BEAMS KPM

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

ACTIVITY B1:

y

20



 18 



 16 

 14 

12

10 

 
 8 


6


4 

2 x



–4 –3 –2 –1 0 1 2 3 4

Curriculum Development Division 41
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

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)

Curriculum Development Division 42
Ministry of Education Malaysia

Basic Essential
Additional Mathematics Skills

UNIT 7

LINEAR INEQUALITIES
Unit 1:
Negative Numbers

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 15
Linear Inequalities 19
Part D2: Perform Computations Involving Multiplication of Linear
Inequalities

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

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.

Curriculum Development Division 1
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

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.

______________________________________________________________________________

Curriculum Development Division 2
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART A:
LINEAR INEQUALITY
OVERALL LESSON NOTES

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 −2 −1 0 1 2 3x

−3 < − 1 1<3
−3 is less than − 1 1 is less than 3

and and

−1 > − 3 3>1
−1 is greater than − 3 3 is greater than 1

______________________________________________________________________________

Curriculum Development Division 3
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

3.0 Properties of Inequalities Algebraic Form
(a) Addition Involving Inequalities
Arithmetic Form

12  8 so 12  4  8  4 If a > b, then a  c  b  c
2  9 so 2  6  9  6 If a < b, then a  c  b  c

(b) Subtraction Involving Inequalities

Arithmetic Form Algebraic Form

7 > 3 so 7  5  3  5 If a > b, then a  c  b  c
2 < 9 so 2  6  9  6 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) If a > b and c > 0 , then ac > bc
12 > 9 so 12  9 If a > b and c > 0, then a  b

33 cc

2  5 so 2(3)  5(3) If a  b and c  0 , then ac  bc
8  12 so 8  12 If a  b and c  0 , then a  b
cc
22

______________________________________________________________________________

Curriculum Development Division 4
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

(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) If a > b and c < 0, then ac < bc
6 < 7 so 6(−3) > 7(−3) If a < b and c < 0, then ac > bc
16 > 8 so If a > b and c < 0, then a  b
16  8
10 <15 so 4 4 cc
10  15 If a < b and c < 0, then a  b
5 5
cc

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,...

______________________________________________________________________________

Curriculum Development Division 5
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

A number line is normally used to represent all the solutions of an inequality.

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.

(ii) x > 2 The open dot
means the value
o
2 is not
−2 −1 0 1 2 included.

x
34

(iii) x  3 The solid dot
means the value
3 is included.

−2 −1 2 x

0 1 3 4

______________________________________________________________________________

Curriculum Development Division 6
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART B:

POSSIBLE SOLUTIONS FOR A
GIVEN LINEAR INEQUALITY IN

ONE UNKNOWN

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.

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.

______________________________________________________________________________

Curriculum Development Division 7
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART B:
POSSIBLE SOLUTIONS FOR
A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN

EXAMPLES

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:

x

−2 −1 0 1 2 3 4 5 6 7 8 9 10
The possible integers are: 5, 6, 7, …

(2) x  3
Solution:

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 x

34

The possible integers are: – 4, − 5, −6, …

(3)  3  x 1
Solution:

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 x

34

The possible integers are: −2, −1, 0, and 1.

______________________________________________________________________________

Curriculum Development Division 8
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

TEST YOURSELF B
Draw a number line to represent the following inequalities:

(a) x > 1

(b) x  2

(c) x  2

(d) x  3

______________________________________________________________________________

Curriculum Development Division 9
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART C:

COMPUTATIONS INVOLVING
ADDITION AND SUBTRACTION ON

LINEAR INEQUALITIES

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.

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.

______________________________________________________________________________

Curriculum Development Division 10
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART C:
COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION

ON LINEAR INEQUALITIES

LESSON NOTES

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

2<4

x

1 23 4

Adding 1 to both sides of the inequality: The inequality
sign is
2+1<4+1
3<5 unchanged.

x

234 5

______________________________________________________________________________

Curriculum Development Division 11
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

(ii) 4 > 2 4>2

x The inequality
sign is
1 23 4
unchanged.
Subtracting 3 from both sides of the inequality:

4−3>2−3
1>−1

x

−1 0 1 2

EXAMPLES

(1) Solve x  5  14 . Subtract 5 from both sides
of the inequality.
Solution:
Simplify.
x  5  14
x  5  5  14  5
x9

(2) Solve p  3  2. Add 3 to both sides of the
inequality.
Solution:
Simplify.
p32
p33 23
p5

______________________________________________________________________________

Curriculum Development Division 12
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

TEST YOURSELF C

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  d  5 (8) p  2  1

(9) m  1  3 (10) 3  x  8
2

______________________________________________________________________________

Curriculum Development Division 13
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART D:

COMPUTATIONS INVOLVING
DIVISION AND MULTIPLICATION

ON LINEAR INEQUALITIES

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.

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)

______________________________________________________________________________

Curriculum Development Division 14
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART D1:
COMPUTATIONS INVOLVING
MULTIPLICATION AND DIVISION ON LINEAR INEQUALITIES

LESSON NOTES

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

2<4

x The inequality
sign is
1 23 4
Multiplying both sides of the inequality by 3: unchanged.

2 3<4 3 x

6 < 12

6 8 10 12 14

______________________________________________________________________________

Curriculum Development Division 15
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

(ii) − 4 < 2
−4<2

−4 − 2 x

02

Dividing both sides of the inequality by 2:

−4  2<2  2 The inequality
sign is
−2 <1
unchanged.

−2 − 1 x

0 12

2. When both sides of the inequality is multiplied or divided by a negative number, the
inequality sign is reversed.

Examples:

(i) 4 < 6

4<6

x The inequality
sign is reversed.
3 456

Dividing both sides of the inequality by −1:

4  (−1) > 6 

(−1)
−4>−6

x

−6 −5 −4 −3
______________________________________________________________________________

Curriculum Development Division 16
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

(ii) 1 > −3

1 > −3

x

−3 −2 −1 0 1
Multiply both sides of the inequality by −1:

(− 1) (1) < (−1) (−3) The inequality
sign is reversed.
1 3

−1 0 x

1 23

EXAMPLES

Solve the inequality 3q  12 . Divide each side of the The inequality
Solution: inequality by −3. sign is reversed.

(i) 3q  12 Simplify.
 3q  12
3 3
q  4

______________________________________________________________________________

Curriculum Development Division 17
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

TEST YOURSELF D1

Solve the following inequalities:

(1) 7 p  49 (2) 6x  18

(3) −5c > 15 (4) 200 < −40p

(5) 3d  24 (6)  2x  8

(7) 12  3x (8) 25  5y

(9)  2m  16 (10)  6b  27

______________________________________________________________________________

Curriculum Development Division 18
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART D2:
PERFORM COMPUTATIONS INVOLVING
MULTIPLICATION OF LINEAR INEQUALITIES

EXAMPLES

Solve the inequality  x  3 . Multiply both sides of the
2 inequality by −2.

Solution: Simplify.

 x 3.
2

 2( x )  (2)3
2

x  6

The inequality
sign is reversed.

______________________________________________________________________________

Curriculum Development Division 19
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

TEST YOURSELF D2
1. Solve the following inequalities:

(1) − d  3 (2) n  8
8 2

(3) 10   y (4) 6  b
5 7

(5) 0  12  x (6) 8  x  0
8 6

______________________________________________________________________________

Curriculum Development Division 20
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART E:

FURTHER PRACTICE ON
COMPUTATIONS INVOLVING

LINEAR INEQUALITIES

LEARNING OBJECTIVES
Upon completion of Part E, pupils will be able perform computations
involving linear inequalities.

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.

______________________________________________________________________________

Curriculum Development Division 21
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

PART E:
FURTHER PRACTICE ON COMPUTATIONS

INVOLVING LINEAR INEQUALITIES
TEST YOURSELF E1

Solve the following inequalities:
1. (a) m  5  0

(b) x  2  6

(c) 3 + m > 4

2. (a) 3m < 12

(b) 2m > 42

(c) 4x > 18

______________________________________________________________________________

Curriculum Development Division 22
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

3. (a) m + 4 > 4m + 1

(b) 14  m  6  m

(c) 3  3m  4  m

4. (a) 4  x  6

(b) 15  3m  12

(c) 3  x  5
4

______________________________________________________________________________

Curriculum Development Division 23
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

(d) 5x  3 18

(e) 1 3p  10
(f) x  3  4

2
(g) 3  x  8

5
(h) p  2  4

3

______________________________________________________________________________

Curriculum Development Division 24
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

EXAMPLES

What is the smallest integer for x if 5x  3 18?

Solution: A number line can
be used to obtain the

answer.

5x  3  18

5x  18  3 x3

5x  15 O

x3 0 1 2 34 5 x
6

x = 4, 5, 6,…

Therefore, the smallest integer for x is 4.

______________________________________________________________________________

Curriculum Development Division 25
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

TEST YOURSELF E2
1. If 3x 1 14, what is the smallest integer for x?

2. What is the greatest integer for m if m  7  4m 1?

3. If x  3  4 , find the greatest integer value of x.
2

4. If p  2  4 , what is the greatest integer for p?
3

5. What is the smallest integer for m if 3  m  9 ?
2

______________________________________________________________________________

Curriculum Development Division 26
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

ACTIVITY

1
23
4
56

78
9

10

11 12

______________________________________________________________________________

Curriculum Development Division 27
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

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. x  1
2

x  2

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.

______________________________________________________________________________

Curriculum Development Division 28
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

ANSWERS

TEST YOURSELF B:

(a) x
−3 −2 −1 0 1 2 3

(b) x
−3 −2 −1 0 1 2 3

(c) x
−3 −2 −1 0 1 2 3

x
(d) − 3 − 2 − 1 0 1 2 3

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) m  5 (10) x  5

2

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) b  9
2

TEST YOURSELF D2: (3) y  50 (4) b  42 (5) x  96 (6) x  48
(1) d  24 (2) n  16

______________________________________________________________________________

Curriculum Development Division 29
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities

TEST YOURSELF E1:

1. (a) m  5 (b) x  8 (c) m  1

2. (a) m  4 (b) m  21 (c) x  9
2

3. (a) m  1 (b) m  4 (c) m  1
2

4. (a) x  10 (b) m 1 (c) x  8 (d) x  3 (e) p  3 (f) x  2 (g) x  25 (h) p 10

TEST YOURSELF E2: (3) x  13 (4) p  9 (5) m  14
(1) x  6 (2) m  1

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

______________________________________________________________________________

Curriculum Development Division 30
Ministry of Education Malaysia

Basic Essential
Additional Mathematics Skills

UNIT 8

TRIGONOMETRY
Unit 1:
Negative Numbers

Curriculum Development Division
Ministry of Education Malaysia

TABLE OF CONTENTS 1
2
Module Overview 6
Part A: Trigonometry I 11
Part B: Trigonometry II 15
Part C: Trigonometry III 19
Part D: Trigonometry IV 21
Part E: Trigonometry V 25
Part F: Trigonometry VI 29
Part G: Trigonometry VII 33
Part H: Trigonometry VIII
Answers

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

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.

Curriculum Development Division 1
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

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.

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.

Curriculum Development Division 2
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

LESSON NOTES

θ

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.

Curriculum Development Division 3
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

EXAMPLES

Example 1:

θ

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:

θ

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.

Curriculum Development Division 4
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

TEST YOURSELF A

Identify the opposite, adjacent and hypotenuse sides of the following right-angled triangles.

1. 2. 3.

Opposite side = Opposite side = Opposite side =
Adjacent side = Adjacent side = Adjacent side =
Hypotenuse side = Hypotenuse side = Hypotenuse side =

4. 5. 6.

Opposite side = Opposite side = Opposite side =
Adjacent side = Adjacent side = Adjacent side =
Hypotenuse side = Hypotenuse side = Hypotenuse side =

Curriculum Development Division 5
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

PART B:

TRIGONOMETRY II

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.

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.

Curriculum Development Division 6
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

LESSON NOTES

Definition of the Three Trigonometric Functions

opposite side Acronym:
(i) sin  = hypotenuse side
SOH:
adjacent side Sine – Opposite - Hypotenuse
(ii) cos  = hypotenuse side
Acronym:
opposite side
(iii) tan  = adjacent side CAH:
Cosine – Adjacent - Hypotenuse

Acronym:

TOA:
Tangent – Opposite - Adjacent

θ

opposite side AB
sin  = hypotenuse side = AC

cos  = adjacent side BC
=
hypotenuse side AC

opposite side AB
tan  = adjacent side = BC

Curriculum Development Division 7
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

EXAMPLES

Example 1:

θ

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.

opposite side AB
Thus sin  = hypotenuse side = AC

cos  = adjacent side BC
hypotenuse side = AC

opposite side AB
tan  = adjacent side = BC

Curriculum Development Division 8
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

Example 2:

θ
You have to identify the
opposite, adjacent and
hypotenuse sides.

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, opposite side WU
sin  = hypotenuse side = TW

cos  = adjacent side TU
hypotenuse side = TW

opposite side WU
tan  = adjacent side = TU

Curriculum Development Division 9
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

TEST YOURSELF B

Write the ratios of the trigonometric functions, sin , cos  and tan  , for each of the diagrams
below:

1. 2. θ 3.

θ
θ

θ

sin  = sin  = sin  =
cos  = cos  = cos  =
tan  = tan  = tan  =

4. 5. 6.
θ θ
θ

sin  = sin  = sin  =
cos  = cos  = cos  =
tan  = tan  = tan  =

Curriculum Development Division 10
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

PART C:

TRIGONOMETRY III

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.

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.

Curriculum Development Division 11
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

LESSON NOTES

Since sin  = opposite Since cos  = adjacent Since tan  = opposite
hypotenuse hypotenuse adjacent

then  = sin-1 opposite then  = cos-1 adjacent then  = tan-1 opposite
hypotenuse hypotenuse adjacent

1 degree = 60 minutes 1 minute = 60 seconds

1o = 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.)

EXAMPLES

Find the angle  in degrees and minutes. Example 2:
Example 1:

θ θ

sin  = o  2 cos  = a = 3

h5 h5

 = sin-1 2  = cos-1 3

5 5
= 23o 34 4l = 53o 7 48
= 23o 35 = 53o 8
(Note that 7 48 is rounded off to 8)
(Note that 34 41 is rounded off to 35)

Curriculum Development Division 12
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry

Example 3: Example 4:
θ
θ
tan  = o = 7 cos  = a = 5

a6 h7

 = tan-1 7  = cos-1 5

6 7
= 49o 23 55 = 44o 24 55
= 49o 24 = 44o 25

Example 5: Example 6:
θ
θ
sin  = o = 4
tan  = o = 5
h7
a6
 = sin-1 4
 = tan-1 5
7
= 34o 50 59 6
= 34o 51 = 39o 48 20
= 39o 48

Curriculum Development Division 13
Ministry of Education Malaysia

Basic Essentials Additional Mathematics Skills (BEAMS) Module θ
Unit 8: Trigonometry

TEST YOURSELF C
Find the value of  in degrees and minutes.
1. 2.

θ

3. 4.

θ
θ

5. 6.
θ θ

Curriculum Development Division 14
Ministry of Education Malaysia