Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 15 Curriculum Development Division Ministry of Education Malaysia PART D: TRIGONOMETRY IV TEACHING AND LEARNING STRATEGIES Pupils may face problem in finding the length of the side of a right-angled triangle given one angle and any other side. Strategy: By referring to the sides given, choose the correct trigonometric ratio to write the relation between the sides. 1. Find the length of the unknown side with the aid of a calculator. LEARNING OBJECTIVE Upon completion of Part D, pupils will be able to find the angle of a right-angled triangle given the length of any two sides.
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 16 Curriculum Development Division Ministry of Education Malaysia Find the length of PR. With reference to the given angle, PR is the opposite side and QR is the adjacent side. Thus tangent ratio is used to form the relation of the sides. tan 50o = 5 PR PR = 5 tan 50o Find the length of TS. With reference to the given angle, TR is the adjacent side and TS is the hypotenuse side. Thus cosine ratio is used to form the relation of the sides. cos 32o = 8 TS TS cos 32o = 8 TS = 8 cos32o LESSON NOTES
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 17 Curriculum Development Division Ministry of Education Malaysia Find the value of x in each of the following. Example 1: tan 25o = 3 x x = 3 tan 25o = 6.434 cm Example 2: sin 41.27o = 5 x x = 5 sin 41.27o = 3.298 cm Example 3: cos 34o 12 = 6 x x = 6 cos 34o 12 = 4.962 cm Example 4: tan 63o = 9 x x = 9 tan 63o = 17.66 cm EXAMPLES
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 18 Curriculum Development Division Ministry of Education Malaysia Find the value of x for each of the following. 1. 2. 3. 4. 5. 6. TEST YOURSELF D 10 cm 6 cm 13 cm
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 19 Curriculum Development Division Ministry of Education Malaysia PART E: TRIGONOMETRY V TEACHING AND LEARNING STRATEGIES Pupils may face problem in relating the coordinates of a given point to the definition of the trigonometric functions. Strategy: Teacher should use the Cartesian plane to relate the coordinates of a point to the opposite side, adjacent side and the hypotenuse side of a right-angled triangle. LEARNING OBJECTIVE Upon completion of Part E, pupils will be able to state the definition of trigonometric functions in terms of the coordinates of a given point on the Cartesian plane and use the coordinates of the given point to determine the ratio of the trigonometric functions.
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 20 Curriculum Development Division Ministry of Education Malaysia In the diagram, with reference to the angle , PR is the opposite side, OP is the adjacent side and OR is the hypotenuse side. r y OR PR hypotenuse opposite sin r x OR OP hypotenuse adjacent cos x y OP PR adjacent opposite tan LESSON NOTES θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 21 Curriculum Development Division Ministry of Education Malaysia PART F: TRIGONOMETRY VI TEACHING AND LEARNING STRATEGIES Pupils may face difficulties in determining that the sign of the x-coordinate and y-coordinate affect the sign of the trigonometric functions. Strategy: Teacher should use the Cartesian plane and use the points on the four quadrants and the values of the x-coordinate and y-coordinate to show how the sign of the trigonometric ratio is affected by the signs of the x-coordinate and y-coordinate. Based on the A – S – T – C, the teacher should guide the pupils to determine on which quadrant the angle is when given the sign of the trigonometric ratio is given. (a) For sin to be positive, the angle must be in the first or second quadrant. (b) For cos to be positive, the angle must be in the first or fourth quadrant. (c) For tan to be positive, the angle must be in the first or third quadrant. LEARNING OBJECTIVE Upon completion of Part F, pupils will be able to relate the sign of the trigonometric functions to the sign of x-coordinate and y-coordinate and to determine the sign of each trigonometric ratio in each of the four quadrants.
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 22 Curriculum Development Division Ministry of Education Malaysia First Quadrant sin = y r (Positive) cos = x r (Positive) tan = y x (Positive) (All trigonometric ratios are positive in the first quadrant) Second Quadrant sin = y r (Positive) cos = x r (Negative) tan = y x (Negative) (Only sine is positive in the second quadrant) Third Quadrant sin = y r (Negative) cos = x r (Negative) tan = y y x x (Positive) (Only tangent is positive in the third quadrant) Fourth Quadrant sin = y r (Negative) cos = x r (Positive) tan = y x (Negative) (Only cosine is positive in the fourth quadrant) LESSON NOTES θ θ θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 23 Curriculum Development Division Ministry of Education Malaysia Using acronym: Add Sugar To Coffee (ASTC) sin is positive sin is negative cos is positive cos is negative tan is positive tan is negative A – All positive T – only tan is positive C – only cos is positive S – only sin is positive
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 24 Curriculum Development Division Ministry of Education Malaysia State the quadrants the angle is situated and show the position using a sketch. 1. sin = 0.5 2. tan = 1.2 3. cos = −0.16 4. cos = 0.32 5. sin = −0.26 6. tan = −0.362 TEST YOURSELF F
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 25 Curriculum Development Division Ministry of Education Malaysia PART G: TRIGONOMETRY VII TEACHING AND LEARNING STRATEGIES Pupils may face problem in calculating the length of the sides of a right-angled triangle drawn on a Cartesian plane and determining the value of the trigonometric ratios when a point on the Cartesian plane is given. Strategy: Teacher should revise the Pythagoras Theorem and help pupils to recall the right-angled triangles commonly used, known as the Pythagorean Triples. LEARNING OBJECTIVE Upon completion of Part G, pupils will be able to calculate the length of the side of right-angled triangle on a Cartesian plane and write the value of the trigonometric ratios given a point on the Cartesian plane
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 26 Curriculum Development Division Ministry of Education Malaysia The Pythagoras Theorem: (a) 3, 4, 5 or equivalent (b) 5, 12, 13 or equivalent (c) 8, 15, 17 or equivalent The sum of the squares of two sides of a right-angled triangle is equal to the square of the hypotenuse side. PR2 + QR2 = PQ2 LESSON NOTES
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 27 Curriculum Development Division Ministry of Education Malaysia 1. Write the values of sin , cos and tan from the diagram below. OA2 = (−6)2 + 82 = 100 OA = 100 = 10 sin = 8 4 10 5 y r cos = 6 3 10 5 x r tan = 8 4 6 3 y x 2. Write the values of sin , cos and tan from the diagram below. OB2 = (−12)2 + (−5)2 = 144 + 25 = 169 OB = 169 = 13 sin = 5 13 y r cos = 12 13 x r tan = 5 5 12 12 EXAMPLES θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 28 Curriculum Development Division Ministry of Education Malaysia Write the value of the trigonometric ratios from the diagrams below. 1. sin = cos = tan = 2. sin = cos = tan = 3. sin = cos = tan = 4. sin = cos = tan = 5. sin = cos = tan = 6. sin = cos = tan = TEST YOURSELF G θ θ θ θ θ θ θ B(5,4) B(5,12) x y
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 29 Curriculum Development Division Ministry of Education Malaysia PART H: TRIGONOMETRY VIII TEACHING AND LEARNING STRATEGIES Pupils may find difficulties in remembering the shape of the trigonometric function graphs and the important features of the graphs. Strategy: Teacher should help pupils to recall the trigonometric graphs which pupils learned in Form 4. Geometer’s Sketchpad can be used to explore the graphs of the trigonometric functions. LEARNING OBJECTIVE Upon completion of Part H, pupils will be able to sketch the trigonometric function graphs and know the important features of the graphs.
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 30 Curriculum Development Division Ministry of Education Malaysia (a) y = sin x The domain for x can be from 0o to 360o or 0 to 2 in radians. Important points: (0, 0), (90o , 1), (180o , 0), (270o , −1) and (360o , 0) Important features: Maximum point (90o , 1), Maximum value = 1 Minimum point (270o , −1), Minimum value = −1 (b) y = cos x Important points:(0o , 1), (90o , 0), (180o , −1), (270o , 0) and (360o , 1) Important features: Maximum point (0o , 1) and (360o , 1), Maximum value = 1 Minimum point (180o , −1) Minimum value = 1 LESSON NOTES
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 31 Curriculum Development Division Ministry of Education Malaysia (c) y = tan x Important points: (0o , 0), (180o , 0) and (360o , 0) Is there any maximum or minimum point for the tangent graph?
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 32 Curriculum Development Division Ministry of Education Malaysia 1. Write the following trigonometric functions to the graphs below: y = cos x y = sin x y = tan x 2. Write the coordinates of the points below: (a) (b) A(0,1) TEST YOURSELF H y = cos x y = sin x
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 33 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF A: 1. Opposite side = AB Adjacent side = AC Hypotenuse side = BC 2. Opposite side = PQ Adjacent side = QR Hypotenuse side = PR 3. Opposite side = YZ Adjacent side = XZ Hypotenuse side = XY 4. Opposite side = LN Adjacent side = MN Hypotenuse side = LM 5. Opposite side = UV Adjacent side = TU Hypotenuse side = TV 6. Opposite side = RT Adjacent side = ST Hypotenuse side = RS TEST YOURSELF B: 1. sin = AB BC cos = AC BC tan = AB AC 2. sin = PQ PR cos = QR PR tan = PQ QR 3. sin = YZ YX cos = XZ XY tan = YZ XZ 4. sin = LN LM cos = MN LM tan = LN MN 5. sin = UV TV cos = UT TV tan = UV UT 6. sin = RT RS cos = ST RS tan = RT TS ANSWERS
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 34 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF C: 1. sin = 1 3 = sin-1 1 3 = 19o 28 2. cos = 1 2 = cos-1 1 2 = 60o 3. tan = 5 3 = tan-1 5 3 = 59o 2 4. cos = 5 8 = cos-1 5 8 = 51o 19 5. tan = 7.5 9.2 = tan-1 7.5 9.2 = 39o 11 6. sin = 6.5 8.4 = sin-1 6.5 8.4 = 50o 42 TEST YOURSELF D: 1. tan 32o = 4 x x = 4 tan 32o = 6.401 cm 2. sin 53.17o = 7 x x = 7 sin 53.17o = 5.603 cm 3. cos 74o 25 = 10 x x = 10 cos 74o 25 = 2.686 cm 4. sin 55 1 3 o = 6 x x = 1 3 6 sin55 o = 7.295 cm 5. tan 47o = 13 x x = 13 tan 47o = 13.94 cm 6. cos 61o = 10 x x = 10 cos61o = 20.63 cm
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 35 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF F: 1. 1 ST and 2nd 2. 1 st and 3rd 3. 2 nd and 3rd 4. 1 st and 4th 5. 3 rd and 4th 6. 2 nd and 4th TEST YOURSELF G: 1. sin = 4 5 cos = 3 5 tan = 4 3 2. sin = 12 13 cos = 5 13 tan = 12 5 3. sin = 4 5 cos = 3 5 tan = 4 3 4. sin = 4 5 cos = 3 5 tan = 4 3 5. sin = 8 17 cos = 15 17 tan = 8 15 6. sin = 5 13 cos = 12 13 tan = 5 12
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 36 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF H: 1. y = tan x y = sin x y = cos x 2. (a) A (0, 1), B (90o , 0), C (180o , 1), D (270o , 0) (b) P (90o , 1), Q (180o , 0), R (270o , 1), S (360o , 0)