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8/9/2019 BEAMS_Unit 8 Trigonometry
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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
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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
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Unit 8: Trigonometry
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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 specificskills. 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.
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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.
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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
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Example 1:
AB is the side facing the angle , thusAB is the opposite side.
BCis the side next to the angle , thusBCis the adjacent side.
ACis the side facing the right angle and it is the longest side, thus ACis 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
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Unit 8: Trigonometry
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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
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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 canbe used in defining the trigonometric ratios.
LEARNING OBJECTIVE
Upon completion ofPart B, pupils will be able to state the definition
of the trigonometric functions and use it to write the trigonometricratio from a right-angled triangle.
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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:
SineOpposite - HypotenuseAcronym:
CAH:
CosineAdjacent - HypotenuseAcronym:
TOA:
TangentOpposite - Adjacent
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Example 1:
AB is the side facing the angle , thusAB is the opposite side.
BCis the side next to the angle , thusBCis the adjacent side.
ACis the side facing the right angle and is the longest side, thus ACis 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
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Example 2:
WUis the side facing the angle, thus WUis the opposite side.
TUis the side next to the angle, thus TUis the adjacent side.
TWis the side facing the right angle and is the longest side, thus TWis the hypotenuse
side.
Thus, sin =opposite side
hypotenuse side=
WU
TW
cos = adjacent sidehypotenuse side
= TUTW
tan =opposite side
adjacent side=
WU
TU
You have to identify the
opposite, adjacent and
hypotenuse sides.
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Unit 8: Trigonometry
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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
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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 ofPart C, pupils will be able to find the angle ofa right-angled triangle given the length of any two sides.
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Find the angle in degrees and minutes.
Example 1:
sin =2
5
o
h
= sin-1 25
= 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
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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
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Unit 8: Trigonometry
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Find the value of in degrees and minutes.
1. 2.
3. 4.
5. 6.
TEST YOURSELF C
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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 theangle of a right-angled triangle given the length of any two
sides.
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Find the length ofPR.
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 ofTS.
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
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Find the value ofx 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
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Unit 8: Trigonometry
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Find the value ofx for each of the following.
1. 2.
3. 4.
5. 6.
TEST YOURSELF D
10 cm
6 cm
13 cm
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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 usethe coordinates of the given point to determine the ratio of thetrigonometric functions.
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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
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PART F:
TRIGONOMETRY VI
TEACHING AND LEARNING STRATEGIES
Pupils may face difficulties in determining that the sign of the x-coordinate
andy-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 thex-coordinate andy-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 secondquadrant.
(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 ofPart F, pupils will be able to relate the sign of the
trigonometric functions to the sign ofx-coordinate andy-coordinate and todetermine the sign of each trigonometric ratio in each of the four quadrants.
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First Quadrant
sin =y
r(Positive)
cos =x
r(Positive)
tan =y
x(Positive)
(All trigonometric ratios are positive in thefirst quadrant)
Second Quadrant
sin =y
r
(Positive)
cos =x
r
(Negative)
tan =y
x(Negative)
(Only sine is positive in the secondquadrant)
Third Quadrant
sin =y
r
(Negative)
cos = xr
(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 fourthquadrant)
LESSON NOTES
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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
AAll positive
Conly cos is positiveTonly tan is positive
S only sin is positive
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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
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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 ofPart G, pupils will be able to calculate the length
of the side of right-angled triangle on a Cartesian plane and write thevalue of the trigonometric ratios given a point on the Cartesian plane
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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
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1. Write the values of sin , cos and tanfrom 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
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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
B(5,4)
B(5,12)
x
y
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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 thegraphs.
Strategy:
Teacher should help pupils to recall the trigonometric graphs which
pupils learned in Form 4. Geometers 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 thetrigonometric function graphs and know the important features of the
graphs.
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(c) y = tanx
Important points: (0o, 0), (180
o, 0) and (360
o, 0)
Is there any
maximum or
minimum point
for the tangent
graph?
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1. Write the following trigonometric functions to the graphs below:
y = cosx y = sinx y = tanx
2. Write the coordinates of the points below:
(a) (b)
A(0,1)
TEST YOURSELF H
y = cos x y =sin x
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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
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TEST YOURSELF C:
1. sin =1
3
= sin-1
1
3= 19
o28
2. cos =1
2
= cos-1
1
2= 60
o
3. tan =5
3
= tan-1
5
3= 59
o2
4. cos =5
8
= cos-1
5
8= 51
o19
5. tan =7.5
9.2
= tan-1
7.5
9.2= 39
o11
6. sin =6.5
8.4
= sin-1
6.5
8.4= 50
o42
TEST YOURSELF D:
1. tan 32o
=4
x
x =4
tan32o
= 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 = 10x
x =10
cos61o
= 20.63 cm
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TEST YOURSELF F:
1. 1ST
and 2nd
2. 1st
and 3rd
3. 2nd
and 3rd
4. 1st
and 4th
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
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TEST YOURSELF H:
1.
y = tanx y = sinx y = cosx
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)