CHAPTER 9: TRIGONOMETRY 11

Important Concepts: Trigonometrical Ratios

Trigonometry II 1

Adjacent side

Hypothenuse

θB

A

C

Sin θ = =

Cos θ = =

Tan θ = =

Oppositeside

1. The unit circle is the circle with radius 1 unit and its centre at origin.

2.

a)Quadrant Angle θ

I 0º < θ < 90ºII 90º < θ < 180ºIII 180º < θ < 270ºIV 270º < θ < 360º

b) sin θ = y = y 1 cos θ = x = x 1

tan θ = y x

1

1

-1

· (x, y)

θ

y

y

x

x

All +sin +

cos +tan +

4.

Quadrant II1800-θQuadrant I

θQuadrant III

θ - 1800

Quadrant IV3600 - θ

900

1800

2700

0º, 3600

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 0 º 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 Quadrant42 º I 19 º70 º 265 º100º II 289 º136 º 126 º197 º 303 º205 º 80 º275 º 150 º354 º 212 º

1 a) Determine whether the values of

a) sin θb) cos θc) tan θ are positive or negative if

oooooo and 360270,270180,18090 ≤≤≤≤≤≤ θθθ

y 1 90º

180 -1 Sin + ALL 1 0 O 360 X Tan + Cos +

-1 270

Trigonometry II 2

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 IVSin 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 º200 º257 º280 º345 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)Cos θ = cos ( 180 - θ)Tan θ = tan ( 180 - θ)

Sin θ = - sin ( θ - 180º )Cos θ = -cos ( θ - 180º )Tan θ = tan ( θ - 180º )

Sin θ = - sin ( 360 - θ )Cos θ = cos ( 360 - θ ) Tan θ = - tan ( 360 - θ )

Trigonometry II 3

Example :

120º

Sin 120º = sin 60ºCos 120º = - cos 60ºTan 120º = - tan 60º

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 143ºTan 98º

Sin 200 º Sin θ = - sin ( 200º - θ) = - sin 20º

Cos 245 ºTan 190 ºSin 285 º Sin θ = - sin ( 360º - θ)

= -sin 55ºCos 300 ºTan 315 º

Finding the value of Sine, Cosine and Tangent of the angle between 90 º and 360º

Exercises 9.4 :

Angle ValueSin 46Cos 57Tan 79Sin 139Cos 154Tan 122Sin 200Cos 187Tan 256Sin 342

Trigonometry II 4

230º 340º

Sin 230º = - sin 50ºCos 230º = - cos 50ºTan 230º = tan 50º

Sin 340º = - sin 20ºCos 340º = cos 20ºTan 340º = - tan 20º

Cos 278

Finding the angle between 0º and 360º when the values of sine, cosine and tangent are given

Exercises 9.5 :

VALUE ANGLESin 1− 0.7654

Sin 1− -0.932

Sin 1− 0.1256

Cos 1− 0.4356

Cos 1− -0.6521

Cos 1− -0.7642

Tan 1− -1.354

Tan 1− 0.7421

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 ºY

ii) y = cos x

X 0º 45 º 90 º 135 º 180 º 225 º 270 º 315 º 360 ºY

iii) y = tan x

X 0º 45 º 90 º 135 º 180 º 225 º 270 º 315 º 360 ºY

Trigonometry II 5

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 =

A. 4

1 B.

2

1 C. 1 D. 0

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

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

A. -6

7 B. -

7

6 C. -0.6 D. 0.6

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

A. 0.8 B. 0.4 C. 1 D. 8.0

4.0

13. In the diagram, ABC is a straight line. The value of sin x is

Trigonometry II 7

B A C xº 15 8

D

A. 15

8 B.

17

8 C.

17

15 D.

15

17

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

A. 13

12− B.

25

12− C.

25

13− D.

25

24−

15. P 15 cm T 6 cm S Q

R

In the diagram, PQR and QTS are straight lines. Given that sin ∠ TRS = 5

3, then

sin ∠ PQT =

A. 15

8 B.

17

8 C.

15

8− D.

17

8−

Trigonometry II 8

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.

In the diagram above, PQR is a straight line. Given that cos 5

3=∠SQP , find tan x.

A. 2

1B.

8

5C.

4

3D.

5

4

18.

In the diagram above, EFGH is a straight line. If sin 5

3=∠JGH , the value of tan x

=

A. 5

4B.

2

1C.

3

1− D. 5

3−

19. Diagram below shows a graph of trigonometric function.

Trigonometry II 9

The equation of the trigonometric function isA. y = sin x B. y = -sin x C. y = cos x D. y = -cos x

20.

The value of cos ϑis

A. 3

4B.

5

3C.

5

3− D. 5

4−

15.4 PAST YEAR SPM QUESTIONS

Trigonometry II 10

Nov 2003, Q11

1. In Diagram 5, GHEK is a straight line. GH = HE.

7 cm 25 cm

Diagram 5 Find the value of tan x◦

A. −12

5 C. −

12

13

B. − 13

12 D. −

5

12

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

Trigonometry II 11

F

G

EK

H

J

13 cm

x◦

Q

x◦

Find the value of cox x◦ =

A. 24

7 C. −

24

7

B. 25

24 D. −

25

24

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

P

7 cm

24 cm

R

S

16 cm

12 cm

x

13 cm

H

E

G

F

It is given that cos x° = 13

5 and tan y° = 2. Calculate the length, in cm, of JKL

A. 22 C. 44B. 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

SWhat is the value of cos θ° ?

A. 5

4 C. −

5

3

B. 5

3 D. −

5

4

July 2004, Q13

Diagram 6

Trigonometry II 13

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

The value of sin r° is

A. 5

3 C. −

5

3

B. 5

4 D.

4

3−

July 2005, Q12

Trigonometry II 14

P (-3, 4)

r

y

x0

-1

1

090 18000

y

x

D

10. Which of the following graphs represents y = sin 2x for 0° ≤ x° ≤ 180?

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 207B 117 and 297 D 153 and 333

NOV 2005, Q11

13. It is given that cos ϑ = -0.721 and 00 360180 ≤≤ϑ . Find the value of ϑ.

A. 219o 27’B. B. 230o33’C. 309o27’D. D. 320o33’

Trigonometry II 15

2

1

090 180

00

y

x

A

-1

1

090 18000

y

x

B

1

0

-1

90 18000

y

x

C

NOV 2005, Q12

14. In Diagram 6, QRS is a straight line

Diagram 6

What is the value of cos 0ϑA.

5

4

B. 5

3

C. 5

3−

D. 5

4−

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

A. 13

5C.

12

5−

B. 12

13D.

5

12−

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. 3.56B. 3.44 D. 4.91

Trigonometry II 17

NOV 2006, Q11

18. In Diagram 5, S is the midpoint of straight line QST.

The value of cos xo is

A. 3

4C.

4

3

B. 5

4D.

5

3

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.

A. 13

5− C. 13

12−

Trigonometry II 18

B. 12

5− D. 12

13−

NOV 2006, Q1320. Which of the following represents the graph of y = cos x for 00 1800 ≤≤x ?

A.

B.

C.

Trigonometry II 19

D.

SPM 2007

13. Which of the following graphs represents 0sin 0 180y x for x= ≤ ≤A. B.

Trigonometry II 20 0o 90o 180o

1

-1

C.

11. In diagram below, USR and VQTS are straight lines.

It is given that TS = 29 cm, PQ = 13 cm, QR = 16 cm and 0 8sin

17x = .

Find the value of 0tan y

A. 12

5B.

5

12

C. 5

12− D.

12

5−

12. In Diagram ., O is the origin and JOK is a straight on a Cartesian plane.

Trigonometry II 21

0o 90o 180o

1

-1

V

P

U

T

S

R

x0

y0

K(3,4)

J

The value of cosθ is

A. 4

5− B.

3

5−

C. 3

5D.

4

5

Trigonometry II 22