chapter 15 i trigonometry ii student

8/12/2019 Chapter 15 i Trigonometry II Student

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Trigonometry II 1

CHAPTER 15: TRIGONOMETRY

Important Concepts: Trigonometrical Ratios

Exercise 3 :

In Diagram below,ABCandBDEare two right-angled triangles.

Solution :

IfAB = 9 cm,BD= 8 cm,DE= 10 cm, andBE= CE,

calculate the value of sinx.

Example 4

In the diagram above, QRS is a straight line . Given

Adjacent side

Hypotenuse

B

A

C

Sin =eHypothenus

sideOpposite =AC

AB

Cos =eHypothenus

sideAdjacent=

AC

BC

Tan =sideAdjacent

sideOpposite=

BC

AB

Opposite

side

9 cm 8 cmA B D

C

10 cm

E

x

Q

S

P

TR

5 cm


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Trigonometry II 2

cos13

5QPS and tan4

3RTS , find

the length ofRT in

Solution

Exercise4

Based on the figure above, calculate the length of FL , in cm .

Example 5

In the diagram aboveFGH is a straight line . Given that tan 1HEF , thus tan HGE

Solution

5 cm

H

F

G

E7 cm

H F

L

48o

10 cm


3/41

Trigonometry II 3

Exercise 5

In the diagram above, BCD is a straight line and AB is perpendicular to line BCD .Find BAC .

Example 6

In the diagram above , QRS is a straight line .

Given that sin xo =13

12, find the value of

cos yo.

Exercise 6

In the diagram above ,FGH is a straight line . find the value of cosxo.

RS

o

xo

P

14 cm

8 cm

A

B C D

30o

E

FG

G

xo

5 cm4 cm


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Trigonometry II 4

Exercise :

1 Diagram 1 shows a rectanglePQRS,where diagonal SQ= 17 cm andPQ= 15 cm.

Calculate the value of cosx.

2 Diagram 2 shows a triangleKLM.

Diagram 2

Calculate the length, in cm, ofKLifML= 60 cm and cosx=3

1 .

3 In Diagram 3,Mis the midpoint ofBCin a right-angled triangleABC.

DIAGRAM 13

Diagram 3

Given that cosx=25

7 andAB= 50 cm, calculate the length, in cm, ofMC.

x

M

K L

60 cm

B

M

CA

x

17cm

15cmP Q

RS

x

Diagram 1


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Trigonometry II 5

4 In Diagram 4,MNLPis a trapezium.

Diagram 4

IfNL = 2MN, sinx=17

15 ,NL= 8 cm and OP= 15 cm, find the perimeter, in cm, of

the trapezium.

5 In Diagram 5,BCDE is a rectangle andABMis a triangle.

Diagram 5

AMDandABCare straight lines andAB= 6 cm, andED= 20 cm. Given that cosx=3

1 ,

find the length, in cm, ofAD.

P

OM

LN

15 cm

x

8 cm

DE

CB

A

M

x


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Trigonometry II 6

6 In Diagram 6,ABCD is a rectangle.

Diagram 6

IfAB= 9 cm andBX=

3

2AB. IfBC= 16 cm, and BY =

2

1BC, find siny.

7 In Diagram 7 cosx=13

5 .

B

x y

A C D

Diagram 7

IfACD is a straight line andAC= 5 cm, find the length ofBD when siny= .2

1

8 In Diagram 8ABCis an equilateral triangle.

C

X

yA B

Diagram 8

IfAXYis a right-angled triangle, find the value of siny.

BAX

CD

Y

y

Y


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Trigonometry II 7

9 In Diagram 9,ABCDis a trapezium.

15 cm

D C

24 cm

y

A B22 cm

Diagram 9

IfAD= 24 cm,DC = 15 cm andAB= 22 cm, find the value of cosy.

10 In Diagram 10, the area of squareABCDis 49 cm and the area of triangle CDEis 84

cm.

A D

x

B C E

Diagram 10

Given that BCE is a straight line, find the value of cos x.


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Trigonometry II 8

TRIGONOMETRY II

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

2.

a)Quadrant Angle

I 0 < < 90II 90 < < 180III 180 < < 270IV 270 < < 360

b) sin = y = y1

cos = x = x1

tan = yx

1

1

-1

(x, y)

y

y

x

x

All +sin +

cos +tan +

3. 0 90 180 270 360

sin 0 1 0 -1 0cos 1 0 -1 0 1tan 0 Undefined 0 Undefined 0

30 45 60sin 1 / 2 1/ 2 3 / 2cos 3 / 2 1/ 2 1 / 2tan 1 / 3 1 3

4.

Quadrant II

1800-Quadrant I

Quadrant III - 1800

Quadrant IV

3600 -

900

1800

2700

0, 360

y

x3600

y

x3600

y

x3600180

0 270

090

0

27001800900900 1800 2700

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

1

-1

1 1

-1-1


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Trigonometry II 9

15.1 Identifying the Quadrants and the Angles in A Unit Circle.

1. The x-axis and the y-axis divides the unit circle with centre origin into 4 quadrants as shown in

the diagram below

y1 90

180 -1 II I 1 0 O 360 X

III IV

-1 270

Examples and exercises :

State the quadrant for the following angles in the table below.

Angle Quadrant Angle Quadrant

42 I 19

70 265

100 II 289

136 126

197 303

205 80

275 150

354 212

REMEMBER

QUADRANT I 0 < < 90QUADRANT II 90 < < 180

QUADRANT III 180 < < 270QUADRANT IV 270 < < 360


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Trigonometry II 10

15.1 a) Determine whether the values of

a) sin b) cos c) tan are positive or negative if

oooooo and 360270,270180,18090

y1 90

180 -1 Sin + ALL 1 0O 360 X

Tan + Cos +

-1 270

Examples :

i) Sin 142 ii) cos 232 iii) tan 299

142 is in quadrant II cos 232 is in quadrant III tan 299 is in quadrant IVSin is positive in Quadrant II Cos is negative in quadrant III tan is negative in quadrant IV

Exercises :

Angle Quadrant Value (Positive/ Negative)

Sin Cos Tan

75 I + + +

120 II + - -

160 200

257

280

345


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Trigonometry II 11

15.1 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 diagrambelow :

QUADRANT II QUADRANT III QUADRANT IV

( 90 180 ) ( 180 270) (270 360)Sin = sin ( 180 - )Cos = cos ( 180 - )Tan = tan ( 180 - )

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

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

Example :

120

Sin 120 = sin 60Cos 120 = - cos 60Tan 120 = - tan 60

EXERCISES :

Find the values of the angles in quadrant I which correspond to the following values of angles inother quadrants.

ANGLE CORRESPONDING ANGLE IN

QUADRANT I

Sin 125 Sin = sin ( 180 - 125)= sin 55

Cos 143

Tan 98

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

Cos 245

Tan 190

Sin 285 Sin = - sin ( 360 - )= -sin 55

Cos 300

Tan 315

230 340

Sin 230 = - sin 50Cos 230 = - cos 50

Tan 230 = tan 50

Sin 340 = - sin 20Cos 340 = cos 20Tan 340 = - tan 20


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Trigonometry II 12

15.1 c)Find the value of Sine, Cosine and Tangent of the angle between 90 and 360

TIPS :If a calculator is used, press either , or

Followed by the value of the angle and then

Example :

a) sin 145 b) cos 220 = c) tan 92.5

Exercises :

Angle Value

Sin 46

Cos 57

Tan 79

Sin 139

Cos 154

Tan 122

Sin 200

Cos 187

Tan 256

Sin 342

Cos 278Tan 305

=

Press display

Sin Sin1 sin 1

4 sin 145 sin 145

= 0.573 576

Press display

cos cos2 cos 2

2 cos 220 cos 220

= - 0.951 056

Press display

tan tan9 tan 9

2 tan 92. tan 92.

5 tan 92.5= 0.573 576

SIN COS TAN


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Trigonometry II 13

15.1 d ) Find the angle between 0 and 360 when the values of sine, cosine and tangent are

Given

TIPS :

If a calculator is used, press either SIN 1 , COS 1 or TAN 1

Followed by the value of the angle and then

Examples :

a) Sin1 0. 94 b) Cos

1 -0.64 c) Tan

10.625

Press display Press Display Press Display

Shift Sin Sin 1 Shift Cos Cos

1 Shift Tan Tan

1

0 Sin 1 0 (-) - 0 Tan

10

. Sin 1 0. 0 Cos

1-0 . Tan

10.

9 Sin 1 0.9 . Cos

1-0. 625 Tan

10.625

4 Sin 1 0.94 64 Cos

1-0.64 = 32.00

= 70.05 15 = 129.79

Sin1 0. 94 = 70.05 Cos

1-0.64 = 129.79 Tan

1 0.625 = 32.00

Exercises :

VALUE ANGLE

Sin1 0.7654

Sin1 -0.932

Sin10.1256

Cos10.4356

Cos1-0.6521

Cos1-0.7642

Tan1-1.354

Tan 1

0.7421

Tan 1

1.4502

=


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Trigonometry II 14

15.1 e) Determine The Value Of Sin , Cos And Tan For Special Angles

A A

45

3030

2 2 1 23

4560 60

B D C B 1 C

1 1

Using The Right-Angled Triangle Bad, Using Isosceles Triangle

Sin 30 =2

1 Sin 45 =

Cos 30 =2

3 Cos 45 =

Tan 30 =3

1 tan 45=

sin 60 =2

3

cos 60 =

tan 60 =


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Trigonometry II 15

15.2 Graphs Of Sine, Cosine And Tangent15.2.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 0 45 90 135 180 225 270 315 360

Y

ii) y = cos x

X 0 45 90 135 180 225 270 315 360

Y

iii) y = tan x

X 0 45 90 135 180 225 270 315 360

Y

Examples :

The diagram shows graphs y = sin x for 0 360 . Find the value of y when yis a) 90

b) 270

c) 360

SOLUTION : a) y = 1

b) y = -1

c) y = 0


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Trigonometry II 16

15.3 Questions Based on the Examination Format.

1. Which of the following is equal to cos 35 ?

A. cos 145 C. cos 235 B. cos 215 D. cos325

2. Find the value of sin 150 + 2 cos 240 - 3 tan 225

A. -3.5 B. -1.5 C. 1.5 D. 2.5

3. Sin 30 + cos 60 =

A.4

1 B.

2

1 C. 1 D. 0

4. Given that sin 45 = cos 45 = 0.7. Find the value of 3 sin 315 - 2 cos 135

A. -3.5 B. -1.5 C. 1.5 D. 2.5

5. Given that cos = 0.9511 and 0 360, find the value of

A. 18 B. 162 C. 218 D. 300

6. Given that tan = 05774 and 0 360, find the value of

A. 30 , 210 B. 152 , 210 C.30 , 330 D. 30 , 150

7. Given that sin = -0.7071 and 90 270, find the value of

A. 135 B. 225 C. 45 D. 315

8. Given that Sin x = 0.848 and 90 x 180 , find the value of x

A. 108 B. 122 C. 132 D. 158

9. Given that tan y = -2.246 and 0 360 , find the value of y

A. 66 , 246 B. 114 ,246 C. 114 , 294 D.246 , 294


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Trigonometry II 17

10.

y(0,1)

( -1,0) (1,0)

O X

(0.87,-0.50)

( -1,0)

The diagram shows the unit circle. The value of tan is

A. -1.74 B. -0.57 C. -0.50 D. 0.87

11.

y1

-1 1O X

P

-1

The diagram shows the unit circle. If P is (-0.7, -0.6), find the value of Sin

A. -6

7 B. -

7

6 C. -0.6 D. 0.6

12

y1

-1 1O X

R (0.8, -0.4)

-1

The diagrams shows a unit circle and R (0.8, -0.4). find the value of cos

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

4.0


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Trigonometry II 18

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

BA C

x

15 8

D

A.15

8 B.

17

8 C.

17

15 D.

15

17

14.

T13 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 SQ

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


19/41

Trigonometry II 19

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 cos5

3SQP , find tan x.

A.21 B.

85 C.

43 D.

54

18.

In the diagram above, EFGH is a straight line. If sin5

3JGH , the value of tan x =

A.5

4 B.

2

1 C.

3

1 D.

5

3


20/41

Trigonometry II 20

19. Diagram below shows a graph of trigonometric function.

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 is

A.3

4 B.

5

3 C.

5

3 D.

5

4


21/41

Trigonometry II 21

15.4 PAST YEAR SPM QUESTIONS

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 ?

F

G

EK

H

J

13

x


22/41

Trigonometry II 22

Nov 2004, Q 11

3. In Diagram 5, PRS is a straight line

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

Q

P

7 cm

24 cm

R

S


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Trigonometry II 23

Nov 2004, Q13

5. In diagram 7, JKL is a straight line.

Diagram 7

It is given that cos x =13

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

A. 22 C. 44

B. 29 D. 58

Nov 2005, Q11

6. It is given that cos = 0.7721 and 180 360. Find the value of

A. 219 27 C. 309 27B. 230 33 D. 320 33

Nov 2005, Q12

7. In Diagram 6, QRS is a straight line.

4 cm

Q P

3 cm

R

Diagram 6

S

What is the value of cos ?

A.5

4 C.

5

3

B.5

3 D.

5

4


24/41

Trigonometry II 24

16 cm

12 cm

x

13 cm

H

E

G

F

JUNE 2004, Q13

Diagram 6

8. Diagram 6 shows a quadrilateral EFGH. Find the value of x.

A. 33 01 C. 49 28B. 40 33 D. 50 54

JUNE 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

P(-3, 4)

r

y

x0


25/41

Trigonometry II 25

y

NM

P

Q

x 0

0

-1

1

090 18 0

00

y

x

D

JUNE 2005, Q12

10. Which of the f ollowing graphs represents y = sin 2x for 0 x 180,?

JUNE 2005, Q13

11. In Diagram 5, MPQ is a straight line.

Diagram 5

Given cos x=25

24, find the value of tan y.

A.24

7 B.

7

24 C.

24

7 D.

7

24

-1

1

090 180

00

y

x

B1

0

-1

90 18000

y

x

A

2

1

090 180

00

y

x

C


26/41

Trigonometry II 26

JUNE 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 = -0.721 and00 360180 . Find the value of .

A. 19o27

B. 230o33

C. 309o27

D. 320o33

NOV 2005, Q12

14. In Diagram 6, QRS is a straight line

Diagram 6

What is the value of cos0

A.

5

4

B.5

3

C.5

3

D.5

4


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Trigonometry II 27

JUNE 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.

A.13

5 C.

12

5

B.12

13 D.

5

12

JUNE 2006, Q12

16. Which of the following represents part of the graph of y = tan x?

A. C.

B. D.


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Trigonometry II 28

JUNE 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

NOV 2006, Q11

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

The value of cos xois

A.3

4 C.

4

3

B.5

4 D.

5

3


29/41

Trigonometry II 29

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.135

C.1312

B.12

5 D.

12

13

NOV 2006, Q13

20. Which of the following represents the graph of y = cos x for00 1800 x ?

A.


30/41

Trigonometry II 30

B.

C.

D.


31/41

Trigonometry II 31

JUNE 2007, Q13

21. In Diagram 7, SPQ and PRU are right angle triangles. STQ and PTU are straight lines.

It is given that cos yo

= 13

12

and PQ = QR . Calculate the length incm, of PTU

A. 25.54

B. 27.67

C 65.94

D. 70.17

JUNE 2007,Q14

22. In Diagram 8. PRS is a straight line,

S

R

U

P Q

T

20O y

o

Diagram 7

P 12 cm Q

xo

R

S

Diagram 8

h cm


32/41

Trigonometry II 32

If tan xo=4

3 , then the value of h is

A. 5

B. 15

C. 16

D. 20

JUNE 2007 , Q15

23, Which of the following represents the graph of y = sin x for 0o x 369o

A.

B.

C.

0.5

060 180

y

x360O

0.5

045 180

y

x360O

0.5

090 180

0

y

x360O


33/41

Trigonometry II 33

D.

NOV 2007, Q11

In diagram 6, USR and VQTS are straight lines,

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

17

8 ,

Find the value of tan yo

A.5

12

B.125

C.12

5

D.5

12

1

0

y

x360O

-1

180o

V Digram 6

U

S

R

xo

T

P

Qy

o


34/41

Trigonometry II 34

NOV 2007, Q12

24.

In Diagram 7, O is the origin and JOK is a straight line on a Cartesian plane.

The value of cos is

A. -5

4

B. -5

3 .

C.5

3

D.5

4

NOV 2007, Q13

25. Which of the following graphs represents y = Sin x for 00 1800 x ?A.

B.

K(3,4)

J

0x

y

1

090 180

y

x

-1

1

090 180

y

x

-1


35/41

Trigonometry II 35

C.

JUNE 2008,Q 12

27. In Diagram 7, RTU is a right angled triangle RST and TUV are straight lines

It is given that RS = 28 cm, TU = 15 cm and tan RUV = -5

12

Find the length, in cm, of SU.

A. 23

B. 22.63

T U V

R

S

Digram 7

1

0 90 180

y

x

-1

1

090

O

180

y

x

-1


36/41

Trigonometry II 36

C. 17

D 15.73

JUNE 2008,Q13

28. Given that sin x = -2

1, 270900 x find the value of 3 cos x.

A. -2

3

B.2

3

C 23

D. 23

JUNE 2008,Q14

28. Which graph represents y = cos x for00 3600 x ?

A.

B

1

0180

o

y

x360O

-1

1

0180o

y

x360O

-1


37/41

Trigonometry II 37

C

D

NOV 2008,Q11

30. Digram 6 shows a right angled triangle PQR.

Given sin xo=

2

1, find the value of h.

A.o

k

30tan

B. k tan 30O

C.o

k

60cos

D. k cos 60o

1

0180o

y

x360O

-1

1

0180o

y

x

360

O

-1

P k Q

xo

R

h

Diagram 6


38/41

Trigonometry II 38

NOV 2008, Q13

31. Which of the following represents the graph of y = tan x for 00 3600 x ?

A.

B.

C

D.

1

0

y

x360O

-1

180o

1

0

y

x360O

-1

180o

1

0

y

x360O

-1

180o

1

0

y

x360O

-1

180o


39/41

Trigonometry II 39

ANSWERS

Chapter 15 Trigonometry

Exercise 1:

Sin x =13

12, Cos x =

13

5, Tan x =

5

12

Exercise 2: CD = 17 Exercise 3: Sin x =9

12=

3

4 Exercise 4 : FL = 9cm

Exercise 5 : BAC = 28.96

Exercise 6 :5

3

Exercise:

1. Cos x =

17

8

2. KL = 20 cm 3. MC = 12.5 cm4. Perimeter = 48 5. AD = 18 + 60 = 78

6. Sin y =10

8or

5

4

7. BD = 24 8. Sin y = Sin 30

Sin 30

= 0.59. Cos y =

96.22

7

10. Cos x =25

24

15.1a)

Angle Quadrant Angle Quadrant42 I 19 I

70 I 265 III

100 II 289 IV

136 II 126 II

197 III 303 IV

205 III 80 I

275 IV 150 II

354 IV 212 III

15.1 b.

Angle Quadrant Value (Positive/ Negative)

Sin Cos Tan75 I + + +

120 II + - -

160 II

200 III

257 III 280 IV

345 IV


40/41

Trigonometry II 40

15.1 c

ANGLE CORRESPONDING ANGLE IN

QUADRANT I

Cos 143 Cos 37

Tan 98 Tan 82

Cos 245 Cos 65

Tan 190 Tan 10

Cos 300 Cos 60

Tan 315 Tan 45

15.1 d

Angle Value

Sin 46 0.7193398

Cos 57 0.5446390

Tan 79 5.1445

Sin 139 0.6560

Cos 154 -0.8987

Tan 122 -1.6003

Sin 200 -0.3420

Cos 187 -0.9925

Tan 256 4.01078

Sin 342 -0.30901

Cos 278 0.13917

Tan 305 -1.42814

15.1 e

VALUE ANGLE

Sin1 0.7654 49..94

Sin1 -0.932 68.74

Sin10.1256 7.215

Cos10.4356 64.17

Cos1-0.6521 49.29

Cos1-0.7642 40.16

Tan1-1.354 53.55

Tan 1

0.7421 36.57

Tan 1

1.4502 55.411

15.3: EXAMINATION FORMAT QUESTIONS

No Answer No Answer

1 D 11 C

2 A 12 B

3 C 13 C

4 B 14 D


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5 A 15 B

6 A 16 D

7 A 17 A

8 B 18 C

9 C 19 B

10 B 20 D

15.4

Questions Answers Questions Answers

NOV 2003, Q11 A

NOV 2003, Q12 D

JUN 2004, Q13 C NOV 2004, Q11 D

JUN 2004, Q14 B NOV 2004, Q12 A

NOV 2004, Q13 B

JUN 2005, Q12 C NOV 2005, Q11 A

JUN 2005, Q13 B NOV 2005, Q12 DJUN 2005, Q11 C

JUN 2006, Q15 D NOV 2006, Q11 B

JUN 2006, Q16 A NOV 2006, Q12 C

JUN 2006, Q17 C NOV 2006, Q13 B

JUN 2007, Q13 A NOV 2007, Q11 D

JUN 2007, Q14 D NOV 2007, Q12 B

JUN 2007, Q15 D NOV 2007, Q13 C

JUN 2008, Q12 C NOV 2008, Q11 B

JUN 2008, Q13 A NOV 2008, Q12 A

JUN 2008, Q14 C NOV 2008, Q13 D

chapter 15 i trigonometry ii student

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