BEAMS KPM

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

PART D:

TRIGONOMETRY IV

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.

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.

Curriculum Development Division 15
Ministry of Education Malaysia

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

LESSON NOTES

Find the length of PR. Find the length of TS.

With reference to the given angle, PR is the With reference to the given angle, TR is the
opposite side and QR is the adjacent side. adjacent side and TS is the hypotenuse
side.
Thus tangent ratio is used to form the
relation of the sides. Thus cosine ratio is used to form the
relation of the sides.
tan 50o = PR
5 cos 32o = 8
TS
PR = 5  tan 50o
TS  cos 32o = 8

TS = 8
cos 32o

Curriculum Development Division 16
Ministry of Education Malaysia

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

EXAMPLES

Find the value of x in each of the following.

Example 1: Example 2:

tan 25o = 3 sin 41.27o = x
x 5

x= 3 x = 5  sin 41.27o
tan 25o = 3.298 cm

= 6.434 cm Example 4:

Example 3:

cos 34o 12 = x tan 63o = x
6 9

x = 6  cos 34o 12 x = 9  tan 63o
= 4.962 cm = 17.66 cm

Curriculum Development Division 17
Ministry of Education Malaysia

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

TEST YOURSELF D

Find the value of x for each of the following.
1. 2.

3. 4.
10 cm 6 cm

5. 6.
13 cm

Curriculum Development Division 18
Ministry of Education Malaysia

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

PART E:

TRIGONOMETRY V

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.

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.

Curriculum Development Division 19
Ministry of Education Malaysia

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

LESSON NOTES

θ

In the diagram, with reference to the angle , PR is the opposite side, OP is the adjacent side
and OR is the hypotenuse side.

sin  opposite  PR  y
hypotenuse OR r

cos  adjacent  OP  x
hypotenuse OR r

tan  opposite  PR  y
adjacent OP x

Curriculum Development Division 20
Ministry of Education Malaysia

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

PART F:

TRIGONOMETRY VI

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.

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.

Curriculum Development Division 21
Ministry of Education Malaysia

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

LESSON NOTES Second Quadrant
First Quadrant

θθ

sin  = y (Positive) sin  = y (Positive)

r r

cos  = x (Positive) cos  = x (Negative)

r r

tan  = y (Positive) tan  = y (Negative)

x x

(All trigonometric ratios are positive in the (Only sine is positive in the second
first quadrant) quadrant)

Third Quadrant Fourth Quadrant

θθ

sin  = y (Negative) sin  = y (Negative)

r r

cos  = x (Negative) cos  = x (Positive)

r r

tan  = y  y (Positive) tan  = y (Negative)

x x x

(Only tangent is positive in the third (Only cosine is positive in the fourth
quadrant) quadrant)

Curriculum Development Division 22
Ministry of Education Malaysia

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

Using acronym: Add Sugar To Coffee (ASTC)

sin  is positive  cos  is positive  tan  is positive 

sin  is negative  cos  is negative  tan  is negative 

S – only sin  is positive A – All positive

T – only tan  is positive C – only cos  is positive

Curriculum Development Division 23
Ministry of Education Malaysia

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

TEST YOURSELF F

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

Curriculum Development Division 24
Ministry of Education Malaysia

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

PART G:

TRIGONOMETRY VII

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

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.

Curriculum Development Division 25
Ministry of Education Malaysia

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

LESSON NOTES

The Pythagoras Theorem:

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

(a) 3, 4, 5 or equivalent (b) 5, 12, 13 or equivalent (c) 8, 15, 17 or equivalent

Curriculum Development Division 26
Ministry of Education Malaysia

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

EXAMPLES

1. Write the values of sin , cos  and tan 2. Write the values of sin , cos  and tan 

from the diagram below. from the diagram below.

θ
θ

OA2 = (−6)2 + 82 OB2 = (−12)2 + (−5)2
= 100 = 144 + 25
= 169
OA = 100
= 10 OB = 169
= 13
sin  = y  8  4
sin  = y   5
r 10 5
r 13
cos  = x  6   3
cos  = x  12
r 10 5
r 13
tan  = y  8   4
tan  = 5  5
x 6 3
12 12

Curriculum Development Division 27
Ministry of Education Malaysia

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

TEST YOURSELF G

Write the value of the trigonometric ratios from the diagrams below.

1. 2. 3.

y
B(5,4)

B(5,12) θ

θ θθ

sin  = sin  = x
cos  = cos  =
tan  = tan  = sin  =
4. 5. cos  =
tan  =
θ θ
6.

θ

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

Curriculum Development Division 28
Ministry of Education Malaysia

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

PART H:

TRIGONOMETRY VIII

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.

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.

Curriculum Development Division 29
Ministry of Education Malaysia

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

LESSON NOTES

(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) 30
Important features: Maximum point (0o, 1) and (360o, 1),

Maximum value = 1 Minimum point (180o, −1)
Minimum value = 1

Curriculum Development Division
Ministry of Education Malaysia

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

(c) y = tan x

Important points: (0o, 0), (180o, 0) and (360o, 0)
Is there any
maximum or

minimum point
for the tangent

graph?

Curriculum Development Division 31
Ministry of Education Malaysia

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

TEST YOURSELF H

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: (b)
(a)
y = sin x
y = cos x

A(0,1) 32

Curriculum Development Division
Ministry of Education Malaysia

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

ANSWERS

TEST YOURSELF A:

1. Opposite side = AB 2. Opposite side = PQ 3. Opposite side = YZ
Adjacent side = XZ
Adjacent side = AC Adjacent side = QR Hypotenuse side = XY

Hypotenuse side = BC Hypotenuse side = PR

4. Opposite side = LN 5. Opposite side = UV 6. Opposite side = RT
Adjacent side = ST
Adjacent side = MN Adjacent side = TU Hypotenuse side = RS

Hypotenuse side = LM Hypotenuse side = TV

TEST YOURSELF B: 2. sin  = PQ 3. sin  = YZ

1. sin  = AB PR YX

BC cos  = QR cos  = XZ

cos  = AC PR XY

BC tan  = PQ tan  = YZ

tan  = AB QR XZ

AC 5. sin  = UV 6. sin  = RT

4. sin  = LN TV RS

LM cos  = UT cos  = ST

cos  = MN TV RS

LM tan  = UV tan  = RT

tan  = LN UT TS

MN

Curriculum Development Division 33
Ministry of Education Malaysia

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

TEST YOURSELF C: 2. cos  = 1

1. sin  = 1 2

3  = cos-1 1 = 60o

 = sin-1 1 = 19o 28 2

3 4. cos  = 5

3. tan  = 5 8

3  = cos-1 5 = 51o 19

 = tan-1 5 = 59o 2 8

3 6. sin  = 6.5

5. tan  = 7.5 8.4

9.2  = sin-1 6.5 = 50o 42

 = tan-1 7.5 = 39o 11 8.4

9.2

TEST YOURSELF D: 2. sin 53.17o = x
1. tan 32o = 4 7

x x = 7  sin 53.17o = 5.603 cm
x = 4 = 6.401 cm
4. 1o = 6
tan 32o sin 55
3. cos 74o 25 = x 3x

10 x= 6 = 7.295 cm
x = 10  cos 74o 25
sin 55 1 o
= 2.686 cm 3
5. tan 47o = x
6. cos 61o = 10
13 x
x = 13  tan 47o = 13.94 cm
x = 10 = 20.63 cm
cos 61o

Curriculum Development Division 34
Ministry of Education Malaysia

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

TEST YOURSELF F: 2. 1st and 3rd 3. 2nd and 3rd
1. 1ST and 2nd

4. 1st and 4th 5. 3rd and 4th 6. 2nd and 4th

TEST YOURSELF G: 2. sin  = 12 3. sin  = 4

1. sin  = 4 13 5

5 cos  = 5 cos  =  3

cos  = 3 13 5

5 tan  = 12 tan  =  4

tan  = 4 5 3

3 6. sin  =  5

4. sin  =  4 5. sin  =  8 13

5 17 cos  = 12

cos  =  3 cos  =  15 13

5 17 tan  =  5

tan  = 4 tan  = 8 12

3 15

Curriculum Development Division 35
Ministry of Education Malaysia

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

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)

Curriculum Development Division 36
Ministry of Education Malaysia