10 trigonometry

10C H A P T E R

Trigonometry

What you will learn10.1 Introducing trigonometry

10.2 Finding the side length of a right-angled triangle

10.3 Further problems involving side lengths

10.4 Finding the angle

10.5 Mixed application problems

10.6 Angles of elevation and depression

10.7 Bearings

10.8 Problems involving two triangles

CUAT029-C10[352-395].qxd 6/5/06 9:51 PM Page 352

VELSMeasurement, chance and data

Students estimate and measure lengthand angle.Students use trigonometric ratios, sine,cosine and tangent to obtain lengths ofsides, angles and the area of right-angled triangles.

Working mathematicallyStudents choose, use and developmathematical models and proceduresto investigate and solve problems set ina wide range of practical, theoreticaland historical contexts.Students select and use technology invarious combinations to assist inmathematical inquiry, to manipulateand represent data, to analysefunctions and carry out symbolicmanipulation.

The Eureka TowerThe Eureka Tower in Melbourne is 292 metres tall. Building the towerinvolved many logistic andmathematical problems. One of themost dangerous and challengingproblems was how to dismantle andremove safely the giant 180-tonnecrane in the lift well on top of thebuilding. To do this another identicalcrane was placed on the top of thebuilding to sit opposite the existingone to remove the first crane piece bypiece. The second crane was thenremoved using a smaller but powerfulrecovery crane which is easier todismantle. This whole process requiredcareful planning and calculation ofangles and lengths of triangles toensure that no damage was done tothe Eureka Tower or other surroundingbuildings. Calculations involvingtriangles, angles and lengths requirethe use of trigonometry.

CUAT029-C10[352-395].qxd 6/5/06 9:51 PM Page 353

Essential Mathematics VELS Edition Year 9

Do now

354

Skillsheet

T EACHE R

1 Round off each number correct to four decimal places.

a 0.456 78 b 0.345 69 c 0.045 62 d 0.279 97

2 Round off each number correct to two decimal places.

a 4.234 b 5.678 c 76.895 d 23.899

3 Round off each number to the nearest metre.

a 4.6 m b 34.67 m c 0.678 m d 32.89 m

4 Solve each of the following equations to find x.

a 3x 6 b 4x 12 c 5x 60 d 8x 48

e f g h

i j k l

5 Solve each of the following equations to find x correct to two decimal places.

a b c d

6 Solve each of the following equations to find x correct to one decimal place.

a b c d

e f g h

7 Find the value of each pronumeral.

a b c

d e f

Answers1 a 0.4568 b 0.3457 c 0.0456 d 0.2800 2 a 4.23 b 5.68 c 76.90 d 23.90 3 a 5 b 35 c 1 d 334 a 2 b 3 c 12 d 6 e 12 f 30 g 28 h 182 i 9 j k 12 l 8 5 a 15.04 b 15.60 c 1.38 d 6.30 6 a 0.6 b 0.6c 2.1 d 0.5 e 0.6 f 2.0 g 9.1 h 7.1 7 a 32 b 18 c 152 d 60 e 20 f 50

52

80

x70 x

30

x

x28

x

18x

32

2.456x

0.34529.34

x 3.24

17x

8.43.8x

6.9

14x

2732x

154x

73x

5

x

1.235 5.1

x

0.3456 4

x

2.1 7.43

x

3.2 4.7

3x4

62x3

86x5

34x3

12

x

13 14

x

7 4

x

5 6

x

3 4

CUAT029-C10[352-395].qxd 7/3/08 1:01 PM Page 354 ppg SATHYA:CUP-Australia-1:CUP-AUS:Reprint:VELS 09: Aptara (PPG-Quark)

10.1 Introducing trigonometryTrigonometry deals with the relationship betweenthe sides and the angles of triangles. Trigonometryenables us to calculate lengths and angles whichmay be difficult or impossible to measure directly.Trigonometry is used in the fields of science,engineering, surveying, astronomy, navigation andarchitecture.

When using trigonometry, it is important toname the sides of a right-angled triangle correctly.If the symbol is used to represent one angle,then the other sides can be named according towhether they are opposite or adjacent to that angle.

a b c

adjace

nt side opposite sidehypotenuse

adja

cent

sid

e

opposite side

hypotenuse

adjacent side

oppo

site

sid

e

hypo

tenus

e

Key ideas

The three trigonometric ratios are defined as:

sine of angle (or sin )

cosine of angle (or cos )

tangent of angle (or tan )

In summary:Label each side of the triangle with O (opposite side), A (adjacent side) and H (hypotenuse).

Decide which two sides are involved in the problem by using:

SOH CAH TOA

tan oppositeadjacent

cos adjacent

hypotenusesin

oppositehypotenuse

length of the opposite side

length of the adjacent

length of the adjacent sidelength of the hypotenuse

length of the opposite sidelength of the hypotenuse

For right-angled triangles, the basic trigonometric ratios are called sine, cosine and tangentand these are derived from the unit circle (a circle with radius one unit) which will bediscussed in greater depth in Year 10 mathematics.

A

HO

Chapter 10 Trigonometry 355

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 355

Example 1

ExplanationSolution

Copy this triangle and label the sides as opposite to (O), adjacent to (A) or hypotenuse (H).

Draw the triangle and label the sides as hypotenuse (H),opposite (O) and adjacent (A).

Example 2

ExplanationSolution

Write trigonometric ratios (in fraction form) for each of the following triangles.

a b c

5

3

4

9

7

5

a

b

c tan OA

35

sin OH

49

cos AH

57

Side length 7 is the longestside so it is (H). Side length5 is adjacent to angle so it is (A).

O

A

H

Side length 9 is the longestside so it is (H). Side length4 is opposite angle so it is (O).

Side length 5 is the adjacent side to angle soit is (A). Side length 3 isopposite angle so it is (O).

3 (O)

(H)

(A) 5

9 (H)

4 (O)

(A)

5 (A)

(H) 7(O)


CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 356

357Chapter 10 Trigonometry

10AExercise

1Example 1 Copy each of these triangles and label the sides as opposite to (O), adjacent to (A)or hypotenuse (H).

a b c d

e f g

2 For the triangle shown, state which number corresponds to:

a the hypotenuseb the side opposite angle c the side opposite angle d the side adjacent to angle e the side adjacent to angle

3 Write a trigonometric ratio (in fraction form) for each of the following triangles andsimplify where possible.

a b c d

e f g h

4 Copy each of these triangles and mark the angle that will enable you to write a ratiofor sin .

a b c d

2Example

4

3

5

35

2 4

6

7 n

m

33

4a

5a

2x

4y

3x

2y

H O HOHO

H

O

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 357


5 For each of these triangles, write a ratio (in fraction form) for sin , cos and tan .

a b c

6 For the triangle shown on the right, write a ratio (in fraction form) for:

a sin b sin c cos d tan e cos f tan

7 This triangle has angles 90, 60 and 30 and side lengths 1, 2 and

a Write a ratio for:i sin 30 ii cos 30 iii tan 30iv sin 60 v cos 60 vi tan 60

b What do you notice about the following pairs of ratios?i cos 30 and sin 60 ii sin 30 and cos 60

8 a Measure all the side lengths of this triangle to the nearestmillimetre.

b Use your measurements from part a to find an approximate ratiofor:i cos 40 ii sin 40 iii tan 40iv sin 50 v tan 50 vi cos 50

c Do you notice anything about the trigonometric ratios for 40 and 50?

9 For each of the following:

i Use Pythagoras theorem to find the unknown side.ii Find sin , cos and tan .a b c d

10 a Draw a right-angled triangle and mark one of the angles as . Then mark in thelength of the opposite side as 15 units and the length of the hypotenuse as 17 units.

b Find the length of the adjacent side using Pythagoras theorem.c Determine the ratios for sin , cos and tan .

11 Triangle ABC has a right angle at B and angle C is . Distance AB is 4 cm anddistance AC is 5 cm.

a Draw the triangle. b Find distance BC.c Write the ratios for sin , cos and tan

8

6

9

12

7

24

3

4

23.

53

4

24

10

26

5

13

12

810

6

30

6012

3

40

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 358


Enrichment

12 a Given that is acute and cos find sin and tan . Hint: Use Pythagoras

theorem.

b For each of the following draw a right-angled triangle then use it to find theother trigonometric ratios.

i sin ii cos iii tan 1

13 For triangle ABC on the right, find:

a distance ACb a ratio for sin , cos and tan c a ratio for sin , cos and tan

14 Investigate the Pythagorean Identity for trigonometry and test it by:

a choosing a ratio for sin b calculating cos using Pythagoras theorem andc testing the identity.Describe what an identity is and research other identities.

12

12

45

,

C

A B6

10

Th

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 359

CAS calculatorGraphics calculatorScientific calculator

Using technology to determine trigonometric ratios


It is difficult to determine trigonometric ratios accurately just by measuring the sides and anglesof a triangle. A scientific, graphics or CAS calculator can be used to obtain the accurate values.Before entering angles you need to make sure that the calculator is in degree mode.Example: Use a calculator to find the value of each of the following, correct to four decimal places.a cos 30 b cos 54 c tan 89

360

Exercise1 Use a calculator to find the value of each of the following, correct to four decimal places.

a sin 10 b tan 30 c cos 40 d tan 60 e tan 80 f cos 90g tan 10 h sin 70 i cos 60 j sin 40 k cos 80 l cos 50

2 Use a calculator to find the value of each of the following, correct to three decimal places.a sin 12 b tan 34 c cos 44 d tan 69 e tan 82 f cos 88g tan 14 h sin 72 i cos 68 j sin 64 k cos 86 l cos 12

a Press sin 30. Press SIN, type 30) and Press 2nd SIN, type 30) and press press . .

b Press cos 54. Press COS, type 54) and Press 2nd COS, type 54) and press . press . Use to get the

decimal approximation.

c Press tan 89. Press TAN, type 89) and Press 2nd TAN, type 89) and press . press to get the decimal

approximation.

This gives the answer 0.5 or .12

This gives the answer which rounds up to 0.5878.0.58778

This gives the answer 57.28996 which rounds up to 57.2900.

TI-nspire

APPE NDI

X

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 360

10.2 Finding the side length of a right-angled triangle


There are many situations wherewe need to calculate the side lengthof a right-angled triangle, forexample the height of a building orthe width of a river.

Example 3

ExplanationSolution

Find x in the equation correct to two decimal places.cos 20 x

3,

2.82 x 3 cos 20

cos 20 x

3Multiply both sides of the equation by 3.Evaluate.

Example 4

For each triangle find the value of x correct to two decimal places.

a b c

10

24

x

442

x7

38

x

Key ideas

If the size of the angle and the length of one side of a right-angled triangleare given, the length of any other side can be found using SOH CAH TOA.For example, for the diagram shown:

or cos 30 y

6sin 30

x

6

6

30x

y

CUAT029-C10[352-395].qxd 6/9/06 2:56 PM Page 361

4Example


10BExercise

3Example 1 In each of the following find the value of x correct to two decimal places.

a b c

d e f

g h i

2 For the triangles given below, find the value of x correct to two decimal places.a b c d

e f g h

x

8 sin 27.3tan 11.4

x

2x

4 cos 8.7

cos 34 x

14x

9 sin 16tan 39

x

11

tan 87 x

5cos 43

x

7sin 20

x

3

ExplanationSolution

a

b

x 4 tan 42 3.60

c

x 10 cos 24 9.14

cos 24 x

10

cos 24 AH

tan 42 x

4

tan 42 OA

4.31 x 7 sin 38

sin 38 x

7

sin 38OH

Since (O) and the (H) are given,the sin ratio must be used.

Multiply both sides by 7.Evaluate.

Since (O) and the (A) are given, thetan ratio must be used.


Since (A) and the (H) are given,the cos ratio must be used.


7 (H)

(A)38

(O) x

(H)

42

x (O)

4 (A)

1x

23 2 x

43

4

x

175

x

18

x

42

8 x

3512

x42

20

x25

32

(O)

x (A)

10 (H)

24


i j k l

m n o p

3 Determine the height of each of these triangles correct to two decimal places.a b c

4 Andrew walks 3.2 km up a hill which is inclined at 12 to thehorizontal. How high (correct to two decimal places) is heabove his starting point?

5 Leonie wanted to measure the width of a river. Sheplaced two markers, A and B, 72 m apart along thebank. C is a point directly opposite marker B. Leoniemeasured angle CAB to be 32. Find the width of theriver correct to two decimal places.

6 One end of a 12.2-m rope is tied to a boat. The other endis tied to an anchor, which is holding the boat steady inthe water. If the anchor is making an angle of 34 withthe vertical, how deep is the water correct to two decimalplaces?

7 An escalator is 11 m long and slopes at an angle of 42to the horizontal. How high up (correct to two decimalplaces) is a shopper who is at the top of the escalator?


x

4021

x

19

5.8

x

222.5

x

342.4

15 m

40

12.5 m

23

325.2 m

x

436.2

x

1634

x

30

17

x

63 45

12

3.2 km

11 m

42

3412.2 m

C

A B32

width

72 m

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 363


Enrichment

8 An isosceles triangle has a base length of 24 cm and base angles of 42.

a Find the height correct to two decimal places.b Use Pythagoras theorem to find the value

of x correct to one decimal place.c Find the perimeter and area of the large

triangle correct to the nearest cm.

9 In this diagram you should see three right-angled triangles.

a Find the length BC correct to two decimal places.b Find the length AD correct to two decimal places.c Find length AC correct to one decimal place.d Investigate how changing the length DC changes

the answers to parts a to c above.

60

2

A CB

D

24 cm

42

x

Th

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 364


10.3 Further problems involving side lengths

Key ideas

If the unknown value of a trigonometric ratio is in the denominator you need to rearrange theequation to make the pronumeral the subject.

For example, for the triangle shown:

which gives x 5

cos 30

cos 30 5

x

When finding the hypotenuse length or other side length of a triangle, the unknown valuesometimes appears in the denominator of the equation.

305

x

Example 5

ExplanationSolution

Find x in the equation correct to two decimal places.cos 35 2x

,

2.44

x 2

cos 35

x cos 35 2

cos 35 2x

Multiply both sides of the equation by x.

Divide both sides of the equation by cos 35.

Evaluate and round off to two decimal places.

Example 6

Find the values of the pronumerals correct to two decimal places.a b

x

35

5

x

y

2819

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 365


ExplanationSolution

a

b

y 40.47 y 21637.63 1637.63 y 2 x2 192

35.73

x 19

tan 28

x tan 28 19

tan 28 19x

tan 28 OA

8.72

x 5

sin 35

x sin 35 5

sin 35 5x

sin 35 OH

Since (O) and the (A) are given, use sin .Multiply both sides of the equation by x.

Divide both sides of the equation by sin 35.


Since (O) and (A) aregiven use tan .

Multiply both sides of the equation by x.

Divide both sides of the equation by tan 28.


Find y by using Pythagoras theorem andsubstitute the exact value of x stored in yourcalculator.

Alternatively, y can be found by using sin .

x (A)

19 (O)(H) y

28

x (H)

35

(O) 5

(A)

10CExercise

1 For each of the following equations find the value of x correct to two decimal places.

a b c

d e f

g h i

2 Find the value of x correct to two decimal places.

a b c d

5.2x

sin 54cos 37 4.7x

32x

tan 49

cos 88 3x

2x

tan 67sin 73 4x

tan 11 5x

sin 27 3x

cos 35 2x

x9

42

x8

29

x45 39 x

14

28

x (A)

19 (O)(H) y

28

5Example

6aExample

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 366


e f g h

i j k l 3

3 Find the value of each pronumeral correct to one decimal place.

a b c d

e f g h

4 A ladder is inclined at an angle of 32 to thehorizontal. If the ladder reaches 9.2 m up a wall,what is the length of the ladder, correct to onedecimal place?

5 A kite is flying at a height of 27 m. If the string isinclined at 42 to the horizontal, find the length of thestring correct to the nearest metre.

6 A glider flying at a height of 800 m descends at anangle of 12 to the horizontal. How far (to thenearest metre) has it travelled in descending to theground?

7 A 100-m mine shaft is dug at an angle of 15 to the horizontal. How far (to the nearest metre):

a below ground level is the end of the shaft?b is the end of the shaft horizontally from the

opening?

yx

2314.2

12

12.1

n

myx27

9.6y

x

8.342

yx

8

40ba

4

30b

a

7

43

x

y

4

32

42

27 m

distance in descent

800 m

12

ground

100 m

15

x14

26

x

21

44x

2653

x

649

x

15

44 x

2647

x2528 x

7

8

6bExample

CUAT029-C10[352-395].qxd 6/5/06 9:52 PM Page 367


Enrichment

9 Lena wants to find the height of a tree. From a distance of 25 metres from the base of the tree she measures the angle shown as 32.

a Given that Lena is 1.64 metres tall find the height of the tree correct to two decimal places.

b What effect would it have on the calculated height of the tree if the horizontal distance (25 metres) was overestimated or underestimated by 50 cm?

c What effect would it have on the calculated height of the tree if the angle wasoverestimated or underestimated by 0.5?

d What do you notice?

10 a For the diagram shown find the value of x.b What is the value of x if the given side length is

i doubled?ii halved?Describe any pattern you observe.

c What is the value of x if the side length is left unchanged but the angle is:i doubled?ii halved?Describe any pattern you observe.

32

30

2x

8 In the shed shown on the right, how long will one of thesloping timber beams be if they are each inclined at anangle of 24 to the horizontal? Give your answer correct totwo decimal places.

24

75 m

Th

CUAT029-C10[352-395].qxd 6/9/06 2:59 PM Page 368

369Chapter 10 Trigonometry 369369

10.4 Finding the angle Sometimes it is useful to know the angles in aright-angled triangle. For example, you mayneed to know the angle a wire makes with avertical pole or the bearing a plane might betravelling on.

Example 7

ExplanationSolution

Find the value of to the level of accuracy indicated.

a sin 0.3907 (nearest degree) b tan (one decimal place)12

a sin 0.3907 0.3907

23sin1 Use the sin1 key on your calculator.

Round off to the nearest whole number.

Key ideas

Given two side lengths of a right-angled triangle you can find an anglewithin the triangle. 3 2

If you know which trigonometric ratio is relevant you can use one of the keys below to work outthe the angle.

To find the angle if sin 0.5446:

on a scientific on a graphics on a CAS calculator,calculator, press: calculator, press: press:

sin1 0.5446 2nd sin1 (0.5446) ENTER sin1 (0.5446) ENTER

cos1 sin1 tan1

CUAT029-C10[352-395].qxd 6/5/06 9:53 PM Page 369


Example 8

ExplanationSolution

Find the value of to the nearest degree.

37

sin1 a 610

b

6

10

sin OH

Since (O) and the (H) are given, use sin .

Use the key on your calculator.

Round off to the nearest degree.

sin1

106

6 (O)

(H) 10

(A)

10DExercise

1 Find the value of to the nearest degree.

a sin 0.5 b cos 0.5 c tan 1d tan 0.5774 e sin 0.7071 f sin 0.8660g cos 1 h tan 0.8397 i cos 0j tan 0.8391 k sin 1 l cos 0.3420m cos 0.9948 n tan 0.1763 o cos 0.4540

2 Evaluate each of the following to the nearest degree.

a sin1 (0.6884) b cos1 (0.9763) c tan1 (0.8541)d sin1 (0.4305) e tan1 (1.126) f cos1 (0.997)g cos1 (0.1971) h sin1 (0.1817) i sin1 (0.7051)

7aExample

b

26.6

tan1 a 12b

tan 12

Use the tan1 key on your calculator.Dont forget to close the bracketsRound off to one decimal place.

CUAT029-C10[352-395].qxd 6/5/06 9:53 PM Page 370

371Chapter 10 Trigonometry 371

3 Find the angle correct to two decimal places.

a sin b sin c sin

d cos e cos f cos

g tan h tan i tan 10

4 a The sine of angle is 0.9205. What is the value of angle to the nearest degree?b The cosine of angle is 0.6235. What is the value of angle to the nearest

degree?

5 Which trigonometric ratio should be used to solve for ?

a b c d

6 Find the value of to the nearest degree.

a b c d

e f g h

7 A road rises at a grade of 3 in 10. Find the angle (to the nearest degree) the road makes with the horizontal.

8 A ramp is 6 m long and 2 m high. Find the angle (correct to two decimal places) the ramp makes with the ground.

9 When a 2.8-m long seesaw is at its maximumheight it is 1.1 m off the ground. What angle(correct to two decimal places) does the seesawmake with the ground?

24

18

32

24

7

11

6 20

43

12

12

5

29 22

26 19

1526

1421

7 m 5 m

89 m

53

23

79

45

12

78

17

35

3

10

2 m

6 m

8Example

7bExample

CUAT029-C10[352-395].qxd 6/5/06 9:53 PM Page 371

10 Adam, who is 1.8 m tall, holds up a plank of wood 4.2 m long. Find the angle that the plank makes with the ground correct one decimal place.

11 A childrens slide has a length of 5.8 m. The vertical ladder is 2.6 m above the ground.Find the angle the slide makes with the ground correct to one decimal place.

12 The leaning tower of Pisa was 54.6 m tall when itwas built in the 12th century. Today the tower isleaning over and it is about 440 cm out of line atthe top. Find its inclination to the vertical correctto two decimal places.

Enrichment

13 a If sin x cos x and x is acute what is the value of angle x?b For each of the following equations find two different values for

if .

i sin ii cos iii tan 1

c Investigate angles larger than 180 which satisfy the equations in part b.

14 Solve each triangle, that is, find the length of all sides and the value of all anglescorrect to one decimal place.

a b12.2

7.4

7.3

6.2

12

12

0 180


1.8 mplank (4.2 m)

Th

CUAT029-C10[352-395].qxd 6/5/06 9:53 PM Page 372


Some problems may involve finding more than one length or angle. You may, for example,want to know the length and height of an escalator as well as the angle it makes with theground.

10.5 Mixed application problems

Key ideas

To solve application problems involving trigonometry:

Draw a diagram and label the key information.

Identify and draw the appropriate right-angled triangles separately.

Solve using trigonometry to find the missing measurements.

Express your answer in words.

Example 9

ExplanationSolution

A flagpole is supported by a wire running from thetop of the pole to a point on the ground 6.2 m fromthe base of the pole. If the wire makes an angle of 36with the ground, find the height of the pole correct toone decimal place.

Let the height of the flagpole be h metres.

4.5So the height of the flagpole is 4.5 m.

h 6.2 tan 36

tan 36 h

6.2

tan 36 OA

(H)

(A)

(O) h m

6.2 m

36

Define the unknown length.

Draw a diagram.

Since the opposite side (O) and theadjacent side (A) are given, use tan .

Multiply both sides by 6.2.Evaluate.Express the answer in words.

CUAT029-C10[352-395].qxd 6/5/06 9:53 PM Page 373


Example 10

ExplanationSolution

A plane flying at 1500 m starts to climb at an angle of 15 to the horizontal when the pilot sees a mountain peak 2120 m high, 2400 m away from him in a horizontal direction. Will the pilot clear the mountain?

From triangle MNP:

MN 2400 tan 15 643

Since m the plane will clearthe mountain peak.

MN 620

tan 15 MN

2400

Draw a diagram.

The plane will clear the mountain if

That is: m

Since (O) and (A) are given use tan .

Multiply by 2400.Round off to the nearest metre.Write the answer in words.It will be (643 620) 23 metres above themountain peak when it reaches it.

MN 620MN (2120 1500) m

10EExercise

1 A flagpole is supported by a wire running from the top of the pole to a point on theground 4.6 m from the base of the pole. If the wire makes an angle of 28 with theground, find the height of the pole correct to two decimal places.

2 A large advertising balloon is tied to the roof of a 20-mhigh building by a 50-m rope which makes an angle of42 with the horizontal. Find the height of the balloonabove the ground correct to two decimal places.

3 A ramp for wheelchairs and prams runs from street levelto the entrance of a building, which is 0.8 m above streetlevel. How long (correct to two decimal places) is theramp if it makes an angle of 10 with the horizontal?

1500 m

2400 m

M

PO

2120 m

1500 m

2400 m

N

O

P

H

A

M

15

9Example

CUAT029-C10[352-395].qxd 6/6/06 6:03 AM Page 374


4 A train travels up a slope, making an angle of 7with the horizontal. When the train is at a heightof 3 m above its starting point, find the distance ithas travelled up the slope, to the nearest metre.

5 Toro the human cannon ball is catapulted into theair at an angle of 70 to the horizontal. Whatdistance (to the nearest metre) will he havetravelled when he has reached a height of 30 m?

6 A pendulum on a grandfather clock is 70 cm longand swings through a total angle of 12. What isthe straight-line distance (to the nearest cm)between the extreme positions of the bob on theend of the pendulum as it swings?

7 A ski lift travelling up a mountain is inclined at 15 to the horizontal. If the ski lift is560 m long, how high (to the nearest metre) is the top of the ski lift vertically from thefoot of the mountain?

8 Madeline tries to swim across a 40-m-wide river. The current pushes her off course atan angle of 26 to her direct route across the river. How far (to the nearest metre) doesshe actually swim to reach the other side?

9 A ship out at sea observes a lighthouse on top of an 82-m cliff. If the ship is 180 mfrom the base of the cliff find the value of the observation angle from horizontal to thenearest degree.

10 A removalist van has a ramp, which is used to move furniture from ground level toinside the van. If the floor of the van is 1.2 m off the ground and the ramp is 2.4 m inlength what angle (to the nearest degree) does the ramp make with the ground?

11 An escalator rises 3 metres for every 7 metres horizontally. Give your answer for eachof the following correct to one decimal place.

a What angle does the escalator make with the horizontal ground?b If the total height of the escalator is 6 m, how long is the escalator?

12 A road has a steady gradient of 1 in 10.

a What angle does the road make with the horizontal? Give your answer to the nearest degree.

b A car starts from the bottom of the inclined road and drives 2 km along the road.How high vertically is the car? Give your answer correct to the nearest metre anduse your answer from part a.

13 A plane flying at 1850 m starts toclimb at an angle of 18 to thehorizontal when the pilot sees amountain peak 2450 m high, 2600 maway from him in a horizontaldirection. Will the pilot clear themountain?

10Example

1850 m

2600 m P

2450 m


14 A house is to be built using the design shown on the right. The eaves are 600 mm and the house is 7200 mm wide, excluding the eaves. Calculate the length (to the nearest mm) of a sloping edge of the roof, which is pitched at 25 to the horizontal.

15 A garage is to be built with measurements as shown in the diagram on the right. Calculate the sloping length and pitch of the roof if the eaves extend 500 mm on each side. Give your answers correct to the nearest unit.

16 A roof has a horizontal span of 10.2 m and a pitch of 24. Find the height of the roof, correct to the nearest millimetre.

17 A kite is 160 cm long and 75 cm wide. The shorter edges each make an angle of 60 with the horizontal, as shown in the diagram on the right. Find the value of x and y correct to the nearest cm.

18 The chains on a swing are 3.2 m long and the seat is 0.5 m off the ground when it is inthe vertical position. When the swing is pulled as far back as possible, the chains makean angle of 40 with the vertical. How high off the ground, to the nearest cm, is theseat when it is at this extreme position?

Enrichment

19 An aeroplane takes off and climbs at an angle of 20 to the horizontal, at 190 km/halong its flight path for 15 minutes.

a Find correct to two decimal places:i the distance the aeroplane travels ii the height the aeroplane reaches

b If the angle at which the plane climbs is twice the given angle but its speed ishalved, will it reach a greater height after 15 minutes?

c If the plane speed is doubled and its climbing angle is halved, will the planereach a greater height after 15 minutes?

20 The residents of Skeville live 12 km from an airport. They maintain that any planeflying lower than 4 km disturbs their peace. Each Sunday they have an outdoorconcert from 12.00 noon till 2.00 pm.

a Will a plane taking off from the airport at an angle of 15 over Skeville disturb the residents?

b When the plane in part a is directly above Skeville, how far (to the nearest m) has itflown?

c If the plane leaves the airport at 11.50 am on Sunday and travels at an average speed of 180 km/h, will it disturb the start of the concert?

d Investigate what average speed the plane can travel at in order not to disturb the concert.

7200 mm

600 mm600 mm

2700 mm1820 mm

3200 mm

x cm

y cm75 cm

160 cm

60


Th

CUAT029-C10[352-395].qxd 6/5/06 9:53 PM Page 376


10.6 Angles of elevation and depressionAngles of elevation and depression are measured from the horizontal, for example the angleto the top of a building from ground level or the angle from the top of a building down tothe ground.

Key ideas

The angle of elevation or depression of a point, Q, from another point, P, is given by the angle theline PQ makes with the horizontal.

Angles of elevation or depression are always measured from the horizontal.

It is important to note that the angle of elevation of Qfrom P is equal to the angle of depression of P from Q,because they are alternate angles.

Q

P horizontal

angle of elevation

line o

f sigh

t

Q

P horizontal

angle of depression

line of sight

Q

P

Example 11

ExplanationSolution

The angle of elevation of the top of a towerfrom a point on the ground 30 m away fromthe base of the tower is 28. Find the heightof the tower to the nearest metre.

Let the height of the tower be h m.

h 30 tan 28 16

The height is 16 m.

h

30

tan 28 OA

Since (O) and (A) are given, use

Multiply both sides by 30.Simplify to the nearest metre.Write the answer in words.

(A)

(O)h m

30 m28

(H)

tan .

angle of elevation

h m

30 m28

line o

f sigh

t

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 377


Example 12

ExplanationSolution

From the top of a vertical cliff Andrea spots a boat out at sea. If the top of the cliff is 42 mabove sea level and the boat is 90 m away from the base of the cliff, find Andreas angle ofdepression to the nearest degree.

The angle of depression is 25.

25

4290

tan OA

Draw a diagram and label all the givenmeasurements.

Since (O) and (A) are given, use tan .

Use the key on your calculator andround off to the nearest degree.

Express the answer in words.

tan1

42 m

90 m

10FExercise

1 The angle of elevation of the top of a tower from a point on theground 40 m away from the base of the tower is 36. Find theheight of the tower to the nearest metre.

2 The angle of elevation from the point on the ground to the top ofa building is 65. If the horizontal distance to the building is84 m find the height of the building to the nearest metre.

3 Tran is 34 m away from a tree and the angle of elevation of the top of the tree from theground is 53. What is the height of the tree to the nearest metre?

4 The angle of elevation of the top of a castle wall from a point on the ground 25 maway from the base of the castle wall is 32. Find the height of the castle wall to thenearest metre.

5 From a point on the ground Emma measures the angle ofelevation of an 80-m tower to be 17. Find how far fromthe base of the tower Emma is, to the nearest metre.

36 40 m

h mline of

sight

angle of elevation

6584 m

17

80 m

11Example

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 378

6 From a pedestrian overpass Terry spots a landmark at an angle of depression of 32. Howfar away (to the nearest metre) is the landmark from the base of the 24-m-high overpass?

7 From a lookout tower David spots a bushfire at an angle of depression of 25. If thelookout tower is 42 m high, how far away (to the nearest metre) is the bushfire fromthe base of the tower?

8 Angela is 1.5 m tall. How long (correct to one decimal place) is her shadow when theangle of elevation of the Sun is 58?

9 From the top of a vertical cliff Bruce spots a swimmer out at sea. If the top of the cliffis 38 m above sea level and the swimmer is 50 m away from the base of the cliff, findBruces angle of depression to the nearest degree.

10 From the top of a viewing platform 20 m high a wombat is spotted in the bush belowat a horizontal distance of 15 m. Find the angle of depression from the viewingplatform to the wombat to the nearest degree.

11 From a ship a person is spotted floating in the sea 200 m away. If the viewing positionon the ship is 20 m above sea level find the angle of depression from the ship toperson in the sea to the nearest degree.

12 A power line is stretched from a pole to the top of a house. The house is 4.1 m highand the power pole is 6.2 m high. The horizontal distance between the house and thepower pole is 12 m. Find the angle of elevation of the top of the power pole from thetop of the house to the nearest degree.

Enrichment

13 Chau observes a plane flying directly overhead at a height of 820 m. Twenty secondslater, the angle of elevation of the plane from Chau is 32.

a How far (to the nearest metre) did the plane fly in 20 seconds?b What is the planes speed in km/h correct to two decimal places?

14 a Bertha observes a stationary hot air balloonhovering at a height of 120 m at an angle ofelevation of 24 measured from her eye level.Her eyes are 1.5 m above the ground.

i How far away (to the nearest metre) is theballoon along her line of vision?

ii How far does Bertha need to walk to bedirectly underneath the balloon?

b Bertha starts walking towards the balloon. After20 seconds she stops and looks up at the balloon.The balloons angle of elevation is now 32.i How far has Bertha walked during this time?ii How much further does Bertha need to

walk to be directly under the balloon?Give all answers correct to the nearest unit.

12Example


Th

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 379


A compass can be used to show the direction in whichsomeone may wish to travel or the bearing of one objectfrom another.

Bearings can be expressed as:surveyors bearingstrue bearings

10.7 Bearings

Key ideas

SSuurrvveeyyoorrss bbeeaarriinnggss are based on the compass directions north, south,east and west. We usually start at south or north, and move east or west.Each bearing is described as a number of degrees east or west fromnorth or south. e.g. S25 W or N30 E

TTrruuee bbeeaarriinnggss describe the angle in a clockwise direction from north.They are written using three digits, for example 008, 032 or 144.

To describe the true bearing of an object positioned at A from anobject positioned at O, we need to start at O, face north then turnclockwise through the required angle to face the object at A.

40

N

E

S

W

N 40 E

N

E

S

W

360 true 000 true

090 true120 T

180 true

270 true

N

N

bearing ofA from O

bearing ofO from A

O

A

Example 13

ExplanationSolution

For the following diagram write:

a the surveyors bearingb the true bearing

a S42E Start from south andturn 42 towards east

42

48

42

N

E

S

W

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 380


Example 14

ExplanationSolution

For the diagram shown write

a the true bearing of A from Ob the true bearing of O from A

a The bearing of A from O is 120 T.b

Therefore the bearing of O from A is:(360 60) T 300 T

N

S

EAW

N

120

3060

60

300

S

EOW

Start at O, face north and turn clockwiseuntil you are facing A.

Start at A, face north and turn clockwiseuntil you are facing O.

N

120

S

E

A

OW

Example 15

ExplanationSolution

A bushwalker walks 3 km on a true bearing of 060 T from point A topoint B. Find how far (correct to one decimal place) east point B isfrom point A.

Let the distance travelled towards the eastbe d km.

d 3 cos 30 2.6

The distance east is 2.6 km.

cos 30 d

3

30

3 km

d km

Define the distance required.

Draw a triangle.

Since the adjacent side (A) and thehypotenuse (H) are given use cos .Multiply both sides of the equation by 3.Evaluate and round off to one decimal place.Express the answer in words.

N

60

S

E3 km

B

AW

b 180 42 138 Angle is 138 T 138 T

180 42 = 13842

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 381


Example 16

ExplanationSolution

A fishing boat starts from point O and sails75 km on a bearing of 160 to point B.

a How far east (to the nearest metre) ofits starting point is the boat?

b What is the bearing of O from B?

a Let the distance travelled towards theeast be d km.

d 75 sin 20 26

The boat has travelled 26 km to the eastof its starting point.

b The bearing of O from B is(360 20)T 340 T

sin 20 d

75

Define the distance required.

Draw a diagram and label all the givenmeasurements.Since (O) and (H) are given use sin .

Multiply both sides of the equation by 75.Evaluate and round off to the nearest metre.

Write the answer in words.

Start at B, face north then turn clockwise to face O.

E

B

W

N

160

S 75 km

O

E

B

W

N

160

70

20

340

20S

75 km

EW

N

S

O

d km

10GExercise

1 For each of the following diagrams write:

i the surveyors bearingii the true bearinga b c d

2 For each diagram shown write:

i the true bearing of A from O ii the true bearing of O from A.a b c d

18

O

AEW

N

S30

O

A

EW

N

S38

O

A

EW

N

S

40

O

A

EW

N

S

52

EW

N

S

20EW

N

S

35EW

N

S

32EW

N

S

13Example

14Example

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 382


3 A bushwalker walks 4 km on a true bearing of 55 from point A topoint B. Find how far east point B is from point A correct to twodecimal places.

4 A yacht sails 80 km on a true bearing of 048. Find how fareast of its starting point the yacht is correct to two decimalplaces.

5 After walking due east, then turning and walking due south, a hiker is 4 km S32Efrom her starting point. Find how far she walked in a southerly direction correct to onedecimal place.

6 A four-wheel-drive vehicle travels for 32 km on a true bearing of 200. How far west(to the nearest km) of its starting point is it?

7 A fishing boat starts from point O and sails 60 km on a true bearingof 140 to point B.

a How far east of its starting point is the boat, to the nearestkilometre?

b What is the bearing of O from B to the nearest degree?

8 Two towns, A and B, are 12 km apart. The true bearing of B from Ais 250.

a How far is B west of A correct to the nearest km?b Find the bearing of A from B to the nearest degree.

9 A cyclist starts from O and rides 7.6 km on a bearing of N20W to point A.

a How far (to the nearest metre) west has she travelled from her starting point?b What is the bearing of O from A correct to one decimal place?

N

S

W E

324 km

N

S

W E

4880 km

15Example

16Example

A

BN

W E

S

554 km

S

W E

B

N

140

60 km

O

S

12 km

250AW E

B

N

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 383


10 A submarine travels 720 km on a bearing of 130, then travels 40 km due east.

a How far east (to the nearest km) of its starting point is the submarine?b Find how far south (to the nearest km) of its starting point is the submarine.

11 A helicopter flies on a bearing of S40E for 210 km, then flies due east for 175 km.How far east (to the nearest km) has the helicopter travelled from its starting point?

12 Christopher walks 5 km south, then walks N36 E until he is due east of his startingpoint. How far is he from his starting point to the nearest kilometre?

13 Two cyclists leave from the same startingpoint. One cyclist travels due west while theother travels on a bearing of S22W. Aftertravelling for 18 km, the second cyclist is duesouth of the first cyclist. How far (to thenearest metre) has the first cyclist travelled?

Enrichment

14 A plane flies on a bearing of 168 for twohours at an average speed of 310 km/h.How far (to the nearest kilometre):a has the plane travelled?b to the south of its starting point has

the plane travelled?c to the east of its starting point has

the plane travelled?

15 A pilot intends to fly directly to Anderly which is 240 km due north of his starting point.The trip usually takes 50 minutes. Due to a storm, the pilot changes course and flies ona true bearing of 320 for 150 km, at an average speed of 180 km/h, to Boxleigh.

a Find to the nearest kilometre how fari north the plane has travelled from its starting pointii west the plane has travelled from its starting point

b How many kilometres is the plane off course?c From Boxleigh the pilot flies directly to Anderly at 240 km/h.

i Compared to the usual route, how many extra kilometres has the pilottravelled in reaching Anderly?

ii Compared to the usual trip, how many extra minutes did the trip to Anderlytake?

Th

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 384


Problems involving the solution of two triangles arise in various situations such as viewinga tower from two different positions.

10.8 Problems involving two trianglesKey ideas

In some situations it may be necessary to find missing values ontwo triangles to obtain the answer we are looking for.

For example, to find the value of y in triangle ABD shown onthe right,you need to:1 find x in triangle BCD2 find y in triangle ABD

D

A B C

xy

35 36

10

Example 17

ExplanationSolution

Find the values of the pronumerals in the diagram shown,correct to two decimal places.

Draw triangle BCD; find x.Sides (O) and (A) are given so use tan .

Multiply both sides by 6.Evaluate and round to two decimal places.

Draw triangle ABD (use the exact value of xstored in your calculator).Sides (O) and (H) are given so use sin .

Multiply both sides by y.

Divide both sides by sin 40.

Evaluate using the stored value of x and round to two decimal places.

D

A B C

xy

40 32

6

D

x

32B C6

D

40B

yx

A

5.83

y x

sin 40

y sin 40 x

sin 40 xy

3.75 x 6 tan 32

tan 32 x

6

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 385


Example 18

ExplanationSolution

A ship (at P) is 9 km due east of a lighthouse (L). Thecaptain takes bearings from two landmarks, M and Q, whichare due north and due south of the lighthouse respectively.The bearings of the M and Q from the ship are N44W andS73W respectively. How far apart (correct to two decimalplaces) are the two landmarks?

ML 9 tan 46 9.320 km

LQ 9 tan 17 2.752 km

ML LQ 9.320 2.752 12.07 km

The distance between the two landmarksis 12.07 km.

tan 17 LQ

9

tan 46 ML

9

Distance between M and Q ML LQDraw triangle LMP.Sides (O) and (A) are given so use tan .

Multiply both sides by 9.Calculate and round to three decimal places orstore the exact answer in your calculator.

Draw triangle LPQ.Sides (O) and (A) are given so use tan .

Multiply both sides by 9.Calculate and round to three decimal places.Calculate the distance between M and Q andround off to two decimal places.Answer in words.

N

M

L

Q73

17

44

9 km

S

EPW

M

L P9 km46

L

Q

P

17

9 km

10HExercise

1 Find the values of the pronumerals in each diagram, correct to two decimal places.

a b c

2 Find the length of AC and of BC in each diagram, correct to two decimal places.

a b c 18

A

B DC

40

14

2050

A

D C

12 cm

27 35

38

C

B

A

D

4236

12

D

y

x

A B C1722

16

A

yx

B D C

35 28

8

A

C DB

yx

17Example

CUAT029-C10[352-395].qxd 6/8/06 9:55 AM Page 386


3 A flagpole is secured by ropes as shown in the diagramon the right. Find correct to one decimal place the

a height of the flagpoleb distance (BC) from the base of the flagpole to the

base of the rope on the right

4 An engineer made themeasurements shownwhen she was designing abridge. Find PO and PBcorrect to two decimalplaces.

5 An observer is 50 m horizontally from a hot air balloon.The angle of elevation to the top of the balloon is 60 andto the bottom of the balloons basket is 40. Find theheight (to the nearest metre)

a of AB and ATb of the balloon and its basket

6 PQ and RS are the walls of two buildings which are 100 mapart. Regina is standing at point T, midway between thetwo buildings. From her eye level the angle of elevation ofQ is 20 and the angle of elevation of S is 32. Her eyes are1.5 m above the ground. Calculate correct to one decimalplace:

a the height of the wall QPb the height of the wall RS

7 A ship (at P) is 24 km due east of a lighthouse(L). The captain takes bearings from twolandmarks, M and Q, which are due north and duesouth of the lighthouse respectively. The bearingsof M and Q from the ship are N38W and S64W respectively. How far apart (correct to two decimal places) are the two landmarks?

4258

P

BQO

20 15.5

50 m A

B

T

O40 60

100 mP R

QS

T 32201.5 m

18Example

A

B CD32 28

5 m

N

M

L

Q 64

38

24 km

S

EPW

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 387


8 From the top of a 90-m cliff the angles ofdepression of two boats in the water, in thesame direction, are 25 and 38 respectively.What is the distance between the two boatsto the nearest metre?

9 A shark is observed on theseabed at an angle ofdepression of 26 from asubmarine which is 32 mabove the seabed. Sometime later, the same sharkis seen from the sameposition, in the samedirection, but at an angle of depression of 34. How far (to the nearest metre) has theshark travelled since the first sighting?

10 An aeroplane starts at point O and flies due east for90 km to point A. B is a town 60 km from O on atrue bearing of 042. Find the distance (to thenearest km) of the:

a aeroplane from B when it is at point C as shown on the diagram

b aeroplane from town B when it has reached its destination A

11 Vicky cycles 22 km from O to A on a bearing of N20W, then turns and cycles 45 kmdue east to B. X is the point on AB that is due north of O. Find to one decimal place:

a AXb OXc OB

12 Town A and Town B are 7 km apart on a coastline that runs eastwest. A yacht, C, is atsea on a true bearing of 055 from town A, and on a bearing of 325 from town B.Calculate:

a angle ACBb the distance of the yacht from town Ac the distance of the yacht from town Bd the distance of the yacht from the

nearest point on the coastGive all answers correct to one decimalplace.

38 25

90 m

26

32 m

34

42

N

C

E

B

OA

S

W

60 km

90 km

EEW

W

N N

S S

A B

C

7 km

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 388


13 Winnie is standing on the edge of the beach (at W )directly opposite a windsurfer in the water (at S ). Winniewalks 40 m along the edge of the beach, at which pointshe thinks the windsurfer is at an angle of 60 fromwhere she is standing.

a How far is Winnie from the windsurfer?b Winnie walks further along the edge of the beach

until the windsurfer is now at an angle of about 30. How much further did she walk?

14 A television antenna is on top of a building. From a point on the ground 30 m from thebuilding, the angle of elevation of the top of the antenna is 54 and of the bottom ofthe antenna is 50.

a Find the height of the antenna to the nearest metre.b If antennas over 8 m tall are not allowed, is this antenna too tall?

W

S

306040 m

Enrichment

15 A yacht race starts and finishes at point O. The yachts must pass around the outsideof the buoys at points O, A and B. Buoy A is 12 km from O at a true bearing of 042,and buoy B is due north of O and at a true bearing of 325 from A. What is the totallength of the race correct to two decimal places?

16 a Two fishing boats, the Anchor and the Barrier, leave port at the same time. The Anchor travels on a true bearing of 120 for 75 km while the Barrier travels on atrue bearing of 170 for 60 km.

i Find how many kilometres east each of the boats has travelled correct to twodecimal places.

ii Find how many kilometres south each of the boats has travelled correct totwo decimal places.

b The Barrier is in distress and its radio is not working, so the crew decides torelease a flare which is visible from a distance of 30 km.

Will the distress signal be visible from the Anchor? Investigate what the minimum visible flare distance is in this situation.

Th

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 389


Esse

ntia

l M

athe

mat

ics

VEL

S P

roje

cts

W O R K I N G

Calculating heights or widthsIt is not always possible or practical to measure the height of an object directly, for examplethe height of a skyscraper or a tall tree. In this investigation you will use trigonometry andan inclinometer to find such heights. You will need to work in groups of three or four asdirected by the teacher.

Constructing an inclinometerAn inclinometer is an instrument used to measure angles of elevation and depression. Usethe diagram below to help you to construct an inclinometer.

You will need:a drinking strawa weighta piece of cardboard cut into a semicircleand marked off accurately every 10stringtape

The height of a landmarkSelect a landmark, such as a tall building, a tree or a bridge which you cannot measure theheight of directly. The base of the landmark must be accessible.

a Clearly describe the situation and draw a diagram to illustrate it.b Estimate the height of your chosen landmark.c Measure the angle of elevation and the

horizontal distance from the point where you are standing.

d Calculate the height of your chosen landmark. (Remember to take into account the height of your eye level.)

e Repeat parts a to d for different viewing postions, one closer and two further away from the landmark. Present your results in a table.

f Find the average of the heights to obtain a more accurate value for the height. Compare this value with the height you estimated earlier.

g Why didnt you get the same answer for the height each time? Which measurement is the most important, the distance or angle? What happens if you are out by just a small amount on one or both measurements? Investigate and discuss.

TrigonometryMathematically

stringweight

read off and subtractthis angle from 90 togive the angle tobe measured

drinkingstraw

90

eye

angle to bemeasured

horizontal distance

line of sight

angle of elevation

Th

PL

ID

E

Com

A

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 390


Esse

ntia

l M

athe

mat

ics

VEL

S P

roje

cts

Height of an inaccessible landmarkUse your inclinometer and your knowledge of trigonometry to determine the height of anobject which is inaccessiblefor example, a tree, a house or a tower with an inaccessiblebase.a i Choose two points which are

perpendicular to each other. For each position measure the angle of elevation, the distance between the two points and the angles a and b as shown on the diagram (along the ground).

ii Draw a diagram.iii Calculate the height of the landmark,

using your collected measurements and the three right-angled triangles in your diagram.

b Repeat your measurements and calculations from two other directions and at different distances from the landmark.

c Discuss your results and any possible sources of error.

The width of a rivera Identify a landmark, P, on a riverbank on the side opposite where you are standing.b Mark a point, Q, directly opposite P, on the riverbank on your side of the river.c Mark another point, R, a fixed distance on the riverbank from Q on your side of the

river such that .d Measure distance QR with a tape measure or other measuring device.e Measure angle PRQ. Calculate the approximate width of the river.f Repeat your measurements and calculations for two different distances of QR.g Write a paragraph discussing your findings and any possible sources of error.

PQR 90

distance between the two points

a b

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 391

Rev

iew

Chapter summary

Right-angled trianglesFor right-angled triangles, the basic trigonometric ratios are called sine, cosine and tangent.

SOH CAH TOA

Using a calculatorBefore entering angles in degree mode, you need to make sure that the calculator is set fordegrees.

Finding the angleGiven two side lengths of a right-angled triangle you can find an angle within the triangle byusing the inverse trigonometric keys on your calculator.

Angles of elevation or depression

BearingsSurveyors bearings use the compass directions north, south, east and west. Start at south ornorth, and turn east or west.

True bearings describe the angle in a clockwise direction from north.

tan1sin1cos1

tan OA

cos AH

sin OH


A

HO

S180 true

360 true N 000 true

270 true W E 090 true

To describe the bearing of an objectpositioned at A from an object positionedat O, we need to start at O, face north thenturn clockwise through the required angleto face the object at A.

line o

f sigh

t

angle of elevation

horizontalP

Q P

line of sight

angle of depression

horizontal

Q

CUAT029-C10[352-395].qxd 6/5/06 9:54 PM Page 392

Review

Multiple-choice questions

1 For the triangle shown

A B C

D E

2 The value of tan 32 correct to four decimal places is

A 0.5514 B 0.6249 C 0.6248 D 0.624 E 6.24

3 In the diagram the value of x correct to two decimal places is

A 40 B 13.61 C 4.70

D 9.89 E 6.474 Which of the following could be used to find the value of x in the triangle shown?

A 9cos 23 B 23cos 9 C

D E

5 The value of x correct to four decimal places is

A 0.8255 B 0.83 C 7.9

D 9.4336 E 7.96 The length of x in the triangle is given by

A 8 sin 46 B 8 cos 46 C

D E

7 The value of a in the diagram correct to one decimal place is

A 5.15 B 23.5 C 24.7

D 4.9 E 2.35

8 A ladder is inclined at an angle of 28 to the horizontal. If the ladder reaches 8.9 m upthe wall, the length of the ladder correct to the nearest metre is

A 19 m B 4 m C 2 m

D 33 m E 24 m

9 The value of in the diagram correct to two decimal places is

A 0.73 B 41.81 C 48.19

D 33.69 E 4.181

10 To calculate the value of you need to evaluate

A B C

D E sin1 a 23bcos1 a 3

2b

sin1 a 32btan1 a 2

3btan1 a 3

2b

cos 468

8

sin 46

8

cos 46

sincos 23

9cos 23

9

9

cos 23

sin c

bsin

bc

sin ac

sin ca

sin a

b


ac

b

x8

36

x23

9

172.7x

8

46

x

2411a

28

ladder8.9 m

8

12

23

CUAT029-C10[352-395].qxd 6/5/06 9:55 PM Page 393

Rev

iew

Short-answer questions

1 Find the value of each of the following, correct to two decimal places.

a sin 40 b tan 66 c cos 442 Find the value of each pronumeral, correct to two decimal places.

a b c

3 A ramp runs from street level to the entrance of a buildingwhich is 0.7 m above street level. How long is the ramp if it makes an angle of 8 with the horizontal, to one decimalplace?

4 The angle of elevation of the top of a lighthouse from apoint on the ground 40 m away from its base is 35. Findthe height of the lighthouse to two decimal places.

5 A train travels up a slope, making an angle of 7 with thehorizontal. When the train is at a height of 3 m above itsstarting point, find the distance it has travelled up theslope, to the nearest metre.

6 A yacht sails 80 km on a true bearing of 048.

a How far east of its starting point is the yacht correct to

two decimal places?

b How far north of its starting point is the yacht correct

to two decimal places?

7 From a point on the ground, Geoff measures the angle of elevation of a 120-m tower tobe 34. How far from the base of the tower is Geoff, correct to two decimal places?

8 A ship leaves Coffs Harbour and sails 320 km east. It then changes direction and sails240 km due north to its destination. What will the ships bearing be from Coffs Harbourwhen it reaches its destination, correct to two decimal places?

9 From the roof of a skyscraper, Aisha spots a car at an angle of depression of 51 fromthe roof of the skyscraper. If the skyscraper is 78 m high how far away is the car fromthe base of the skyscraper, correct to one decimal place?

10 Penny wants to measure the width of a river. She places two markers, A and B, 10 mapart along one bank. C is a point directly opposite marker B. Penny measures angleBAC to be 28. Find the width of the river to one decimal place.

11 An aeroplane takes off and climbs at an angle of 15 to the horizontal, at 210 km/halong its flight path for 15 minutes. Find correct to two decimal places:

a the distance the aeroplane travels b the height the aeroplane reaches

9 1142 29

yx

14

54


8

0.7 m

3540 m

7

3 m

E

N

48 80 km

S

W

CUAT029-C10[352-395].qxd 6/5/06 9:55 PM Page 394

Review

Extended-response questions

1 From the top of a 100-m cliff Skevi sees a boat out at sea at an angle of depression of12.

a Draw a diagram for this situation.

b Find how far out to sea the boat is to the nearest metre.

c A swimmer is 2 km away from the base of the cliff and in line with the boat. What is

the angle of depression to the swimmer to the nearest degree?

d How far away is the boat from the swimmer to the nearest metre?2 A pilot takes off from Amber Island and

flies for 150 km at 040 true to BarterIsland where she unloads her first cargo.She intends to fly to Dream Island but abad thunderstorm between Barter andDream Islands forces her to fly off coursefor 60 km to Crater Atoll on a bearing of060 true before turning on a bearing of140 true and flying for 100 km until shereaches Dream Island where she unloadsher second cargo. She then takes off andflies 180 km on a bearing of 55 true toEmerald Island.

a How many extra kilometres did she fly

trying to avoid the storm? Round to

the nearest km.

b From Emerald Island she flies directly

back to Amber Island. How many

kilometres did she travel on her return

trip? Round to the nearest km.


MC

TEST

D&D

TEST

CUAT029-C10[352-395].qxd 6/5/06 9:55 PM Page 395

ContentsCHAPTER 10 TrigonometryDo now10.1 Introducing trigonometry10.2 Finding the side length of aright-angled triangle10.3 Further problems involvingside lengths10.4 Finding the angle 10.5 Mixed application problems10.6 Angles of elevation and depression10.7 Bearings10.8 Problems involving two trianglesWorking MathematicallyReview

Answers

10 trigonometry

Documents

b c d6

b cd e fanswers1

angles of triangles

decimal places

number correct

f g h7

angles of elevation

angles whichmay