mathematics stage 5 ms5.1.2...

Mathematics Stage 5

MS5.1.2 Trigonometry

Part 3 Applying trigonometry

Number: M43683 Title: MS5.1.2 Trigonometry

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Extracts from Mathematics Syllabus Years 7-10 ©Board of Studies, NSW 2002 Unit overview pp iii-iv, Part 1 p 3, Part 2p 3, Part 3 p 3-4

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Part 3 Applying trigonometry 1

Contents – Part 3

Introduction – Part 3..........................................................3

Indicators ...................................................................................3

Preliminary quiz.................................................................5

Choosing the ratio .............................................................9

General problems............................................................13

Gradient of a line.....................................................................16

Elevation and depression................................................19

Further elevation and depression....................................23

Reviewing trigonometry...................................................27

Suggested answers – Part 3 ...........................................29

Exercises – Part 3 ...........................................................33

2 MS5.1.2 Trigonometry


Introduction – Part 3

This third part of trigonometry deals with deciding which of the three

trigonometric functions to use in answering a question, and to solve

problems where trigonometry can be applied. It is also extended to cover

angles of elevation and depression.

Throughout these notes diagrams are given. Occasionally incomplete

diagrams are provided to encourage students to add relevant information

to them which will aid them in answering the question.

Indicators

By the end of Part 3, you will have been given the opportunity to work

towards aspects of knowledge and skills including:

• selecting and using appropriate trigonometric ratios in right-angled

triangles to find unknown sides, including the hypotenuse

• selecting and using appropriate trigonometric ratios on right-angled

triangles to find unknown angles correct to the nearest degree

• identifying angles of elevation and depression

• solving problems involving angles of elevation and depression when

given a diagram.

By the end of Part 3, you will have been given the opportunity to work

mathematically by:

• solving problems in practical situations involving right-angled

triangles

• interpreting diagrams in questions involving angles of elevation and

depression


• relating the tangent ratio to the gradient of a line.Source: Extracts from outcomes of the Mathematics Years 7–10 syllabus

<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf > (accessed 04 November 2003).© Board of Studies NSW, 2002.


Preliminary quiz

Before you start this part, use this preliminary quiz to revise some skills

you will need.

Activity – Preliminary quiz

Try these.

1 Use your calculator to find, correct to 3 decimal places,

a sin 29° _____________________________________________

b tan 10° _____________________________________________

c cos 53° ____________________________________________

d4 × cos25°

5 _________________________________________

2 Find angles for which,

a sin A = 0.563 ________________________________________

b tan B = 12 __________________________________________

c cos C = 3

20 _________________________________________

d tan α = 1 ___________________________________________


3 Use the cosine ratio to calculate the length of the side marked.

E F

G

x

126 m

36°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

4 Use the tangent ratio to calculate the length of the side marked.

e 7.6 cm

38°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

5 Use the sine ratio to calculate the size of the angle between north and

the diagonal line.

5

6

N

E

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


6 Use the tangent ratio to calculate the size of the angle marked.

G H15.3 cm

9.5 cm

h°

F

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

Check your response by going to the suggested answers section.


Choosing the ratio

Until now you have been directed to the trig function (sine, cosine, or

tangent) you needed to use. You have now reached a stage where you

should be able to decide which of these three trig ratios is the appropriate

one to use in any given situation.

SOH CAH TOA

sin opphyp cos adj

hyp tan oppadj op

pos

ite

adjacent

hypotenuse

*

For example, if the opposite side andhypotenuse are being used, the trig.ratio you want is sine.

Follow through the steps in this example. Do your own working in the

margin if you wish.

a Calculate the angle, a° , correct to the nearest degree.

B C

A

12 cm8 cm

a°


b Calculate z correct to one decimal place.

Z Y

X17

cm z

25°

Solution

a The two sides involved are the adjacent (8 cm) and hypotenuse

(12 cm). Therefore use cosine.

cos a° = 8

12 ∴ a° = 48.189685°

= 48° (correct to the nearest degree)

use SHIFT cos

b This time the two sides involved are the opposite (17 cm) and the

hypotenuse (z). Sine is needed.

sin 25° = 17

z

z = 17

sin 25°= 81.8 cm (correct to one decimal place)

In each case, look over your answers to check if they are reasonable.

Activity – Choosing the ratio

Try these.

1 Use the appropriate trigonometric ratio to calculate x in each

triangle.

a

x

50°

80 mm


___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b10

0 m

60°x

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

2 Calculate the size of the marked angle, correct to the nearest degree.

a

38

α

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b

41

9α

___________________________________________________


___________________________________________________

___________________________________________________

___________________________________________________

c7 5

α β

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


You have been practising choosing and using the correct trig ratio in a

right-angled triangle. Now check that you can solve these kinds of

problems by yourself.

Go to the exercises section and complete Exercise 3.1 – Choosing the

ratio.


General problems

You have learned how to use the trigonometric ratios of an acute angle in

a right-angled triangle in two different ways.

• To find the size of the acute angles of the triangle

• To find the length of the side given one of the acute angles.

You will now use this knowledge to solve simple problems. In each case

you will need to use a labelled diagram to represent the situation. In this

set of notes a diagram will always be given, but you may want to add

more information to it.


margin if you wish.

Merill skis 132 metres down an even slope and drops 20 m in

height. What angle (to the nearest degree) does the slope make

with the horizontal?

132 m

α

20 m


Solution

The diagram helps you identify that the sine ratio is required for

this question.

sin α = 20

132 ∴α = 8.715°

= 9° (to the nearest degree)

sin =opp.

hyp.

⎡

⎣⎢

⎤

⎦⎥

Try this calculation. Did you obtain the same answer?

Activity – General problems

Try these.

1 A ladder has its foot 1.3 metres from the base of a wall on level

ground. The ladder makes an angle of 75° with the horizontal.

Calculate the length of the ladder.

y

75°

1.3 m

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


2 A boat is at anchor 500 m from the foot of a cliff 40 m high.

Calculate the angle, θ , which the line of sight to the top of the cliff

makes with the horizontal.

not drawn to scale

θ

40 m

500 m

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


In each of these examples you can see that, regardless of the situation, a

right-angled triangle is involved. Once you have identified that triangle

you then need to decide which of the three trig functions: sine cosine or

tangent you will use.

Use a diagram as a visual representation of the problem. It does not need

to be drawn to scale.

Check that the answer you obtain looks reasonable for the problem you

are solving. It is very easy to mistakenly choose the wrong trig ratio, or

press the wrong key on your calculator.


Gradient of a line

In co-ordinate geometry you established that the slope of the line, m, is

given by m =rise

run.

Also from trigonometry,

tan θ = opposite

adjacent and so, from the

triangle shown, tan θ = rise

run.

This means that m = tan θ . O

run

rise

θ

y

x

The angle in the triangle is the same as the angle between the line and the

positive (right-hand) side of the x-axis. Can you see why? (Think of

parallel lines, and corresponding angles.)

So the gradient of a line is simply the tan of the angle it makes with the

positive side of the x-axis.


margin if you wish.

What angle does the line y = 12

x + 1 make with the x-axis?

Solution

Make a small table of values to draw the line.

x –1 0 1 2

y 0.5 1 1.5 2

Now draw the line on the x-y plane and use it to determine the

gradient.


–1–2 1 2 30

–1

1

2

3 y

x

1

2θ

θ

The gradient, m = 1

2, and so tan θ =

1

2.

Using your calculator, θ = 27° (to the nearest degree).

You can use any two points on the line to establish rise and run.


Try this.

3 Draw the line y = 2x – 3 on the co-ordinate plane and use it to

determine the angle the line makes with the x-axis.

0

1

2

3

–1

–2

–3

–4

–5

–6

1 2 3–1

y

x


Go to the exercises section and complete Exercise 3.2 – General problems.


Elevation and depression

If someone is looking at an object, the straight line from the eye of the

observer to the object is the line of sight.

line of sight

If the object is above the eye level of the observer, then the angle they

raise their eyes, from the horizontal to the line of sight of the object, is

called the angle of elevation.

horizontal at eye level

angle of elevation

line of sight

If the object is below the eye level of the observer, then the angle they

look down, from the horizontal to the line of sight of the object, is called

the angle of depression.

horizontal at eye level

angle of depression

line of sight

Note: the angle of elevation or depression of one point from another is

always measured from the horizontal.


Since these horizontal lines are

parallel, the angle of elevation

must be equal to the angle of

depression, because they are

alternate angles formed by

parallel straight lines.

horizontal

horizontal

angle of depression

angle of elevation

You can use angle of elevation or angle of depression to answer

questions involving trigonometry.


margin if you wish.

From a plane travelling at a height of 6 km, the angle of

depression of an airfield is found to be 14°. How far must the

plane fly to be directly above the airfield?

ground level

angle of depression14°

6 km

P Q

A(airfield)

(directly abovethe airfield)

Solution

∠PQA = 90° because the plane flies horizontally. At Q, it is

directly above A.

∆ PQA is the right-angled triangle. You want to calculate the

length PQ.


tan P = AQ

PQ

tan 14° = 6

PQ

PQ = 6

tan 14°= 24.065

(The two shorter sides involve using the tangent ratio.)

The plane must fly 24 km (to the nearest kilometre).

When answering questions like this, give your answer to an appropriate

number of significant figures. Correct to the nearest kilometre for this

question is reasonable, but writing 24.06469 km is not.

Activity – Elevation and depression

Try these.

1 From the top of a cliff 20 m high the angle of depression of a ship at

sea is 15°. Calculate the distance from the ship to the foot of the

cliff.

C

FS

15°75°

20 m

?15°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


2 A flagpole of height 10 metres casts a shadow of length 16 metres on

level ground. Calculate the angle of elevation of the sun at this time.

10 m

16 m

F

EDx

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Notice that the angle of depression, in the first question, is the angle

outside the triangle. It is the angle measured downwards from the

horizontal.

You could use alternate angles, and work with ∠CSF inside the triangle.

Or you could calculate the complement of 15° (which is 75° ) and work

with ∠FCS. The choice is yours.

C

FS

15°75°

20 m

?15°

The two angles are equal.

The horizontal lines are parallel,and the angles are alternate. The two angles

add up to 90°.

Either way, the answer you will arrive at is the same.

Go to the exercises section and complete Exercise 3.3 – Elevation and

depression.


Further elevation and depression

When looking up at an object, the angle is an angle of elevation. When

looking down at an object, you have an angle of depression.

horizontal at eye level of the man

angle of depression

angle of elevation

horizontal at eye level of the dog

line of sight

Angles of elevation and depression are always measured between the line

of sight and the horizontal.

Sometimes you are not given a labelled diagram to assist you. So you

need to visualise the situation yourself and draw a diagram to help you

get the picture clear. Then you can see what triangle to use.

So you have three main steps in solving trigonometry questions:

• draw a diagram to represent the situation

• locate a right-angled triangle in the diagram

• use trigonometry to calculate the side or angle needed.



margin if you wish.

The angle of elevation of the top of a flagpole from an observer

is 39° when measured 12 m away from the flagpole. The

distance from the ground to the observer’s eyes is 1.8 m.

Calculate the total height of the flagpole.

Solution

12 m

39°

A

CB

E D

You need to calculate the total height of the flag above the

ground (that is, AD).

To find AC, you use the tangent ratio in ∆ABC. To find the

actual height of the flagpole, you add CD to AC.

CD = BE (the height of the observer) = 1.8 m

∴ AD = AC + BE

In ∆ABC, tan B = opposite

adjacent

tan 39° = AC

12AC = 12 × tan 39°

= 9.7 m (correct to 1 decimal place)

∴ AD = 9.7 + 1.8 = 11.5 m


The actual height of the flagpole is 11.5 m (correct to one

decimal place).

Sometimes it may be necessary to add, or subtract, values from the ones

you calculate using trigonometry to arrive at the answer.

Even when a diagram is provided, it may not include all the information.

Feel free to write on the diagram and include other information that may

help you answer the question.

Finally, look over your answer and ask yourself whether it looks

reasonable for the problem.

In the next activity you will be provided with an incomplete diagram.

Mark on it necessary values so you can answer the questions.

Activity – Further elevation and depression

Try these.

1 From the top of a building 60 m high the angle of depression of a

parked car is found to be 28°. Complete the diagram and calculate

the distance the car is parked from the building.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


2 Complete the diagram and use it to calculate the angle of elevation

(to the nearest degree) of the top of a wall 15 m high from a point on

the ground 5 m from the bottom of the wall.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


You have been practising further examples of elevation and depression.

Now check that you can solve these kinds of problems by yourself.

Go to the exercises section and complete Exercise 3.4 – Further elevation

and depression.


Reviewing trigonometry

There is an enormous number of applications for trigonometry. Of

particular interest is the technique of triangulation that is used in

astronomy to measure the distance to nearby stars, in geography to

measure distances between landmarks, and in satellite navigation

systems.

But regardless of where it is used, the basic definitions of sine, cosine

and tangent are the same.

SOH CAH TOA

sin opphyp cos adj

hyp tan oppadj op

pos

ite

adjacent

hypotenuse

In this session you will practice the ideas you learned in using

trigonometry in right-angled triangles.

The following exercises will help you to consolidate trigonometry

applications.

Go to the exercises section and complete Exercise 3.5 – Reviewing

trigonometry.


Suggested answers – Part 3

Check your responses to the preliminary quiz and activities against these

suggested answers. Your answers should be similar. If your answers are

very different or if you do not understand an answer, contact your teacher.

Activity – Preliminary quiz

1 a 0.485 b 0.176 c 0.602 d 0.725

2 a 34° b 85° c 81° d 45°

3 cos 36° =x

126x =126 × cos 36°

=101.9 m

4 tan 38° =e

7.6e = 7.6 × tan 38°

= 5.94 cm

5 sin x° =5

6x = 56°

6 tan h =9.5

15.3h = 32°

Activity – Choosing the ratio

1 a tan =opposite

adjacent

tan 50° = x

80x = 80 × tan 50°

=95.3 mm


b cos =adjacent

hypotenuse

cos 60° = x

100x = 100 × cos 60°

= 50 m

2 a sin α = 3

8 ∴ α = 22°

b cos α = 9

41 ∴ α = 77°

c tan α = 5

7You can use trig ratios to calculate β.

∴ α = 36° Alternatively, β �= 90°– 36°

=54°


1 cos =adjacent

hypotenuse

cos 75° = 1.3

y

y = 1.3

cos 75°= 5.02

The ladder is 5 m long.

2 tan =opposite

adjacent

tan θ = 40

500θ = 85.41°

= 85°

(correct to the nearest degree)

3 The gradient, m = 2 (so tanθ = 2 ) and the angle the line makes is

63° .


Activity – Elevation and depression

1 tan 15° = 20

SF

SF = 20

tan 15°= 74.6 m

(correct to 1 dec. pl.)

2 Let the angle be θ .

tan θ = 10

16θ = 32°

(to the nearest degree)

Activity – Further elevation and depression

1 tan 28° = 60

d

d = 60

tan 28°= 112.8 m

(correct to 1 dec. pl.)

The car is parked 112.8 m

from the building.

28°

d60

m

28°

2 Let the angle be θ .

tan θ = 15

5θ = 72°

(to the nearest degree)

The angle of elevation is 72° .5 m

15 m

θ


Exercises – Part 3

Exercises 3.1 to 3.5 Name ___________________________

Teacher ___________________________

Exercise 3.1 – Choosing the ratio

1 Choose a suitable trigonometric ratio and then calculate x (measured

in centimetres), correct to two decimal places.

a

40030°

x

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b

x

40°75

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


c

x

25°

8

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

d

A C

B

80

48°x

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

e

28

59°

x

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


2 Calculate the value of α (alpha) in degrees in the following triangles.

a

7

α10

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b

2

α6

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

c

36

α

9

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


Exercise 3.2 – General problems

1 A ladder reaches 2.5 m up a vertical wall, and has its foot on level

ground 1 m from the base of the wall. Find the angle the ladder

makes with the ground.

θ

2.5

m

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

2 A 30° – 60° – 90° set square has the side opposite the 60° angle

12 cm long. Find the length of the longest side. (Give your answer

correct to the nearest millimetre. Hint: label the longest side with a

pronumeral.)

60°

30°

12 c

m

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


3 The diagonal BD of a rectangle ABCD makes an angle of 27° with

the side AB, which is 7.0 cm long. Calculate the length of the other

side, correct to two significant figures.

A B

D C

7.0 cm27°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

4 To calculate the width of a river, a man stands at a point A directly

opposite a tree T on the edge of the opposite bank. He walks 120 m

along his bank to another point B and measures ∠ABT = 27°.

Calculate the width of the river. (The banks are straight and parallel

along this section of the river.)

120 m

A

T

B27°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


5 O is the centre of the circle. OD = 3.28 cm, and ∠AOD = 49°.

Calculate the radius of the circle.

O

A D B

r 49°

3.28

cm

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

6 The pitch of a roof is a measure of its steepness. It is often

expressed as the ratio of the rise to the run.

rise

runθ

Drawn to scale

Use a ruler to measure the rise and run on this scale diagram. Use

this information to calculate the pitch angle, θ .

__________________________________________________________


Exercise 3.3 – Elevation and depression

1 From a boat out at sea, Chandra measured the angle of elevation of

the top of a cliff 65 m high to be 27°. How far is the boat from the

foot of the cliff (to the nearest metre)?65

m

27°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

2 Kimberley found the angle of depression of the base of a statue in a

nearby park to be 65° from a 35 m high window. How far from the

building is the statue? (Answer correct to the nearest metre.)

35 m

65°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


3 The altitude of the sun is the angle of elevation of the sun. It is the

angle between the horizontal and the sun’s rays. A building casts a

shadow 40 m long when the altitude of the Sun is 50°. Find the

height of the building, correct to the nearest metre.

40 m50°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

4 Calculate the angle of depression of Mrs Lee’s cottage which is

1.6 km away down a slope from the top of a hill 225 m higher.

1. 6 km

225

m

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


5 Find the elevation of the sun when Olav, who is 1.9 m tall, casts a

shadow 2.6 m long. (Answer to the nearest degree.)

α

2.6 m

1.9

m

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

6 (Harder) You will need to draw a diagram for this one. The angle

of elevation of an aircraft flying at 800 metres above ground level is

64° from a point on the ground. How far is the aircraft from this

point?

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Exercise 3.4 – Further elevation and depression

In each of these only a partial diagram is provided. Label the diagram

appropriately to help you answer the question.

1 From the top of a vertical cliff 40 m above sea level the angle of

depression of a boat is measured to be 18°. Find the distance from

the boat to the bottom of the cliff.

40 m

18°

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

2 An upright stick casts a shadow of length 2 m on level ground. If the

stick is 1 m long, find the angle of elevation of the sun at this time.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


3 From a plane flying at a height of 4 km above a town A, the angle of

depression of a town B is found to be 20°. Find the distance

between the two towns, correct to one tenth of a kilometre.

Town A Town B

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

4 A tent pole is 2.2 metres high. Calculate the angle of elevation of

the top of the pole given the top of the pole is secured to the ground

with a rope 3.5 m long.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


5 A boat is at anchor where the top of the 25 m anchor chain is 16 m

above the seabed. Calculate the angle of depression of the anchor

chain.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

6 The Eiffel Tower was built for the Universal Exhibition in

celebration of the French Revolution and opened in 1889.

Write a trigonometry question

you could answer using the

information on this diagram.

(You do not need to answer the

question.)

324 m

200 m

θ

_______________________________________________________

_______________________________________________________

_______________________________________________________


Exercise 3.5 – Reviewing trigonometry

1 Calculate the lengths or angles marked.

a

4 cm

x30°

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b

35°

d cm

50 cm

F

D E

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

cg

14.6 9.5

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


d

12 mm

A

B C

7 m

m

α°

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

e

5.7 mm

38°

j

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

2 A boy is flying a kite on 45 metres of line which makes an angle of

44° with the horizontal. How much higher is the kite than the boy’s

hand? (Answer to the nearest metre.)

45 m

44°

h

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


3 A fishing boat is held at anchor in the water by 23 metres of chain.

The depth of the water is 18 metres. Find the angle the chain makes

with the sea floor.

1823

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

4 Calculate the angle the line makes with the x-axis.

–1–2 1 2 30

–1

1

2

3 y

x

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


5 One end of a 10-metre rope is tied to the bow of a yacht and the

other end to a point on the edge of a jetty. The rope is taut (stretched

tightly) and makes an angle of 21° with the horizontal. How far out

is the bow from the wharf?

21°10 m

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

6 A 14-metre fire engine ladder has its foot 4 m away from the side of

a building. Calculate the angle the ladder makes with the wall.

4 m

14 m

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7 A fish notices a bug on the

surface of the water 55 cm in

front of it, and 65 cm above it.

65 cm

55 cm

a Calculate the angle of elevation of the bug from the fish.

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b How far away is the bug from the fish?

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mathematics stage 5 ms5.1.2...

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