, minutes and seconds MS5.1.2 using a calculator to find approximations of 5 calculator (page 508) Finding the size of an angle (page 509) WE 12a-b, 13a-b Ex 14F Finding the size of an angle (page 513) (page 505) GC tip – Casio: Converting degrees in decimal form to degrees, minutes and seconds (page 507) Maths Quest challenge: Q1-2 (page 508) 10 Quick Questions 1 (page 509) 10 Quick Questions 2 (page 514) Code puzzle (page 515) SkillSHEET 14.8: Rounding angles to the nearest minute (page 508) SkillSHEET 14.9: Rounding angles to the nearest second (page 508) (page 505) GC program – TI: Converting degrees in decimal form to degrees, minutes and seconds (page 507) Excel: Converting angles (page 508) Mathcad: Finding angles (page 513) Excel: Finding angles (page 513) the trigonometric ratios of a given angle measured in degrees using a calculator to find an angle correct to the nearest degree, given one of the trigonometric ratios of the angle MS5.2.3 using a calculator to find trigonometric ratios of a given approximation for angles measured in degrees and minutes using a calculator to find an approximation for an angle in degrees and minutes, given the trigonometric ratio of the angle MS5.1.2 selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest degree MS5.2.3 using trigonometric 6 Applications (page 516) WE 14a-c, 15 Ex 14G Applications (page 518) Investigation: Using an SkillSHEET 14.10: Using inclinometer to measure Pythagoras’ theorem inaccessible heights (page 518) (page 520) SkillSHEET 14.11: Identifying and using properties of a triangle (page 518) WorkSHEET 14.3 (page 519) Excel: Universal trigonometric calculator (page 519) ratios to find unknown angles in degrees and minutes in right-angled triangles MS5.1.2 selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown sides, including the hypotenuse selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest degree identifying angles of elevation and depression solving problems involving angles of elevation and depression when given a diagram solving problems in practical situations involving right-angled triangles (Applying strategies) 7 interpreting diagrams in questions involving angles of elevation and depression (Communicating) MS5.2.3 finding unknown sides in right-angled triangles where the given angle is measured in degrees and minutes using trigonometric ratios to find unknown angles in degrees and minutes in right-angled triangles drawing diagrams and using them to solve word problems which involve angles of elevation and depression solving practical problems involving angles of elevation and depression (Applying strategies) checking the reasonableness of answers to trigonometry problems 8 Bearings (page 521) WE 16a-b, 17a-b, 18a-c Ex 14H Bearings (page 524) History of mathematics: George Everest (page 526) SkillSHEET 14.12: Mathcad: Converting to Understanding direction true bearings (page 524) (page 524) SkillSHEET 14.13: Drawing a diagram from given directions (page 525) (Reasoning) MS5.2.3 finding unknown sides in right-angled triangles where the given angle is measured in degrees and minutes using trigonometric ratios to find unknown angles in degrees and minutes in right-angled triangles using three-figure bearings (e.g. 035, 225) and compass bearings e.g. SSW drawing diagrams and using them to solve word problems which involve bearings solving simple problems involving three-figure bearings (Applying strategies, Communicating) recognising directions given as SSW, NE etc. (Communicating) interpreting directions given as bearings (Communicating) 9 Summary (page 527) Chapter review (page 528) ‘Test yourself’ multiple choice questions (page 530) Topic tests (2)