Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 41 Curriculum Development Division Ministry of Education Malaysia ACTIVITY B1: –4 –3 –2 –1 0 1 2 3 4 x 2 4 6 8 10 12 14 16 18 20 y
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 42 Curriculum Development Division Ministry of Education Malaysia PART B3: 1. a = 3, b = 16, c = – 3, d = – 18 2. a = 3.5, b = 7, c = – 2.5, d = – 8 3. a = 1.4, b = 2.4, c = – 1.6, d = – 3.8 4. a = 0.7, b = 1.8, c = – 0.5, d = – 1.4 5. a = 0.08, b = 0.16, c = – 0.02, d = – 0.17 6. a = 6, b = 15, c = – 3, d = – 17 7. a = 2, b = 8, c = – 0.5, d = – 8.5 8. a = 1.4, b = 3.6, c = – 0.8, d = – 3.4 9. a = 0.5, b = 1.7, c = – 0.4, d = – 1.6 10. a = 0.06, b = 0.16, c = – 0.07, d = – 0.15 PART B4: 1. (a) 6.4 (b) – 2.8 2. (a) – 12 (b) 13 3. (a) – 2.5 (b) 9 4. (a) 0.6 (b) – 5.4 5. (a) 8 (b) – 6.5 6. (a) – 16 (b) 22 7. (a) 0.7 (b) – 1.3 8. (a) – 0.08 (b) 0.12 9. (a) – 3.5, 1.5 (b) – 3 , 1 10. (a) – 1.6, 0.6 (b) – 2.7, 1.7 11. (a) 2.2 (b) – 3.5 12. (a) – 2.3 (b) – 0.6 (c) 1.4 ACTIVITY B2: k =15, h = 1.1, 8.9 Two possible locations: (1.1, 15), (8.9, 15)
Unit 1: Negative Numbers UNIT 7 LINEAR INEQUALITIES B a s i c E s s e n t i a l A d d i t i o n a l M a t h e m a t i c s S k i l l s Curriculum Development Division Ministry of Education Malaysia
TABLE OF CONTENTS Module Overview 1 Part A: Linear Inequalities 2 1.0 Inequality Signs 3 2.0 Inequality and Number Line 3 3.0 Properties of Inequalities 4 4.0 Linear Inequality in One Unknown 5 Part B: Possible Solutions for a Given Linear Inequality in One Unknown 7 Part C: Computations Involving Addition and Subtraction on Linear Inequalities 10 Part D: Computations Involving Division and Multiplication on Linear Inequalities 14 Part D1: Computations Involving Multiplication and Division on Linear Inequalities 15 Part D2: Perform Computations Involving Multiplication of Linear Inequalities 19 Part E: Further Practice on Computations Involving Linear Inequalities 21 Activity 27 Answers 29
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities 1 Curriculum Development Division Ministry of Education Malaysia MODULE OVERVIEW 1. The aim of this module is to reinforce pupils‟ understanding of the concept involved in performing computations on linear inequalities. 2. This module can be used as a guide for teachers to help pupils master the basic skills required to learn this topic. 3. This module consists of six parts and each part deals with a few specific skills. Teachers may use any parts of the module as and when it is required. 4. Overall lesson notes given in Part A stresses on important facts and concepts required for this topic.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 2 Curriculum Development Division Ministry of Education Malaysia PART A: LINEAR INEQUALITIES LEARNING OBJECTIVE Upon completion of Part A, pupils will be able to understand and use the concept of inequality. TEACHING AND LEARNING STRATEGIES Some pupils might face problems in understanding the concept of linear inequalities in one unknown. Strategy: Teacher should ensure that pupils are able to understand the concept of inequality by emphasising the properties of inequalities. Linear inequalities can also be taught using number lines as it is an effective way to teach and learn inequalities.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 3 Curriculum Development Division Ministry of Education Malaysia PART A: LINEAR INEQUALITY 1.0 Inequality Signs a. The sign “<” means „less than‟. Example: 3 < 5 b. The sign “>” means „greater than‟. Example: 5 > 3 c. The sign “ ” means „less than or equal to‟. d. The sign “ ” means „greater than or equal to‟. 2.0 Inequality and Number Line −3 < − 1 −3 is less than − 1 and −1 > − 3 −1 is greater than − 3 1 < 3 1 is less than 3 and 3 > 1 3 is greater than 1 OVERALL LESSON NOTES − 3 − 2 −1 x 0 1 2 3
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 4 Curriculum Development Division Ministry of Education Malaysia 3.0 Properties of Inequalities (a) Addition Involving Inequalities Arithmetic Form Algebraic Form 12 8 so 12 4 8 4 2 9 so 2 6 9 6 If a > b, then a c b c If a < b, then a c b c (b) Subtraction Involving Inequalities Arithmetic Form Algebraic Form 7 > 3 so 7 5 35 2 < 9 so 2 6 9 6 If a > b, then a c b c If a < b, then a c b c (c) Multiplication and Division by Positive Integers When multiply or divide each side of an inequality by the same positive number, the relationship between the sides of the inequality sign remains the same. Arithmetic Form Algebraic Form 5 > 3 so 5 (7) > 3(7) 12 > 9 so 12 9 3 3 If a > b and c > 0 , then ac > bc If a > b and c > 0, then a b c c 2 5 so 2(3) 5(3) 8 12 so 2 12 2 8 If a b and c 0 , then ac bc If a b and c 0 , then c b c a
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 5 Curriculum Development Division Ministry of Education Malaysia (d) Multiplication and Division by Negative Integers When multiply or divide both sides of an inequality by the same negative number, the relationship between the sides of the inequality sign is reversed. Arithmetic Form Algebraic Form 8 > 2 so 8(−5) < 2(−5) 6 < 7 so 6(−3) > 7(−3) 16 > 8 so 16 8 4 4 10 <15 so 10 15 5 5 If a > b and c < 0, then ac < bc If a < b and c < 0, then ac > bc If a > b and c < 0, then a b c c If a < b and c < 0, then a b c c Note: Highlight that an inequality expresses a relationship. To maintain the same relationship or „balance‟, pupils must perform equal operations on both sides of the inequality. 4.0 Linear Inequality in One Unknown (a) A linear inequality in one unknown is a relationship between an unknown and a number. Example: x > 12 4 m (b) A solution of an inequality is any value of the variable that satisfies the inequality. Examples: (i) Consider the inequality x 3 The solution to this inequality includes every number that is greater than 3. What numbers are greater than 3? 4 is greater than 3. And so are 5, 6, 7, 8, and so on. What about 5.5? What about 5.99? And 5.000001? All these numbers are greater than 3, meaning that there are infinitely many solutions! But, if the values of x are integers, then x 3 can be written as x 4, 5, 6, 7, 8,...
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 6 Curriculum Development Division Ministry of Education Malaysia A number line is normally used to represent all the solutions of an inequality. (ii) x > 2 (iii) x 3 The solid dot means the value 3 is included. The open dot means the value 2 is not included. − 2 − 1 0 1 2 3 x 4 o − 2 − 1 0 x 1 2 3 4 To draw a number line representing x 3 , place an open dot on the number 3. An open dot indicates that the number is not part of the solution set. Then, to show that all numbers to the right of 3 are included in the solution, draw an arrow to the right of 3.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 7 Curriculum Development Division Ministry of Education Malaysia PART B: POSSIBLE SOLUTIONS FOR A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN TEACHING AND LEARNING STRATEGIES Some pupils might have difficulties in finding the possible solution for a given linear inequality in one unknown and representing a linear inequality on a number line. Strategy: Teacher should emphasise the importance of using a number line in order to solve linear inequalities and should ensure that pupils are able to draw correctly the arrow that represents the linear inequalities. LEARNING OBJECTIVES Upon completion of Part B, pupils will be able to solve linear inequalities in one unknown by: (i) determining the possible solution for a given linear inequality in one unknown: (a) x h (b) x h (c) x h (d) x h (ii) representing a linear inequality: (a) x h (b) x h (c) x h (d) x h on a number line and vice versa.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 8 Curriculum Development Division Ministry of Education Malaysia PART B: POSSIBLE SOLUTIONS FOR A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN List out all the possible integer values for x in the following inequalities: (You can use the number line to represent the solutions) (1) x > 4 Solution: The possible integers are: 5, 6, 7, … (2) x 3 Solution: The possible integers are: – 4, − 5, −6, … (3) 3 x 1 Solution: The possible integers are: −2, −1, 0, and 1. −8 −5 −2 x −7 −6 −4 −3 −1 0 1 2 3 4 EXAMPLES −2 1 4 x −1 0 2 3 5 6 7 8 9 10 −8 −5 −2 x −7 −6 −4 −3 −1 0 1 2 3 4
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 9 Curriculum Development Division Ministry of Education Malaysia Draw a number line to represent the following inequalities: (a) x > 1 (b) x 2 (c) x 2 (d) x 3 TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 10 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES Some pupils might have difficulties when dealing with problems involving addition and subtraction on linear inequalities. Strategy: Teacher should emphasise the following rule: 1) When a number is added or subtracted from both sides of the inequality, the inequality sign remains the same. LEARNING OBJECTIVES Upon completion of Part C, pupils will be able perform computations involving addition and subtraction on inequalities by stating a new inequality for a given inequality when a number is: (a) added to; and (b) subtracted from both sides of the inequalities. PART C: COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION ON LINEAR INEQUALITIES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 11 Curriculum Development Division Ministry of Education Malaysia PART C: COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION ON LINEAR INEQUALITIES Operation on Inequalities 1) When a number is added or subtracted from both sides of the inequality, the inequality sign remains the same. Examples: (i) 2 < 4 Adding 1 to both sides of the inequality: The inequality sign is unchanged. LESSON NOTES 1 x 2 3 4 2 < 4 4 x 2 3 5 2 + 1 < 4 + 1 3 < 5
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 12 Curriculum Development Division Ministry of Education Malaysia (ii) 4 > 2 Subtracting 3 from both sides of the inequality: (1) Solve x 5 14. Solution: 9 5 5 14 5 5 14 x x x (2) Solve p 3 2. Solution: 3 2 3 3 2 3 5 p p p Subtract 5 from both sides of the inequality. Simplify. Add 3 to both sides of the inequality. Simplify. The inequality sign is unchanged. EXAMPLES x −1 0 1 2 1 x 2 3 4 4 > 2 4 − 3 > 2 − 3 1 > − 1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 13 Curriculum Development Division Ministry of Education Malaysia Solve the following inequalities: (1) m 4 2 (2) x 3.4 2.6 (3) x 13 6 (4) 4.5 d 6 (5) 23 m 17 (6) y 78 54 (7) 9 5 d (8) p 2 1 (9) 1 3 2 m (10) 3 8 x TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 14 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES The computations involving division and multiplication on inequalities can be confusing and difficult for pupils to grasp. Strategy: Teacher should emphasise the following rules: 1) When both sides of the inequality is multiplied or divided by a positive number, the inequality sign remains the same. 2) When both sides of the inequality is multiplied or divided by a negative number, the inequality sign is reversed. 3) LEARNING OBJECTIVES Upon completion of Part D, pupils will be able perform computations involving division and multiplication on inequalities by stating a new inequality for a given inequality when both sides of the inequalities are divided or multiplied by a number. PART D: COMPUTATIONS INVOLVING DIVISION AND MULTIPLICATION ON LINEAR INEQUALITIES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 15 Curriculum Development Division Ministry of Education Malaysia PART D1: COMPUTATIONS INVOLVING MULTIPLICATION AND DIVISION ON LINEAR INEQUALITIES 1. When both sides of the inequality is multiplied or divided by a positive number, the inequality sign remains the same. Examples: (i) 2 < 4 Multiplying both sides of the inequality by 3: LESSON NOTES x The inequality sign is unchanged. 1 x 2 3 4 2 < 4 2 3 < 4 3 6 < 12 x 6 8 10 12 14
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 16 Curriculum Development Division Ministry of Education Malaysia (ii) − 4 < 2 Dividing both sides of the inequality by 2: 2. When both sides of the inequality is multiplied or divided by a negative number, the inequality sign is reversed. Examples: (i) 4 < 6 Dividing both sides of the inequality by −1: The inequality sign is reversed. x −6 −5 −4 −3 3 x 4 5 6 The inequality sign is unchanged. −4 x − 2 0 2 4 < 6 4 (−1) > 6 (−1) − 4 > − 6 − 4 < 2 − 4 2 < 2 2 − 2 < 1 −2 − 1 0 1 2 x
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 17 Curriculum Development Division Ministry of Education Malaysia (ii) 1 > −3 Multiply both sides of the inequality by −1: Solve the inequality 3 12 q . Solution: (i) 3 12 q 3 12 3 3 q q 4 Divide each side of the inequality by −3. Simplify. The inequality sign is reversed. EXAMPLES The inequality sign is reversed. 1 > −3 x −3 −2 −1 0 1 (− 1) (1) < (−1) (−3) 1 3 x −1 0 1 2 3
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 18 Curriculum Development Division Ministry of Education Malaysia Solve the following inequalities: (1) 7 49 p (2) 6 18 x (3) −5c > 15 (4) 200 < −40p (5) 3d 24 (6) 2x 8 (7) 12 3x (8) 25 5y (9) 2m 16 (10) 6b 27 TEST YOURSELF D1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 19 Curriculum Development Division Ministry of Education Malaysia PART D2: PERFORM COMPUTATIONS INVOLVING MULTIPLICATION OF LINEAR INEQUALITIES Solve the inequality 3 2 x . Solution: 3 2 x . ) ( 2)3 2 2( x x 6 Multiply both sides of the inequality by −2. Simplify. The inequality sign is reversed. EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 20 Curriculum Development Division Ministry of Education Malaysia 1. Solve the following inequalities: (1) − 3 8 d (2) 8 2 n (3) 5 10 y (4) 6 7 b (5) 0 12 8 x (6) 8 0 6 x TEST YOURSELF D2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 21 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES Pupils might face problems when dealing with problems involving linear inequalities. Strategy: Teacher should ensure that pupils are given further practice in order to enhance their skills in solving problems involving linear inequalities. LEARNING OBJECTIVES Upon completion of Part E, pupils will be able perform computations involving linear inequalities. PART E: FURTHER PRACTICE ON COMPUTATIONS INVOLVING LINEAR INEQUALITIES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 22 Curriculum Development Division Ministry of Education Malaysia PART E: FURTHER PRACTICE ON COMPUTATIONS INVOLVING LINEAR INEQUALITIES Solve the following inequalities: 1. (a) m5 0 (b) x 2 6 (c) 3 + m > 4 2. (a) 3m < 12 (b) 2m > 42 (c) 4x > 18 TEST YOURSELF E1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 23 Curriculum Development Division Ministry of Education Malaysia 3. (a) m + 4 > 4m + 1 (b) 14 m 6m (c) 33m 4 m 4. (a) 4 x 6 (b) 153m 12 (c) 5 4 3 x
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 24 Curriculum Development Division Ministry of Education Malaysia (d) 5x 318 (e) 1 3p 10 (f) 3 4 2 x (g) 8 5 3 x (h) 4 3 2 p
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 25 Curriculum Development Division Ministry of Education Malaysia What is the smallest integer for x if 5x 3 18 ? Solution: 5x 3 18 5x 183 5x 15 O x 3 x = 4, 5, 6,… Therefore, the smallest integer for x is 4. x 3 A number line can be used to obtain the answer. 0 1 2 x 3 4 5 6 EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 26 Curriculum Development Division Ministry of Education Malaysia 1. If 3x 114, what is the smallest integer for x? 2. What is the greatest integer for m if m 7 4m1 ? 3. If 3 4 2 x , find the greatest integer value of x. 4. If 4 3 2 p , what is the greatest integer for p? 5. What is the smallest integer for m if 9 2 3 m ? TEST YOURSELF E2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 27 Curriculum Development Division Ministry of Education Malaysia 1 2 3 4 5 6 7 8 9 10 11 12 ACTIVITY
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 28 Curriculum Development Division Ministry of Education Malaysia HORIZONTAL: 4. 1 3 is an ___________. 5. An inequality can be represented on a number __________. 7. 2 6 is read as 2 is __________ than 6. 9. Given 2x 1 9 , x 5 is a _____________ of the inequality. 11. 3x 12 x 4 The inequality sign is reversed when divided by a ____________ integer. VERTICAL: 1. 2 1 2 x x The inequality sign remains unchanged when multiplied by a ___________ integer. 2. 6x 24 equals to x 4 when both sides are _____________ by 6. 3. x 5 equals to 3x 15 when both sides are _____________ by 3. 6. ___________ inequalities are inequalities with the same solution(s). 8. x 2 is represented by a ____________ dot on a number line. 10. 3x 6 is an example of ____________ inequality. 12. 5 3 is read as 5 is _____________ than 3.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 29 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF B: (a) (b) (c) (d) TEST YOURSELF C: (1) m 6 (2) x 6 (3) x 19 (4) d 1.5 (5) m 6 (6) y 24 (7) d 4 (8) p 3 (9) 2 5 m (10) x 5 TEST YOURSELF D1: (1) p 7 (2) x 3 (3) c 3 (4) p 5 (5) d 8 (6) x 4 (7) x 4 (8) y 5 (9) m 8 (10) 2 9 b TEST YOURSELF D2: (1) d 24 (2) n 16 (3) y 50 (4) b 42 (5) x 96 (6) x 48 − 3 − 2 0 x − 1 1 2 3 − 3 − 2 0 x − 1 1 2 3 − 3 − 2 0 x − 1 1 2 3 − 3 − 2 0 x − 1 1 2 3 ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities ______________________________________________________________________________ 30 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF E1: 1. (a) m 5 (b) x 8 (c) m 1 2. (a) m 4 (b) m 21 2 9 (c) x 3. 1 ( ) 1 ( ) 4 (c) 2 a m b m m 4. ( ) 10 (b) 1 (c) 8 (d) 3 (e a x m x x p x x p ) 3 (f) 2 (g) 25 (h) 10 TEST YOURSELF E2: (1) x 6 (2) m 1 (3) x 13 (4) p 9 (5) m 14 ACTIVITY: 1. positive 2. divided 3. multiplied 4. inequality 5. line 6. Equivalent 7. less 8. solid 9. solution 10. linear 11. negative 12. greater
Unit 1: Negative Numbers UNIT 8 TRIGONOMETRY B a s i c E s s e n t i a l A d d i t i o n a l M a t h e m a t i c s S k i l l s Curriculum Development Division Ministry of Education Malaysia
TABLE OF CONTENTS Module Overview 1 Part A: Trigonometry I 2 Part B: Trigonometry II 6 Part C: Trigonometry III 11 Part D: Trigonometry IV 15 Part E: Trigonometry V 19 Part F: Trigonometry VI 21 Part G: Trigonometry VII 25 Part H: Trigonometry VIII 29 Answers 33
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 1 Curriculum Development Division Ministry of Education Malaysia MODULE OVERVIEW 1. The aim of this module is to reinforce pupils’ understanding of the concept of trigonometry and to provide pupils with a solid foundation for the study of trigonometric functions. 2. This module is to be used as a guide for teacher on how to help pupils to master the basic skills required for this topic. Part of the module can be used as a supplement or handout in the teaching and learning involving trigonometric functions. 3. This module consists of eight parts and each part deals with one specific skills. This format provides the teacher with the freedom of choosing any parts that is relevant to the skills to be reinforced. 4. Note that Part A to D covers the Form Three syllabus whereas Part E to H covers the Form Four syllabus.
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 2 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES Some pupils may face difficulties in remembering the definition and how to identify the correct sides of a right-angled triangle in order to find the ratio of a trigonometric function. Strategy: Teacher should make sure that pupils can identify the side opposite to the angle, the side adjacent to the angle and the hypotenuse side through diagrams and drilling. PART A: TRIGONOMETRY I LEARNING OBJECTIVE Upon completion of Part A, pupils will be able to identify opposite, adjacent and hypotenuse sides of a right-angled triangle with reference to a given angle.
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 3 Curriculum Development Division Ministry of Education Malaysia Opposite side is the side opposite or facing the angle . Adjacent side is the side next to the angle . Hypotenuse side is the side facing the right angle and is the longest side. LESSON NOTES θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 4 Curriculum Development Division Ministry of Education Malaysia Example 1: AB is the side facing the angle , thus AB is the opposite side. BC is the side next to the angle , thus BC is the adjacent side. AC is the side facing the right angle and it is the longest side, thus AC is the hypotenuse side. Example 2: QR is the side facing the angle , thus QR is the opposite side. PQ is the side next to the angle , thus PQ is the adjacent side. PR is the side facing the right angle or is the longest side, thus PR is the hypotenuse side. EXAMPLES θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 5 Curriculum Development Division Ministry of Education Malaysia Identify the opposite, adjacent and hypotenuse sides of the following right-angled triangles. 1. Opposite side = Adjacent side = Hypotenuse side = 2. Opposite side = Adjacent side = Hypotenuse side = 3. Opposite side = Adjacent side = Hypotenuse side = 4. Opposite side = Adjacent side = Hypotenuse side = 5. Opposite side = Adjacent side = Hypotenuse side = 6. Opposite side = Adjacent side = Hypotenuse side = TEST YOURSELF A
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 6 Curriculum Development Division Ministry of Education Malaysia PART B: TRIGONOMETRY II TEACHING AND LEARNING STRATEGIES Some pupils may face problem in (i) defining trigonometric functions; and (ii) writing the trigonometric ratios from a given right-angled triangle. Strategy: Teacher must reinforce the definition of the trigonometric functions through diagrams and examples. Acronyms SOH, CAH and TOA can be used in defining the trigonometric ratios. LEARNING OBJECTIVE Upon completion of Part B, pupils will be able to state the definition of the trigonometric functions and use it to write the trigonometric ratio from a right-angled triangle.
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 7 Curriculum Development Division Ministry of Education Malaysia Definition of the Three Trigonometric Functions (i) sin = opposite side hypotenuse side (ii) cos = adjacent side hypotenuse side (iii) tan = opposite side adjacent side sin = opposite side hypotenuse side = AB AC cos = adjacent side hypotenuse side = BC AC tan = opposite side adjacent side = AB BC LESSON NOTES Acronym: SOH: Sine – Opposite - Hypotenuse Acronym: CAH: Cosine – Adjacent - Hypotenuse Acronym: TOA: Tangent – Opposite - Adjacent θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 8 Curriculum Development Division Ministry of Education Malaysia Example 1: AB is the side facing the angle , thus AB is the opposite side. BC is the side next to the angle , thus BC is the adjacent side. AC is the side facing the right angle and is the longest side, thus AC is the hypotenuse side. Thus sin = opposite side hypotenuse side = AB AC cos = adjacent side hypotenuse side = BC AC tan = opposite side adjacent side = AB BC EXAMPLES θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 9 Curriculum Development Division Ministry of Education Malaysia Example 2: WU is the side facing the angle, thus WU is the opposite side. TU is the side next to the angle, thus TU is the adjacent side. TW is the side facing the right angle and is the longest side, thus TW is the hypotenuse side. Thus, sin = opposite side hypotenuse side = WU TW cos = adjacent side hypotenuse side = TU TW tan = opposite side adjacent side = WU TU You have to identify the opposite, adjacent and hypotenuse sides. θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 10 Curriculum Development Division Ministry of Education Malaysia Write the ratios of the trigonometric functions, sin , cos and tan , for each of 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 B θ θ θ θ θ θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 11 Curriculum Development Division Ministry of Education Malaysia PART C: TRIGONOMETRY III TEACHING AND LEARNING STRATEGIES Some pupils may face problem in finding the angle when given two sides of a right-angled triangle and they also lack skills in using calculator to find the angle. Strategy: 1. Teacher should train pupils to use the definition of each trigonometric ratio to write out the correct ratio of the sides of the right-angle triangle. 2. Teacher should train pupils to use the inverse trigonometric functions to find the angles and express the angles in degree and minute. LEARNING OBJECTIVE Upon completion of Part C, 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 12 Curriculum Development Division Ministry of Education Malaysia Find the angle in degrees and minutes. Example 1: sin = 2 5 o h = sin-1 2 5 = 23o 34 4l = 23o 35 (Note that 34 41 is rounded off to 35) Example 2: cos = a h = 3 5 = cos-1 3 5 = 53o 7 48 = 53o 8 (Note that 7 48 is rounded off to 8) Since sin = opposite hypotenuse then = sin-1 opposite hypotenuse Since cos = adjacent hypotenuse then = cos-1 adjacent hypotenuse Since tan = opposite adjacent then = tan-1 opposite adjacent 1 degree = 60 minutes 1 minute = 60 seconds 1 o = 60 1 = 60 Use the key D M S or on your calculator to express the angle in degree and minute. Note that the calculator expresses the angle in degree, minute and second. The angle in second has to be rounded off. ( 30, add 1 minute and < 30, cancel off.) LESSON NOTES EXAMPLES θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 13 Curriculum Development Division Ministry of Education Malaysia Example 3: tan = o a = 7 6 = tan-1 7 6 = 49o 23 55 = 49o 24 Example 4: cos = a h = 5 7 = cos -1 5 7 = 44o 24 55 = 44o 25 Example 5: sin = o h = 4 7 = sin-1 4 7 = 34o 50 59 = 34o 51 Example 6: tan = o a = 5 6 = tan-1 5 6 = 39o 48 20 = 39o 48 θ θ θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: Trigonometry 14 Curriculum Development Division Ministry of Education Malaysia Find the value of in degrees and minutes. 1. 2. 3. 4. 5. 6. TEST YOURSELF C θ θ θ θ θ θ