UNIT 8 - RMM ICT GROUP | sekadar luahan rasa · PDF fileBasic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: ... covers the Form Three ... Additional Mathematics Skills

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


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.





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.





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





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





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





PART B:

TRIGONOMETRY II


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.





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





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





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





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





PART C:

TRIGONOMETRY III


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.





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

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.)

LESSON NOTES

EXAMPLES

θ

θ





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





Find the value of in degrees and minutes.

1.

2.

3.

4.

5.

6.

TEST YOURSELF C

θ θ

θ

θ

θ

θ





PART D:

TRIGONOMETRY IV


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.





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





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





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





PART E:

TRIGONOMETRY V


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.





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

oppositesin

r

x

OR

OP

hypotenuse

adjacentcos

x

y

OP

PR

adjacent

oppositetan

LESSON NOTES

θ





PART F:

TRIGONOMETRY VI


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.





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





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

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

S – only sin is positive





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





PART G:

TRIGONOMETRY VII


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





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 + QR

2 = PQ

2

LESSON NOTES





1. Write the values of sin , cos and tan

from the diagram below.

OA2 = (−6)

2 + 8

2

= 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





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





PART H:

TRIGONOMETRY VIII


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.





(a) y = sin x

The domain for x can be from 0o to 360

o or 0 to 2 in radians.

Important points: (0, 0), (90o, 1), (180

o, 0), (270

o, −1) and (360

o, 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), (90

o, 0), (180

o, −1), (270

o, 0) and (360

o, 1)

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

o, 1),

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

Minimum value = 1

LESSON NOTES





(c) y = tan x

Important points: (0o, 0), (180

o, 0) and (360

o, 0)

Is there any

maximum or

minimum point

for the tangent

graph?





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





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





TEST YOURSELF C:

1. sin = 1

3

= sin-1

1

3 = 19

o 28

2. cos = 1

2

= cos-1

1

2 = 60

o

3. tan = 5

3

= tan-1

5

3 = 59

o 2

4. cos = 5

8

= cos-1

5

8 = 51

o 19

5. tan = 7.5

9.2

= tan-1

7.5

9.2 = 39

o 11

6. sin = 6.5

8.4

= sin-1

6.5

8.4= 50

o 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 551

3

o

= 6

x

x = 13

6

sin55o

= 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





TEST YOURSELF F:

1. 1ST

and 2nd

2. 1st and 3

rd

3. 2nd

and 3rd

4. 1st and 4

th

5. 3rd

and 4th

6. 2nd

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





TEST YOURSELF H:

1.

y = tan x y = sin x y = cos x

2. (a) A (0, 1), B (90o, 0), C (180

o, 1), D (270

o, 0)

(b) P (90o, 1), Q (180

o, 0), R (270

o, 1), S (360

o, 0)

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UNIT 8 - RMM ICT GROUP | sekadar luahan rasa · PDF fileBasic Essentials Additional Mathematics Skills (BEAMS) Module Unit 8: ... covers the Form Three ... Additional Mathematics Skills