1. CHAPTER 9: TRIGONOMETRY 11
Important Concepts: Trigonometrical Ratios
A
Sin θ = =
Hypothenuse Cos θ = =
Opposite
side
Tan θ = =
θ
B C
Adjacent side
1. The unit circle is the circle with radius 1 unit and its centre at origin.
y
2. 1
(x, y)
·
θ
-1 1 x
a)
Quadrant Angle θ
I 0º < θ < 90º
4.
II 90º < θ < 180º
III 180º < θ < 270º 900
Quadrant II
IV 270º < θ < 360º
1800-θQuadrant I
y θ
Quadrant III
b) sin θ = y = y θ - 1800
1800 0º, 3600
1 sin + All + Quadrant IV
cos θ = x = x
x 3600 - θ
1
tan θ = y tan + cos +
x
2700
Trigonometry II 1

2. 9.1 Identifying The Quadrants and The Angles In A Unit Circle.
The x-axis and the y-axis divides the unit circle with centre origin into 4 quadrants as shown in
the diagram below
y
1 90º
180 º -1 II I 1 0º
O 360 X
III IV
-1 270 º
Exercises 9.11:
1. State the quadrant for the following angles in the table below.
Angle Quadrant Angle Quadrant
42 º I 19 º
70 º 265 º
100º II 289 º
136 º 126 º
197 º 303 º
205 º 80 º
275 º 150 º
354 º 212 º
1 a) Determine whether the values of
a) sin θ
b) cos θ
c) tan θ are positive or negative if
90 o ≤ θ ≤ 180 o , 180 o ≤ θ ≤ 270 o , and 270 o ≤ θ ≤ 360 o
y
1 90º
180 -1 Sin + ALL 1 0
O 360 X
Tan + Cos +
-1 270
Trigonometry II 2

3. Examples :
i) Sin 142º ii) cos 232 º iii) tan 299 º
142º is in quadrant II cos 232 º is in quadrant III tan 299 º is in quadrant IV
Sin is positive in Quadrant II Cos is negative in quadrant III tan is negative in quadrant IV
Exercises 9.2:
Angle Quadrant Value (Positive/ Negative)
Sin Cos Tan
75 º I + + +
120 º II + - -
160 º
200 º
257 º
280 º
345 º
b)Find the values of the angles in quadrant I which correspond to the following values of
angles in other quadrants.
The relationship between the values of sine, cosine and tangent of angles in Quadrant II, III and
IV with their respective values of the corresponding angle in Quadrant I is shown in the diagram
below :
QUADRANT II QUADRANT III QUADRANT IV
( 90 º ≤ θ ≤ 180 º ) ( 180 º ≤ θ ≤ 270 º) (270 º ≤ θ ≤ 360 º)
Sin θ = sin ( 180 - θ) Sin θ = - sin ( θ - 180º ) Sin θ = - sin ( 360 - θ )
Cos θ = cos ( 180 - θ) Cos θ = -cos ( θ - 180º ) Cos θ = cos ( 360 - θ )
Tan θ = tan ( 180 - θ) Tan θ = tan ( θ - 180º ) Tan θ = - tan ( 360 - θ )
Trigonometry II 3

4. Example :
120º 230º 340º
Sin 120º = sin 60º Sin 230º = - sin 50º Sin 340º = - sin 20º
Cos 120º = - cos 60º Cos 230º = - cos 50º Cos 340º = cos 20º
Tan 120º = - tan 60º Tan 230º = tan 50º Tan 340º = - tan 20º
EXERCISES 9.3 :
Finding the values of the angles in quadrant I which correspond to the following values of
angles in other quadrants.
ANGLE CORRESPONDING ANGLE IN QUADRANT
I
Sin 125º Sin θ = sin ( 180 - 125º)
= sin 55º
Cos 143º
Tan 98º
Sin 200 º Sin θ = - sin ( 200º - θ)
= - sin 20º
Cos 245 º
Tan 190 º
Sin 285 º Sin θ = - sin ( 360º - θ)
= -sin 55º
Cos 300 º
Tan 315 º
Finding the value of Sine, Cosine and Tangent of the angle between 90 º and 360º
Exercises 9.4 :
Angle Value
Sin 46
Cos 57
Tan 79
Sin 139
Cos 154
Tan 122
Sin 200
Cos 187
Tan 256
Sin 342
Trigonometry II 4

5. Cos 278
Finding the angle between 0º and 360º when the values of sine, cosine and tangent are
given
Exercises 9.5 :
VALUE ANGLE
Sin − 0.7654
1
Sin − -0.932
1
Sin − 0.1256
1
Cos − 0.4356
1
Cos − -0.6521
1
Cos − -0.7642
1
Tan − -1.354
1
Tan − 0.7421
1
−
Tan 1 1.4502
15.2 Graphs Of Sine, Cosine And Tangent
15.2.a) For each of the following equations, complete the given table and draw its graph based
on
the data in the table.
i) y = sin x
X 0º 45 º 90 º 135 º 180 º 225 º 270 º 315 º 360 º
Y
ii) y = cos x
X 0º 45 º 90 º 135 º 180 º 225 º 270 º 315 º 360 º
Y
iii) y = tan x
X 0º 45 º 90 º 135 º 180 º 225 º 270 º 315 º 360 º
Y
Trigonometry II 5

6. 15.3 Questions Base On Examination Format.
1. Which of the following is equal to cos 35 º ?
A. cos 145 º C. cos 235 º
B. cos 215 º D. cos325 º
2. Find the value of sin 150 º + 2 cos 240 º - 3 tan 225 º
A. -3.5 B. -1.5 C. 1.5 D. 2.5
3. Sin 30 º + cos 60 º =
1 1
A. B. C. 1 D. 0
4 2
4. Given that sin 45 º = cos 45 º = 0.7. Find the value of 3 sin 315 º - 2 cos 135 º
A. -3.5 B. -1.5 C. 1.5 D. 2.5
5. Given that cos θ = 0.9511 and 0 º ≤ θ ≤ 360, º find the value of θ
A. 18 º B. 162 º C. 218 º D. 300 º
6. Given that tan θ = 05774 and 0 º ≤ θ ≤ 360 º, find the value of θ
A. 30 º , 210 B. 152 , 210 C.30 º, 330 D. 30 º, 150
7. Given that sin θ = -0.7071 and 90 º ≤ θ ≤ 270, º find the value of θ
A. 135 º B. 225 º C. 45 º D. 315 º
8. Given that Sin x = 0.848 and 90 º ≤ x ≤ 180 º , find the value of x
A. 108 º B. 122 º C. 132 º D. 158 º
9. Given that tan y = -2.246 and 0 º ≤ θ ≤ 360 º , find the value of y
A. 66 º, 246 º B. 114 º ,246 º C. 114 º, 294 º D.246 º, 294 º
Trigonometry II 6

7. 10.
y
(0,1)
( -1,0) (1,0)
O θ X
(0.87,-0.50)
( -1,0)
The diagram shows the unit circle. The value of tan θ is
A. -1.74 B. -0.57 C. -0.50 D. 0.87
11.
y
1
-1 1
O X
P
-1
The diagram shows the unit circle. If P is (-0.7, -0.6), find the value of Sin θ
7 6
A. - B. - C. -0.6 D. 0.6
6 7
12
y
1
-1 1
O X
R (0.8, -0.4)
-1
The diagrams shows a unit circle and R (0.8, -0.4). find the value of cos θ
0.4
A. 0.8 B. 0.4 C. 1 D.
0.8
13. In the diagram, ABC is a straight line. The value of sin x is
Trigonometry II 7

8. B
A C
xº
15 8
D
8 8 15 17
A. B. C. D.
15 17 17 15
14. T
13 cm
5 cm
Q S
R
X 7 cm
U
In the diagram, PQRS is a straight line and R is the mid-point of QS. The value of cos x is
−12 −12 −13 − 24
A. B. C. D.
13 25 25 25
15. P
15 cm T 6 cm S
Q
R
3
In the diagram, PQR and QTS are straight lines. Given that sin ∠ TRS = , then
5
sin ∠ PQT =
8 8 −8 −8
A. B. C. D.
15 17 15 17
Trigonometry II 8

9. 16.
Given that PQR is a straight line and tan x = -1, find the length of PR in cm.
A. 6 B. 8 C. 10 D. 12
17.
3
In the diagram above, PQR is a straight line. Given that cos ∠SQP = , find tan x.
5
1 5 3 4
A. B. C. D.
2 8 4 5
18.
3
In the diagram above, EFGH is a straight line. If sin ∠JGH = , the value of tan x
5
=
4 1 1 3
A. B. C. − D. −
5 2 3 5
19. Diagram below shows a graph of trigonometric function.
Trigonometry II 9

10. The equation of the trigonometric function is
A. y = sin x B. y = -sin x C. y = cos x D. y = -cos x
20.
The value of cos ϑis
4 3 3 4
A. B. C. − D. −
3 5 5 5
15.4 PAST YEAR SPM QUESTIONS
Trigonometry II 10

11. Nov 2003, Q11
1. In Diagram 5, GHEK is a straight line. GH = HE.
7 cm 25 cm
F
Diagram 5
Find the value of tan x◦ E
K
G
H x◦
5 13
A. − 13 C. −
12 12
12 J cm 12
B. − D. −
13 5
Nov 2003, Q12
2. Which of the following graphs represents y = sin x◦ ?
Nov 2004, Q 11
3. In Diagram 5, PRS is a straight line
Q
Trigonometry II 11

12. 7 cm
P 24 cm
x◦
R
S
Find the value of cox x◦ =
7 7
A. C. −
24 24
24 24
B. D. −
25 25
Nov 2004, Q 12
4. Diagram 6 shows the graph of y = sin x.
The value of p is
A. 90° C. 270
°
B. 180 ° D. 360
°
Nov 2004, Q13
5. In diagram 7, JKL is a straight line.
Diagram 7
Trigonometry II 12

13. 5
It is given that cos x° = and tan y° = 2. Calculate the length, in cm, of JKL
13
A. 22 C. 44
B. 29 D. 58
Nov 2005, Q11
6. It is given that cos θ = −0.7721 and 180° ≤ θ ≤ 360°. Find the value of θ
A. 219° 27’ C. 309° 27’
B. 230° 33’ D. 320° 33’
Nov 2005, Q12
7. In Diagram 6, QRS is a straight line.
4 cm
Q P
3 cm
R
θ
Diagram 6
S
What is the value of cos θ° ?
4 3
A. C. −
5 5
3 4
B. D. −
5 5
July 2004, Q13
H 16 cm
G
12 cm
Diagram 6
x
E
13 cm F
Trigonometry II 13

14. 8. Diagram 6 shows a quadrilateral EFGH. Find the value of x.
A. 33° 01’ C. 49° 28’
B. 40° 33’ D. 50° 54’
July 2004, Q14
9. In Diagram 7, O is the origin of a Cartesian plane.
Diagram 7
y
P (-3, 4)
r
x
0
The value of sin r° is
3 3
A. C. −
5 5
4 3
B. D. −
5 4
July 2005, Q12
Trigonometry II 14

15. 10. Which of the following graphs represents y = sin 2x for 0° ≤ x° ≤ 180?
y
y
2 1
A B
0 x
1 900 1800
-1
0 0 0
x
y 90 180
1
C
y
1
D
0 x
900 180
0
0 x
0
90 1800
-1
-1
July 2005, Q11
12. Given cos x° = - 0.8910 and 0° ≤ x° ≤ 360°, find the values of x.
A 117 and 243 C. 153 and 207
B 117 and 297 D 153 and 333
NOV 2005, Q11
13. It is given that cos ϑ= -0.721 and 180 0 ≤ ≤360 0 . Find the value of ϑ
ϑ .
A. 219o 27’
B. B. 230o33’
C. 309o27’
D. D. 320o33’
Trigonometry II 15

16. NOV 2005, Q12
14. In Diagram 6, QRS is a straight line
Diagram 6
What is the value of cos ϑ0
4
A.
5
3
B.
5
3
C. −
5
4
D. −
5
JULY 2006, Q11
15. Diagram 5 shows a rhombus PQRS
Diagram 5
It is given that QST is a straight line and QS = 10cm.
Find the value of tan xo.
Trigonometry II 16

17. 5 5
A. C. −
13 12
13 12
B. D. −
12 5
JULY 2006, Q12
16. Which of the following represents part of the graph of y = tan x?
A. C.
B. D.
JULY 2006, Q13
17. In Diagram 6, PQR and TSQ are straight lines.
Find the length of ST , in cm.
A. 2.09 C. 3.56
B. 3.44 D. 4.91
Trigonometry II 17

18. NOV 2006, Q11
18. In Diagram 5, S is the midpoint of straight line QST.
The value of cos xo is
4 3
A. C.
3 4
4 3
B. D.
5 5
NOV 2006, Q12
19. In Diagram 6, MPQ is a right angled triangle.
It is given that QN = 13cm, MP = 24cm and N is the midpoint of MNP.
Find the value of tan y0.
5 12
A. − C. −
13 13
Trigonometry II 18

19. 5 13
B. − D. −
12 12
NOV 2006, Q13
20. Which of the following represents the graph of y = cos x for 0 0 ≤x ≤180 0 ?
A.
B.
C.
Trigonometry II 19

20. D.
SPM 2007
13. Which of the following graphs represents y = sin x for 0 ≤ x ≤ 1800
A. B.
1
Trigonometry II 20 0o 90o 180o
-1

21. C.
1
0o 90o 180o
-1
11. In diagram below, USR and VQTS are straight lines.
U
x0
P S
T
y0 R
V
8
It is given that TS = 29 cm, PQ = 13 cm, QR = 16 cm and sin x =
0
.
17
Find the value of tan y 0
12 5
A. B.
5 12
5 12
C. − D. −
12 5
12. In Diagram ., O is the origin and JOK is a straight on a Cartesian plane.
K(3,4)
Trigonometry II J 21

22. The value of cos θ is
4 3
A. − B. −
5 5
3 4
C. D.
5 5
Trigonometry II 22