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Measurement 2: Applications of Trigonometry

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Student Book - Series M-1

sinA = sinBa b

Measurement 2: Applications of trigonometry

Student Book - Series MContents

Topics

Practice Tests

Topic 1 - Review of right-angled triangles __ /__ /__

Topic 1 - Topic test A __ /__ /__

Topic 2 - Solving problems in right-angled triangles __ /__ /__

Topic 2 - Topic test B __ /__ /__

Topic 3 - Compass bearings and true bearings __ /__ /__

Topic 4 - Trigonometric ratios and obtuse angles __ /__ /__

Topic 5 - The sine rule __ /__ /__

Topic 6 - The cosine rule __ /__ /__

Topic 9 - Plane table radial survey __ /__ /__

Topic 7 - The area of a triangle __ /__ /__

Topic 10 - Compass radial survey __ /__ /__

Topic 8 - Miscellaneous problems __ /__ /__

Topic 11 - Offset survey __ /__ /__

Date completed

Author of The Topics and Topic Tests: AS Kalra

iiMeasurement 2: Applications of trigonometry

Mathletics Instant Workbooks – Series M Copyright © 3P Learning

1


Measurement 2: Applications of trigonometryMathletics Instant Workbooks – Series M Copyright © 3P Learning

Topic 1 - Review of right-angled triangles

17CHAPTER 2 – Measurement – Applications of trigonometry

CHAPTER 2Measurement – Applications of trigonometry

Review of right-angled triangles

QUESTION 1 Evaluate correct to three decimal places.

a sin 65° = _________________ b tan 56°18′ = ______________ c cos 88°36′ = _______________

QUESTION 2 If 0° ≤ θ ≤ 90° , find θ to the nearest minute.

a sin θ = 0.7 b cos θ = 0.625 c tan θ = 2.681

QUESTION 3 Find the length of the unknown side, correct to 1 decimal place.

a b c

d e f

QUESTION 4 Find the size of angle θ to the nearest minute.

a b c

EXCEL HSC GENERALMATHEMATICS

pages 32, 33

18 cm25°x

x

34.9 cm60°x72°

12.5 cm

14 cm

28°x

6.5

cma

30°

3 cm

5 cm

θ13

cm5 cm

θ

8 cm

y38

°

17 cm

θ

15 cm

2



Topic 2 - Solving problems in right-angled triangles

18 EXCEL HSC GENERAL MATHEMATICS REVISION & EXAM WORKBOOK

Solving problems in right-angled triangles

QUESTION 1

a Find the length of DC.

b Find the length of CB.

QUESTION 2

a The angle of elevation of the top of a tower AB is 65° from a point C on theground at a distance of 30 metres from the base of the tower. Calculate theheight of the tower to the nearest metre. ___________________________

______________________________________________________________

______________________________________________________________

b A man 1.65 m tall is 28 metres away from a tower 38 m high. What is theangle of elevation of the top of the tower from his eyes? ______________

______________________________________________________________

______________________________________________________________

c From the top of a building 90 metres high, the angle of depression of a carparked on the ground is 63° . Find the distance of the car from the base ofthe building. (Answer correct to 2 decimal places.) __________________

_____________________________________________________________

_____________________________________________________________

QUESTION 3 Give each answer to the nearest centimetre.

a b

Find x. i Find x.

ii Find y.

Measurement – Applications of trigonometry EXCEL HSC GENERALMATHEMATICS

pages 33, 34, 114-116

12 m

D

A

C64°

B28°

6.83 cm58°

4.37 cm

x

h

30 m B

A

C65°

38 m

A

BθM

1.65 m

28 m DC

y

x

39°12 cm

66°

63°

d

90 m

3



Topic 3 - Compass bearings and true bearings


Compass bearings and true bearings

QUESTION 1 For each diagram, write down the true bearing of Q from P.

a b c

d e f

QUESTION 2 A ship sets out from a point A and sails due north to a point B, a distance of 150 km. It thensails due east to a point C. If the bearing of C from A is 048° , find:

a the distance BC b the distance AC

QUESTION 3 A ship leaves port for a destination 80 km east and 70 km north.On what compass bearing should it sail?

QUESTION 4 A ship starts from a port P, sails 226°T for a distance of 120 km. Find:

a how far south of P it is b how far west of P it is.


pages 116, 117

P

Q

N

130°

P

Q

N

51°P

Q

N

126°

P

48°

N

Q

Q

24°

P

N

32°Q

P

N

N

70 km

80 kmA

N

CB

A

48°150 km

N

P

226°

4



Topic 4 - Trigonometric ratios and obtuse angles


Trig ratios and obtuse angles

QUESTION 1 Use your calculator to evaluate the following correct to 2 decimal places.

a sin 120° = ________________ b cos 150° = ______________ c tan 130° = _________________

d cos 141°53′ = _____________ e sin 165°37′ = ____________ f sin 145° = _________________

QUESTION 2 Use your calculator to find the sin, cos and tan of the following angles correct to 4 decimalplaces.

a 120° ; sin 120° = _______________ cos 120° = ________________ tan 120° =________________

b 135° ; sin 135° = _______________ cos 135° = ________________ tan 135° =________________

c 98° ; sin 98° = ________________ cos 98° = _________________ tan 98° =_________________

d 164° ; sin 164° = _______________ cos 164° = ________________ tan 164° =________________

e 175° ; sin 175° = _______________ cos 175° = ________________ tan 175° =________________

f 130° ; sin 130° = _______________ cos 130° = ________________ tan 130° =________________

QUESTION 3 Determine whether the trig ratios of the following angles are positive or negative:

a sin 160° b cos 160° c tan 160° d sin 99°

e tan 35° f sin 145° g cos 10° h tan 95°

i sin 175° j cos 121° k tan 104° l sin 111°

m sin 24° n cos 100° o tan 135° p cos 86°

QUESTION 4 Choose positive or negative to complete each sentence.

a If θ is an obtuse angle then sin θ is

b If θ is an obtuse angle then cos θ is

c If θ is an obtuse angle then tan θ is

QUESTION 5 Find correct to two decimal places:

a

8sin 120

sin 35°

× ° b 15 sin 135

21.25 × °

c

tan 115.8

20°d 72 + 82 – 2 × 7 7 × 8 × cos 127°


page 117

5



Topic 5 - The sine rule (1)


The sine rule(1)

QUESTION 1 Use the sine rule to find the value of the unknown side correct to 2 decimal places.

a b

c d

QUESTION 2 Find:

a b

i the size of ∠ABC. i the size of ∠XYZ.

ii the length of side AC. ii the length of side XZ.


pages 117, 118

7.4 cm81°

54°a cm

48°70°

13.8 cmx cm

123° 19°

x m12.2

m

x m57°

9 m

21°

B

A

C

4 m68°

64°

X

Y Z22°

31°

5 m

6



Topic 5 - The sine rule (2)


The sine rule(2)

QUESTION 1 In the following triangles, find angle θ to the nearest degree.

a b

c d

QUESTION 2

a Use the sine rule to find the size of ∠ABC to the nearest whole degree.

b What is the size of ∠ACB?

QUESTION 3 Find the size of ∠PQR to the nearest minute.


pages 117, 11878

cm

θ

82°60 cm

θ

50°

15 cm

11 c

m

142° θ

15 km9 km

17°θ

29 mm

57 m

m

7 m

C

B

A56°

8 m

PR

Q

68° 40′

11 cm

6.5

cm

7



Topic 6 - The cosine rule (1)


The cosine rule (1)

QUESTION 1 Use the cosine rule to find the value of the unknown side correct to 2 decimal places.

a b

c d

QUESTION 2 In the given diagram:

a use Pythagoras theorem to find the length AC.

b use the cosine rule to calculate the length of CD correct to 2 decimal places.


pages 118, 119

5.8 km43°

7.6 km

87°

x km

x m

5 m17°

12 m

AB

C

9 cm130°

16 cm

a cm

63°14 m

x m

12 m

42°

8 cm

A

B 6 cm C

D

18.6 cm

8



Topic 6 - The cosine rule (2)


The cosine rule (2)

QUESTION 1 In the following triangles, find θ to the nearest degree.

a b

c d

QUESTION 2 Using the diagram shown:

a find ∠ABC to the nearest degree.

b find ∠CAB to the nearest degree.

c what is the size of ∠ACB?


pages 118, 119

4 7

9

C

ABθ

24 cm θA

B

87 cm

C69 cm

7 m

θ10 m

5 m

θ

84 m

m

65 mm

59 mm

7.5 m

A

B

8 m

C10 m

9



Topic 7 - The area of a triangle


Area of a triangle

QUESTION 1 Find the area of each of the following triangles to the nearest square centimetre.

a b

c d

QUESTION 2 Find the area of the parallelograms.

a b

QUESTION 3 The area of this triangle is 52.5 cm2. What is the length of side AC?


pages 27, 28, 120

A

CB6 cm

60°

5 cm

C4 cmB

A

3 cm

40°

A

B

C

45°

9.3 cm

31°

7.26

cm

B C

A

4.6 cm130°

8.7

cm

B

CD

A 15.8 cm

10.2 cm

60°

14 cm

30°

C

A

B

A

D C

B

10.2 cm

12.6 cm

60°

10



Topic 8 - Miscellaneous problems (1)


Miscellaneous problems (1)

QUESTION 1a In �ABC, a = 2.5 cm, b = 3 cm and c = 4 cm. b In ∆ ABC, ∠A = 75° , ∠B = 38° and a = 14.7 cm,

Find ∠A correct to the nearest degree. find b.

QUESTION 2 In the figure below (not drawn to scale), LM = 9 m, NR = 5 m, ∠RLM = 63° , ∠RML = 30° , ∠NRM= 36° , RM = a metres and MN = b metres. Use the sine rule to find a and the cosine rule to findb (correct to 1 decimal place).


pages 117-122

R 5 m36°

a m

30°

9 mM

b m

63°L

N

11



Topic 8 - Miscellaneous problems (2)


Miscellaneous problems (2)

QUESTION 1

a By using the information given in the diagram and the sine rule in �ABC,

show that b = 120 sin 38

sin 32°

°.

b Hence find the value of b correct to 2 decimal places.

c Use the right angled �ADC to find the value of h correct to 2 decimal places.

QUESTION 2 In the diagram P, Q and R represent three towns. Town P is 25 km due west of town Q. The bearingof P from R is 018° and of Q from R is 048° .

a Find the size of ∠PRQ.

b Find the size of ∠PQR.

c Find the distance from P to R.


pages 117-122

120 m BC

110°70°D

38°

b mh m

A

25 kmP Q

R

N

12




Plane table radial survey

QUESTION 1 The diagram shows the result of a plane-table survey.

a Find the size of ∠BPC.

b Find the area of ∆APC correct to 1 decimal place.

c Find the distance from B to C.

QUESTION 2 The diagram below shows the result of a plane table survey ofa paddock.

a Find the size of ∠APC.

b Find the area of ∆BPC correct to 1 decimal place.

c Find the area of the paddock.


pages 122-124

A

42 m

110°

P

B

88°

37 m

C(Figure not to scale)

26 m

135°

95°

20 m

C

B

A

31 m

(Diagram not to scale)

28 mP

Topic 9 - Plane table radial survey

13




Compass radial survey

QUESTION 1 The diagram shows the result of a compass radial surveyconducted on an area of land.

a Find the magnitude of angles BPC and APC.

b Use the cosine rule to find the length in metres of BC, correct to onedecimal place.

QUESTION 2 The diagram shows the results of a compass radial survey ofa triangular area of land.

a Find the size of angle ROQ.

b Find the area of the triangle ROQ.


pages 124, 125

A

88 m

260°P

50 m

BO40°

92 m

145° C

R260°

68 m O56

m

P015°

75 m

155°Q

(Figure not to scale)

Topic 10 - Compass radial survey

14




Offset survey

QUESTION 1 A surveyor conducts an offset survey of a field. The notebookentries appear with measurements in metres.

a Draw a neat sketch (not to scale) of this field.

b Using Pythagoras theorem or otherwise, find the c Find the area of the field.length of the boundary ADB.

QUESTION 2 A scale diagram of an offset survey of a fieldis shown. All measurements are in metres.

a What is the scale used?

b By measurement, find the length of the boundary of thefield DBCA in metres.

c Calculate the area of the field DBCA.


page 34

B

54

45 15 E

D 12 30

20 16 C

0

A

B

25 mD

10 m

30 m

50 mC

20 m

A

Topic 11 - Offset survey


TOPIC TEST SECTION I

Instructions • 10 multiple choice questions• Each question is worth 1 mark• Fill in only ONE CIRCLE• Calculators may be used

Time allowed: 15 minutes Total marks: 10

1 The value of x is given by:

A 43 × cos 28° B 43 × sin 28° C

43cos 28°

D

43sin 28°

2 Find the value of a correct to 2 decimal places if

asin 50

= 8sin 60° °

.

A 7.08 B 8.71 C 9.80 D 12.06

3 If A is an acute angle, find A to the nearest degree in the expression sin A = 25 sin 65

36° .

A 30° B 39° C 47° D 62°

4 Determine the value of a correct to 1 decimal place if a2 = 82 + 92 – 2 × 8 × 9 cos 85° .

A 11.5 B 6.3 C 7.8 D 12.3

5 Find the size of angle A correct to the nearest degree if cos = 5 + 6 – 7

2 5 6

2 2 2A

× ×.

A 89° B 38° C 49° D 78°

6 Using the sine rule, find an expression for the length of AB.

A 12 sin 75

sin 45°

°B

12 sin 45

sin 60°

° C 12 sin 75

sin 60°

° D

sin 4512 sin 75

°°

7 In the diagram what could be used to find the size of ∠ACB?

A sine rule B cosine rule C tangent ratio D none of these

Measurement – Applications of trigonometry

28°

x43

A B

C

12

60° 45°

75°

2.4 m

A

3.2 m

2.7 m BC

15Measurement 2: Applications of trigonometry


Measurement 2: Applications of trigonometryTopic Test PART AInstructions •10 multiple-choice questions •Each question is worth 1 mark •Fill in only ONE CIRCLE •Calculators may be used

Time allowed: 15 minutes Total marks = 10


8 The size of ∠DOA is:

A 99° B 86° C 74° D 58°

9 The diagram shows a sketch of a field drawn from the given notebookentries. All distances are in metres. The distance from E to F is:

A 40 m B 20 m

C 25 m D 30 m

10 In the diagram P is due west of Q and R is due north of Q. ∠RPQ = 38° .The bearing of P from R is:

A 038° B 052° C 218° D 232°

TOPIC TEST SECTION II

Instructions • This section consists of 9 questions• Show all necessary working


11 Find the length of the side marked x. Give the answer to the nearest metre. 3 marks each

a b

B

E

F

A

C

D

38°P

R

Q

106°

A

C

x m26°

7 m

B

P

x m

R10 m

70°Q

12 m

D316° A

042°

B141°

203°C

O

B6040150A

D 2520 C


8 The size of ∠DOA is:

A 99° B 86° C 74° D 58°

9 The diagram shows a sketch of a field drawn from the given notebookentries. All distances are in metres. The distance from E to F is:

A 40 m B 20 m

C 25 m D 30 m

10 In the diagram P is due west of Q and R is due north of Q. ∠RPQ = 38° .The bearing of P from R is:

A 038° B 052° C 218° D 232°

TOPIC TEST SECTION II

Instructions • This section consists of 9 questions• Show all necessary working


11 Find the length of the side marked x. Give the answer to the nearest metre. 3 marks each

a b

B

E

F

A

C

D

38°P

R

Q

106°

A

C

x m26°

7 m

B

P

x m

R10 m

70°Q

12 m

D316° A

042°

B141°

203°C

O

B6040150A

D 2520 C

Instructions •This section consists of 9 questions •Show all necessary working

Time allowed: 45 minutes Total marks = 40

Topic Test PART B



10Total marks achieved for PART A




12 Find the size of the angle θ to the nearest degree. 3 marks each

a b

13 In the diagram Y is 15 km due east of X and 25 km due north of Z. Find thetrue bearing of X from Z. 3 marks

14 In the diagram drawn, ABCD is a quadrilateral in whichAB = 14 m, BC = 10 m, AD = 17.5 m, ∠A = 50° , ∠C = 55° . 2 marks each

a Use the cosine rule to find the length of BD correct to 1decimal place.

b Use the sine rule to find the size of ∠BDC correct to the nearest

degree.

A

B C

14 m13 m

15 m

θ80°

8 cm11 cm

θP

15 km Y

25 km

Z

X

14 mB

10 m

A C50° 55°

D

17.5 m




15 a By using the information given in the diagram and the sine rule in �ABC, show that b = 135 sin 46

sin 34°

°2 marks

b Hence find the value of b correct to 1 decimal place. 1 mark

c Use the right-angled triangle ADC to find the value of h correct to 1 decimal place. 2 marks

16 ABCD is a parallelogram in which AD = BC = 11.2 cm, AC = 13.5 cm and ∠DAC = ∠ACB = 60° .

a Find the area of �ABC to 1 decimal place. 2 marks

b Hence find the area of the parallelogram to the nearest cm2. 1 mark

17 A radial survey of a field is shown.

a Find the size of ∠POQ. 1 mark

h m

A

D CB

b m

80° 100° 46°135 m

AD

B C60°

13.5 cm

11.2 cm

P055°

54 m

40 m

N

S330°

R

35 m

253° 24 m

195°

Q(Not to scale)

O



Total marks achieved for PART B 40


b Calculate the area of the ∆POQ. 2 marks

c Find the size of ∠SOP. 1 mark

18 An offset survey reveals the results at right. All measurements are in metres.

a Draw a sketch of the field. 2 marks

b Calculate the area of the land surveyed. 4 marks

19 A surveyor’s notebook contained the following radial survey. Calculate the distance between the treeand the tower to the nearest metre. 3 marks

B

158

74 140

136 60

90 46

A

052°A

Pole29 m

O

46 m

C

127°

Tower

35 m260°

BTree

measurement 2: applications of trigonometry -...

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