Enrichment

CHAPTER 15: TRIGONOMETRY

Important Concepts:

Trigonometrical Ratios

A Hypothenuse Sin = =

Opposite side B

Cos = =

Adjacent side

C

Tan = =

Example 1 : The figure shows a right-angled triangle.

Sin

= =

Cos = =

15 cm

17 cm

Tan = =

8 cm

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Trigonometry II 1. The unit circle is the circle with radius 1 unit and its centre at origin. 2. 1 y

(x, y)

-1 a) Quadrant I II III IV Angle 0 < < 90 90 < < 180 180 < < 270 270 < < 360 y sin + tan + All + x cos +

1

x

b) sin = y = y1 cos = x = x 1 tan = y x

3. sin cos tan sin cos tan 0 0 1 0 90 1 0 Undefined 30 180 0 -1 0 45 270 -1 0 Undefined 60 360 0 1 0 4. 900 Quadrant II 1800-Quadrant I Quadrant III - 1800 Quadrant IV 3600 - 2700 1 y y 1 1 y

1800

0, 3600

1/2 3 / 2 1 / 3

1/ 2 1/ 2 1

3 / 2 1/2 3

900 1800 2700Trigonometry II

3600

x

900 1800 2700 y = cos x2

3600

x

900

1800 2700 y = tan x

3600

x

y = sin x

Enrichment

-1

-1

-1

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 O III IV -1 270 1 0 360 X

Examples and exercises : State the quadrant for the following angles in the table below. Angle Quadrant 42 I 70 100 II 136 197 205 275 354 Angle 19 265 289 126 303 80 150 212 Quadrant

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

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

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

SIN

,

COS

or

TAN

Followed by the value of the angle and then Example : a) sin 145 b) cos 220 =

=

c) tan 92.5

Press Sin 1 4 5 =

display Sin sin 1 sin 14 sin 145 0.573 576

Press cos 2 2 0 =

display cos cos 2 cos 22 cos 220 - 0.951 056

Press tan 9 2 . 5 =

display tan tan 9 tan 92 tan 92. tan 92.5 0.573 576

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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 either1 SIN

,

1 COS

or

1 TAN

Followed by the value of the angle and then

=

15.1 e) Determine The Value Of Sin , Cos And Tan For Special Angles A 3030 2 3 45 60 B 1 D 1 Using Isosceles Triangle Sin 45 = 60 C B 1 C 2 1 2 A 45

Using The Right-Angled Triangle Bad, Sin 30 = 1 23 2

Cos 30 =

Cos 45 =

Tan 30 =

1 3

tan

45=

sin

60 =

3 2

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15.2 Graphs Of Sine, Cosine And Tangent Examples : 15.3 Questions Base On Examination Format. 1. Which of the following is equal to cos 35 ? A. cos 145 B. cos 215 C. cos 235 D. cos325

2. Find the value of sin 150 + 2 cos 240 - 3 tan 225 A. -3.5 3. Sin 30 + cos A. 1 4 B. -1.5 60 = B. 1 2 C. 1 D. 0 C. 1.5 D. 2.5

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 A. 18 B. 162

0 360, find the value of C. 218 D. 300

6. Given that tan = 05774 and A. 30 , 210

0 360 , find the value of C.30 , 330 D. 30 , 150

B. 152 , 210

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 A. 108 B. 122

90 x 180 , find the value of x C. 132 D. 158

9. Given that tan y = -2.246 and

0 360 , find the value of y

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A. 66 , 246 10. (0,1) ( -1,0) O

B. 114 ,246

C. 114 , 294

D.246 , 294

y

(1,0) (0.87,-0.50) ( -1,0) X

The diagram shows the unit circle. The value of tan is A. -1.74 11. B. -0.57 C. -0.50 D. 0.87

y1 -1 O P -1 The diagram shows the unit circle. If P is (-0.7, -0.6), find the value of Sin A. 12 7 6 B. 6 7 C. -0.6 D. 0.6 1 X

y1 -1 O 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. 0.4 0.8 1 X

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13. In the diagram, ABC is a straight line. The value of sin x is B A x 15 D A.8 1 5

C 8

B.

8 17

C.

15 17

D.

17 15

14. 13 cm Q

T 5 cm 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.12 13

B.

12 25

C.

13 25

D.

24 25

15.

P 15 cm Q T 6 cm S

R In the diagram, PQR and QTS are straight lines. Given that sin sin PQT = A.8 1 5

TRS =

3 , then 5

B.

8 17

C.

8 15

D.

8 17

16.

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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 SQ In the diagram above, PQR is a straight line. Given that cos P = , find tan x. 5 4 1 5 3 A. B. C. D. 2 8 4 5 18.

3 JG In the diagram above, EFGH is a straight line. If sin H = , the value of tan x 5

= A.4 5

B.

1 2

C.

1 3

D.

3 5

19.

Diagram below shows a graph of trigonometric function.

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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 4 A. 3

isB.

3 5

C.

3 5

D.

4 5

15.4 PAST YEAR SPM QUESTIONS

Trigonometry II

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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

E1 3 C. x 1 2 1 2 D. 5

G 5 A.

K

H J

1 2 1 2 B. 1 3

13 cm

Nov 2003, Q12 2. Which of the following graphs represents y = sin x ?

Nov 2004, Q 11

Trigonometry II

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3. In Diagram 5, PRS is a straight line Q 7 cm P

24 cm

R

x

S Find the value of cox x =7 24 24 B. 25

A.

7 24 24 D. 25

C.

Nov 2004, Q 12 4. Diagram 6 shows the graph of y = sin x. The value of p is A. 90 B. 180 C. 270 D. 360

Nov 2004, Q13 5. In diagram 7, JKL is a straight line.

Diagram 7 It is given that cos x = A. 22 5 1 3 and tan y = 2. Calculate the length, in cm, of JKL C. 4412

Trigonometry II

Enrichment

B. 29 Nov 2005, Q11

D. 58

6. It is given that cos = 0.7721 and 180 360. Find the value of A. 219 27 B. 230 33 Nov 2005, Q12 7. In Diagram 6, QRS is a straight line. 4 cm Q 3 cm R S What is the value of cos ? 4 A. 5 3 B. 5 Diagram 6 P C. 309 27 D. 320 33

C.

3 5 4 D. 5

July 2004, Q13

H 12 cm Diagram 6 E x

16 cm

G

13 cm

F

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

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A. 33 01 B. 40 33 July 2004, Q14 9. In Diagram 7, O is the origin of a Cartesian plane.

C. 49 28 D. 50 54y 1

B

0

900

1800

x

Diagram 7

-1

The value of sin r isy P (-3, 4)

r 0 x

3 5 4 B. 5 A.

3 5 3 D. 4 C.

June 2005, Q12 10. Which of the following graphs represents y = sin 2x for 0 x 180?

Trigonometry II

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A

y 2

1

C

y 1

0

90

0

180

0

x

0

900

180

0

x

D

y 1

0 -1 -1

900

1800

x

June2005, Q13 11. In Diagram 5, MPQ is a straight line.

Q P y0

Diagram 5

M

x0

N

Given cos x = 7 24

24 , find the value of tan 25 D. 24 7

y. A. 7 24 B. 24 7 C.

June 2005, Q11 12. Given cos x = - 0.8910 and 0 x 360, find the values of x.

Trigonometry II

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A B

117 and 243 117 and 297

C. 153 and 207 D 153 and 333

NOV 2005, Q11 13. It is given that cos

-0.721 and 180 =

0

6 30

0

. Find the value of

.

A. B. C. D.

219o 27 B. 230o33 309o27 D. 320o33

NOV 2005, Q12 14. In Diagram 6, QRS is a straight line

Diagram 6

What is the value of cos 4 A. 5 3 B. 5 3 C. 5 4 D. 5

0

JUNE 2006, Q11 15. Diagram 5 shows a rhombus PQRS

Trigonometry II

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Diagram 5 It is given that QST is a straight line and QS = 10cm. Find the value of tan xo.5 1 3 1 3 B. 1 2

A.

5 C. 12 12 D. 5

JUNE 2006, Q12 16. Which of the following represents part of the graph of y = tan x? A. C.

B.

D.

JUNE 2006, Q13

Trigonometry II

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17. In Diagram 6, PQR and TSQ are straight lines.

Find the length of ST , in cm. A. 2.09 B. 3.44 NOV 2006, Q11 18. In Diagram 5, S is the midpoint of straight line QST. C. 3.56 D. 4.91

The value of cos xo is 4 A. 3 4 B. 5 NOV 2006, Q12

3 4 3 D. 5 C.

19. In Diagram 6, MPQ is a right angled triangle.

Trigonometry II

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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 5 13 B. D. 12 12 NOV 2006, Q1310 20. Which of the following represents the graph of y = cos x for 0 0 x 80

?

A.

B.

Trigonometry II

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C.

D.

JUNE 2007, Q13 21. In Diagram 7, SPQ and PRU are right angle triangles. STQ and PTU are straight lines.

Trigonometry II

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U S

T 20O P yo Q R

Diagram 7

It is given that cos yo = A. B. C D. 25.54 27.67 65.94 70.17

1 2 and PQ = QR . Calculate the length incm, of PTU 1 3

JUNE 2007,Q14 22. In Diagram 8. PRS is a straight line,

S R xoh cm

PIf tan xo = A. 5

12 cm

Q

3 , then the value ofDiagram 8 h is 4

Trigonometry II

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B. C. D.

15 16 20

JUNE 2007 , Q15 23, A. Which of the following represents the graph of y = sin x for 0o x 369oy

0.5

0

90

0

180

360O

x

B.y

0.50 x

60O

180O

360O

C.

y

0.50 45O x

180O

360O

D.

1

y

0

180o

x 360 22O

Trigonometry II

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U S

NOV 2007, Q11 In diagram 6, USR and VQTS are straight lines,

xo P T R

yo Q V Digram 68 , 17

It is given that TS = 29 cm, PQ = 13 cm, QR = 16 cm and sin xo = Find the value of tan yo A. B. C. D. 1 2 5 5 1 2 5 12 12 5

NOV 2007, Q12 24.

y

K(3,4)Trigonometry II

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Enrichment

0 J

x

In Diagram 7, O is the origin and JOK is a straight line on a Cartesian plane. The value of cos is A. B. C. D. 4 5 3 5 3 5 4 5 -

.

NOV 2007, Q13 10 25. Which of the following graphs represents y = Sin x for 0 0 x 8 A.y 1

0

?

0

90O

180O

x

-1B.1 y

0

90O

180O

x

C.

-1 1

y

0

90O

180O

x

-1Trigonometry II

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y 1

0

90O

180

x

-1

JUNE 2008,Q 12 27. In Diagram 7, RTU is a right angled triangle RST and TUV are straight lines

R

S

T

U Digram 7

V1 2 5

It is given that RS = 28 cm, TU = 15 cm and tan RUV = Find the length, in cm, of SU. A. B. C. D 23 22.63 17 15.73

JUNE 2008,Q13 28. Given that sin x = 1 , 29 00

x 2 0 7

find the value of 3 cos x.

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A.

3 2

3 2

B. C D.

3 2 3 2

JUNE 2008,Q146 28. Which graph represents y = cos x for 0 0 x 3 00

?

A.

1

y

0

180o

360O

x

-1By

1

0

180o

360O

x

-1

C 1

y

0

180o

360O

x

-1

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D 1

y

0

180o

360O

x

-1 NOV 2008,Q11 30. Digram 6 shows a right angled triangle PQR. R

h xo P Given sin xo = A. B. C. D.k tan 30 o

1 , find the value of h. 2

k Diagram 6

Q

k tan 30Ok cos 60 o

k cos 60o

NOV 2008,Q13 31. Which of the following represents the graph of y = tan x for 0 0 x 360 A. 1y0

?

0

180o

x 360O 27

Trigonometry II

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B. 1y

0

180o

x 360O

-1 Cy

1

0

180o

x 360O

-1

D.

1

y

0

180o

x 360O

-1Chapter 15 Trigonometry Exercise 1: 1 2 Sin x = , 1 3

ANSWERS

Cos x =

5 1 2 , Tan x = 1 3 5

Exercise 2: CD = 17 Exercise 5 : BAC = 28.96

Exercise 3: Sin x =

1 2 4 = 9 3

Exercise 4 : FL = 9cm

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Exercise 6 :

3 5

Exercise: 8 17 4. Perimeter = 48 1. Cos x = 7. BD = 24 10. Cos x = 24 25 2. KL = 20 cm 5. AD = 18 + 60 = 78 8. Sin y = Sin 30 Sin 30 = 0.5 3. MC = 12.5 cm 8 4 6. Sin y = or 1 0 5 7 9. Cos y = 22 .96

15.1a) Angle 42 70 100 136 197 205 275 354 15.1 b. Angle 75 120 160 200 257 280 345 15.1 c

Quadrant I I II II III III IV IV Quadrant I II II III III IV IV Sin + + +

Angle 19 265 289 126 303 80 150 212 Value (Positive/ Negative) Cos + + +

Quadrant I III IV II IV I II III

Tan + + +

ANGLE Cos 143 Tan 98 Cos 245 Tan 190 Cos 300 Tan 315

CORRESPONDING ANGLE IN QUADRANT I Cos 37 Tan 82 Cos 65 Tan 10 Cos 60 Tan 45

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15.1 d Angle Sin 46 Cos 57 Tan 79 Sin 139 Cos 154 Tan 122 Sin 200 Cos 187 Tan 256 Sin 342 Cos 278 Tan 305 15.1 e VALUE 1 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 Tan 1 1.4502 15.3: EXAMINATION FORMAT QUESTIONS No Answer No 1 D 11 2 A 12 3 C 13 4 B 14 5 A 15 6 A 16 7 A 17 8 B 18 9 C 19 10 B 20 15.4 SPM NOV 2003, Q11 NOV 2003, Q12 ANSWER A D ANGLE 49..94 68.74 7.215 64.17 49.29 40.16 53.55 36.57 55.411 Answer C B C D B D A C B D Value 0.7193398 0.5446390 5.1445 0.6560 -0.8987 -1.6003 -0.3420 -0.9925 4.01078 -0.30901 0.13917 -1.42814

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Enrichment

NOV 2004, Q11 NOV 2004, Q12 NOV 2004, Q13 NOV 2005, Q11 NOV 2005, Q12 NOV 2006, Q11 NOV 2006, Q12 NOV 2006, Q13 NOV 2007, Q11 NOV 2007, Q12 NOV 2007, Q13 NOV 2008, Q11 NOV 2008, Q12 NOV 2008, Q13 JUN 2004, Q13 JUN 2004, Q14 JUN 2005, Q12 JUN 2005, Q13 JUN 2005, Q11 JUN 2006, Q15 JUN 2006, Q16 JUN 2006, Q17 JUN 2007, Q13 JUN 2007, Q14 JUN 2007, Q15 JUN 2008, Q12 JUN 2008, Q13 JUN 2008, Q14

D A B A D B C B D B C B A D C B C B C D A C A D D C A C

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