maths a - chapter 5

5

syllabussyllabusrrefefererenceenceStrand:Applied geometry

Core topic:Elements of applied geometry

In thisIn this chachapterpter5A Pythagoras’ theorem5B Shadow sticks5C Calculating trigonometric

ratios5D Finding an unknown side5E Finding angles5F Angles of elevation and

depression

Right-angledtriangles and trigonometry

MQ Maths A Yr 11 - 05 57 Wednesday, July 4, 2001 4:39 PM

158

M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

Introduction

Thousands of years before Christ, ancient civi-lisations were able to build enormous structures— like Stonehenge in England and the pyramidsin Egypt. How did they do it? Human ingenuityallows us to achieve many things that at firstseem impossible. How do we determine theheight of a tall object without physically scalingit? Sometimes, even tasks that appear to be verysimple (for example, planning and designing astaircase) present us with unexpected problems.How in fact

do

we go about the practical task ofplanning and designing a staircase?

The theorem of Pythagoras and the geometryof right-angled triangles (trigonometry) can beemployed to answer these questions. As you pro-gress through this chapter, solutions to problemssuch as these will become clear.

To put the mathematics of Pythagoras intoperspective, let us first consider the times inwhich he lived.

MQ Maths A Yr 11 - 05 Page 158 Wednesday, July 4, 2001 4:39 PM

C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y

159

History of mathematicsP Y T H AG O R A S O F S A M O S ( c i rc a 5 8 0 B C – 5 0 0 B C )

During his life:• Taoism is

founded• Kung-Fu-tse

(Confucius) is born

• Buddhism is founded.

Pythagoras was a famous Greek mathemat-ician and mystic but is now best known for his theorem about the sides of a triangle. He was born on Samos Island. It is believed that he was born about 580 BC and died about 500 BC, but because of the way dates were recorded then, various dates are given for his life. Not much was known about his personal life but it was known that he had a wife, son and daughter.

When Pythagoras was a young man, he travelled to Egypt and Babylonia (Mesopo-tamia) where he learned much of his math-ematics and developed an interest in investigating it further.

He founded a cult with the idea that ‘the essence of all things is a number’. This group believed that all nature could be expressed in terms of numbers. They found, however, that some numbers could not be expressed as rational numbers, such as . They kept this information to themselves and there is a story that they killed one member who told some-body else about this fact.

Pythagoras showed that musical notes had a mathematical pattern. He stretched a string tightly and found that it produced a certain sound and then found that if he halved the

length of the string, it produced a sound that was in harmony with the first. He also found that if it was not exactly half, or a multiple of a half then it was a clashing sound. This approach is still used in musical instruments today.

He is credited with the discovery now known as ‘Pythagoras’ theorem’. This states that ‘For a right-angled triangle, the square of the hypotenuse (long side) is equal to the sum of the squares of the two short sides’ and can be written as c2 = a2 + b2 using the diagram below. Other people knew of this idea long before Pythagoras. There is a Babylonian tablet known as ‘Plimpton 322’ (believed to have been made about 1500 years before Pythagoras was born), which has a set of values of the type known as Pythagorean triples.

Some examples of Pythagorean triples are:3, 4 and 58, 15 and 17and a more difficult example: 20, 99 and 101.

For his investigations and demonstrationswith right angles, Pythagoras used a string inwhich he had tied knots.

Questions1. Where was Pythagoras born?2. Where did he travel to and learn most of

his mathematics?3. What is the formula for his famous

theorem?4. What did he start to investigate about

patterns in nature?5. What is the name of the ancient tablet

that contains Pythagorean triples?6. What is a Pythagorean triple?

2

a

bc


160

M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

1

Name the hypotenuse in each of the following triangles.

a b

c d

2

State Pythagoras’ theorem.

3

Calculate the unknown lengths in the following right-angled triangles.

a b c

4

Complete each of the following to form ratios that are equivalent to the ratio 3 : 4 : 5.

a

9 : 12 : ____

b

1.5 : ____ : 2.5

c

____ : 1 : 1.25

d

7.5 : 10 : ____

5

The diagram at right depicts a tree in a field on a sunny day. Copy the diagram.

a

Draw the position of the shadow cast by the tree.

b

Mark the angle of elevation of the sun.

6

Find the value of the unknown angles in the following triangles.

a b

7

Express the following angles in degrees, minutes and seconds.

a

35.5°

b

27.23°

c

68.125°

8

Express the following angles in degrees, correct to 2 decimal places.

a

45°27

′

b

84°35

′

22

″

c

64°28

″

.

9

Define the trigonometric ratios:

a

sine

b

cosine

c

tangent.

C

A

B

ED

F

H

GI

K

J L

6 m

8 m

a

6 m

4 m

b20 cm

a

34°

b

27°



161

10

Copy the diagrams below and label the sides as Opposite, Adjacent or Hypotenusewith respect to the marked angle.

a b c

11

Using your calculator, determine the following (correct to 4 decimal places).

a

sin 45°

b

cos 30°

c

tan 22.5°

d

sin 67°15

′

e

tan 38°12

′

22

″

12

Copy the following diagrams and mark the angles of elevation or depression, from theobserver to the object, on each.

a b

c

13

Solve the following for

x.

a

=

b

=

Ground

Observer

Ship

Sea

Ground

Observer

Bird

5.24.5------- x

1.8-------

5.24.5------- 1.8

x-------


162 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

Right-angled trianglesThe properties of right-angled trianglesenable us to calculate lengths and angleswhich are sometimes not able to be meas-ured. The first property we will investigateis Pythagoras’ theorem.

Pythagoras’ theoremPythagoras’ theorem allows us to calculate the length of a sideof a right-angled triangle, if we know the lengths of the other two sides. Consider LABC at right. AB is the hypotenuse (the longest side). It is opposite the right angle.

Note that the sides of a triangle can be named in either of two ways.

1. A side can be named by the two capital letters given to the vertices at each end. Thisis what has been done in the figure above to name the hypotenuse AB.

2. We can also name a side by using the lower-case letter of the opposite vertex. In thefigure above, we could have named the hypotenuse ‘c’.

Consider the right-angled triangle ABC (below) with sides 3 cm, 4 cm and 5 cm. Squares have been constructed on each of the sides. The area of each square has been calculated (A = S2) and indicated.

Note that the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

25 cm2 = 16 cm2 + 9 cm2

Alternatively: (5 cm)2 = (4 cm)2 + (3 cm)2

Which means: hypotenuse2 = base2 + height2

hypotenuseA

C B

Name the hypotenuse in the triangle at right.

THINK WRITE

The hypotenuse is opposite the right angle.The vertices at each end or the lower- case letter of the opposite vertex can be used to name the side.

The hypotenuse is QR or p.

Q

P

R1

2

1WORKEDExample

A = 9 cm2

A = 25 cm2

C B

A

3 cm4 cm

5 cm

A = 16 cm2Pythagoras calculations

GCpro

gram

Pythagoras


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 163This result is known as Pythagoras’ theorem. Pythagoras’ theorem states:

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. That is,

Hypotenuse2 = base2 + height2

This is the formula used to find the length of the hypotenuse in a right-angled tri-angle when we are given the lengths of the two shorter sides.

In this example, the answer is a whole number because we are able to find exactly. In most examples this will not be possible. In such cases, we are asked to writethe answer correct to a given number of decimal places.

By rearranging Pythagoras’ theorem, we can write the formula to find the length of ashorter side of a triangle.

Since hypotenuse2 = base2 + height2

it follows that base2 = hypotenuse2 − height2

and height2 = hypotenuse2 − base2

The method of solving this type of question is the same as in the previous example,except that here we use subtraction instead of addition. For this reason, it is importantto look at each question carefully to determine whether you are finding the length ofthe hypotenuse or one of the shorter sides.

Find the length of the hypotenuse in the triangle at right.

THINK WRITE

Write the formula. Hypotenuse2 = base2 + height2

Substitute the lengths of the shorter sides.

c2 = 152 + 82

Evaluate the expression for c2. = 225 + 64= 289

Find the value of c by taking the square root.

c = = 17 cm

8 cm

15 cm

c

12

3

4 289

2WORKEDExample EXCEL Spreadsheet

Pythagoras

289

Find the length of side PQ in triangle PQR, correct to 1 decimal place.

THINK WRITE

Write the formula. Base2 = hypotenuse2 − height2

Substitute the lengths of the known sides.

r2 = 162 − 92

Evaluate the expression. = 256 − 81= 175

Find the answer by finding the square root.

r = = 13.2 m

QP

R

16 m 9 m

r

12

3

4 175

3WORKEDExampleMathcad

Pythagoras



Because Pythagoras’ theorem applies only in right-angled triangles, we can use thetheorem to test whether a triangle is right-angled, acute-angled or obtuse-angled. If weassume, as a starting point, that the triangle is right-angled, we could regard the longestside as the hypotenuse. We could then calculate values for hypotenuse2, base2 andheight2. Interpreting the relationships between these three values, we could concludethe following.

If hypotenuse2 = base2 + height2

the triangle is right-angled.

If hypotenuse2 > base2 + height2

the triangle is obtuse-angled (since the hypotenuse is longer than it should be for thatparticular base and height).

If hypotenuse2 < base2 + height2

the triangle is acute-angled (since the hypotenuse is too short for that particular baseand height).

Pythagorean triads (or Pythagorean triples) are sets of 3 numbers which satisfyPythagoras’ theorem. The first right-angled triangle we dealt with in this section hadside lengths of 3 cm, 4 cm and 5 cm. This satisfied Pythagoras’ theorem, so thenumbers 3, 4 and 5 form a Pythagorean triad or triple. In fact, any multiple of thesenumbers, for example 1. 6, 8 and 10

1.5, 2 and 2.5would also form a Pythagorean triad or triple. Some other triads are:

5, 12, 138, 15, 179, 40, 41

Determine whether the triangle shown is right-angled,acute-angled or obtuse-angled.

THINK WRITE

Assume that the longest side is a hypotenuse.Calculate the hypotenuse2 and base2 + height2 separately.

hypotenuse2 = 72

= 49base2 + height2 = 52 + 42

= 25 + 16 = 41

Write down an equality or inequality statement.

hypotenuse2 > base2 + height2

Write a conclusion. Therefore the triangle is obtuse-angled.

4 cm 5 cm

7 cm

1

2

3

4

4WORKEDExample


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 165

Pythagoras’ theorem can be used to solve more practical problems. In these cases, it isnecessary to draw a diagram that will help you decide the appropriate method forfinding a solution. The diagram simply needs to represent the triangle; it does not needto show details of the situation described.

Is the set of numbers 4, 6, 7 a Pythagorean triad?

THINK WRITE

Find the sum of the squares of the two smaller numbers.

42 + 62 = 16 + 36= 52

72 = 4972 ≠ 42 + 62

Find the square of the largest number.

Compare the two results. The numbers form a Pythagorean triad if the results are the same.

Write your answer. 4, 6, 7 is not a Pythagorean triad

1

2

3

4

5WORKEDExample EXCEL Spreadsheet

Triads

The fire brigade attends a blaze in a tall building. They need to rescue a person from the 6th floor of the building, which is 30 metres above ground level. Their ladder is 32 metres long and must be at least 10 metres from the foot of the building. Can the ladder be used to reach the people needing rescue?

THINK WRITE

Draw a diagram and show all given information.

Write the formula after deciding if you are finding the hypotenuse or a shorter side.


Substitute the lengths of the known sides.

c2 = 102 + 302

Evaluate the expression. = 100 + 900= 1000

Find the answer by taking the square root.

c = = 31.62 m

Give a written answer. The ladder will be long enough to make the rescue, since it is 32 m long.

1

10 m

30 mc

Burning building

Fire engine

10 m

30 m c

2

3

4

5 1000

6

6WORKEDExample



Pythagoras’ theorem

1 Name the hypotenuse in each of the following triangles.a b c

2 Find the length of the hypotenuse in each of the following triangles.a b c

3 In each of the following, find the length of the hypotenuse, correct to 2 decimal places.a b c

remember1. Make sure that you can identify the hypotenuse of a right-angled triangle. It is

the side opposite the right angle, and the longest side.2. If you are finding the length of the hypotenuse use

hypotenuse2 = base2 + height2.3. If you are finding the length of a shorter side use either

base2 = hypotenuse2 − height2 or height2 = hypotenuse2 − base2.

4. Pythagoras’ theorem can be used to test whether a triangle is right-angled.(a) if hypotenuse2 = base2 + height2, the triangle is right-angled(b) if hypotenuse2 > base2 + height2, the triangle is obtuse-angled(c) if hypotenuse2 < base2 + height2, the triangle is acute-angled.

5. Pythagorean triads or Pythagorean triples are sets of three numbers which satisfy Pythagoras’ theorem — for example 3, 4, 5.

6. Read the question carefully to make sure that you give the answer in the correct form.

7. Begin problem questions with a diagram and finish with an answer in words.

remember

5AWORKEDExample

1 P

Q R

X

Y

Z

B

A C

WORKEDExample

2

12 cm

5 cmx

150 mm

80 mmm

60 m

11 mz

9 cm

6 cm

a 4.9 m

4.9 m8.6 km 11.3 km


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 1674 Find the length of each unknown shorter side in the right-angled triangles below,

correct to 1 decimal place.a b c

5 In each of the following right-angled triangles, find the length of the side marked witha pronumeral, correct to 1 decimal place.a b

c d

6 Use the converse of Pythagoras’ theorem to determine if the following triangles areright-angled, acute-angled or obtuse-angled.a b c

7

The hypotenuse in LXYZ at right is:

8

Which of the following triangles is definitely right-angled?

A XY B XZC YZ D impossible to tell

A B

C D

WORKEDExample

3

12 cm

6 cm

p 2.2 m 2.9 m

q

4.37 m

2.01 m

t

8 cm

4 cm

m

10.5 cm

24.5 cm z

33 mm

34 mm a

37.25 m

52.75 m

p

WORKEDExample

4

10 cm

6 cm

8 cm 41 cm 40 cm

9 cm

31 m

38 m16 m

mmultiple choiceultiple choiceX

Y

Z

mmultiple choiceultiple choice

24.5 m32.5 m

20.5 m

5 m 24.5 m

25 m

24.5 m 84 m

87.5 m

25.4 m

24.5 m35.3 m



9 Are the following sets of numbers Pythagorean triads?

10 Complete the following Pythagorean triads.

11 For each of the sets which were Pythagorean triads in question 9 state which side theright angle is opposite.

12

Which of the following is a Pythagorean triad?

13

Which of the following is not a Pythagorean triad?

14 A television antenna is 12 m high. To support it, wires are attached to the ground 5 mfrom the foot of the antenna. Find the length of each wire.

15 Susie needs to clean the guttering on her roof. She places her ladder 1.2 m back fromthe edge of the guttering that is 3 m above the ground. How long will Susie’s ladderneed to be (correct to 2 decimal places)?

16 A rectangular gate is 3.5 m long and 1.3 m wide. The gate is to be strengthened by a diagonal brace as shown at right.How long should the brace be (correct to 2 decimal places)?

17 A 2.5-m ladder leans against a brick wall. The foot of the ladder is 1.2 m from the footof the wall. How high up the wall will the ladder reach (correct to 1 decimal place)?

18 Use the measurements in the diagram at right to find the height of the flagpole, correct to 1 decimal place.

19 An isosceles, right-angled triangle has a hypotenuse of 10 cm. Calculate the length ofthe shorter sides. (Hint: Call both shorter sides x.)

a 9, 12, 15 b 4, 5, 6 c 30, 40, 50 d 3, 6, 9e 0.6, 0.8, 1.0 f 7, 24, 25 g 6, 13, 14 h 14, 20, 30i 11, 60, 61 j 10, 24, 26 k 12, 16, 20 l 2, 3, 4

a 9, __, 15 b __, 24, 25 c 1.5, 2.0, __ d 3, __, 5e 11, 60, __ f 10, __, 26 g __, 40, 41 h 0.7, 2.4, __

A 7, 14, 21 B 1.2, 1.5, 3.6 C 3, 6, 9D 12, 13, 25 E 15, 20, 25

A 5, 4, 3 B 6, 9, 11 C 13, 84, 85D 0.9, 4.0, 4.1 E 5, 12, 13

WORKEDExample

5



WORKEDExample

6

3.5 m

1.3 m

7.9 m

2.4 m



Pythagoras’ theoremResources: Paper, pencil, protractor, compass, calculator.

As mentioned previously, Pythagoras’ theorem states:‘In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.’

In this investigation we consider replacing the word square/s with other geometric shapes.

1 What would happen if the word square/s were to be replaced by the word semicircle/s? Draw diagrams and investigate to see whether the statement would still be true in all cases. (Make sure that the triangles you draw are indeed right-angled.)

2 What would happen if we were to replace square/s with circle/s or equilateral triangle/s? Remember that it must work for all situations.

3 What are your conclusions?

4 Prepare a formal report on your investigation. Support your conclusions with detailed diagrams and mathematical evidence.

Constructing a staircaseResources: Pen, paper, calculator.

With an understanding of the theorem of Pythagoras, you should now be able to investigate the staircase problem.

Building stairs prompts the following questions:

1. How much timber should be ordered?

2. Where should the stairs start if we are to get the correct slope?

inve

stigationinvestigatio

n

3 cm

4 cm

5 cm

inve


n



We start by considering the general structure of a staircase.Mrs Kelly has contacted a local builder to construct an external set of stairs from the ground to an existing deck 2550 mm above the ground.1 Use the information in the diagram above

to determine:a the number of treads, 170 mm apart,

required to reach a height of 2550 mm (remember that no treads are necessary at the top or base of the staircase)

b the horizontal distance from the deck where the footings should be cemented (knowing that each tread is 250 mm wide)

c the length of each stringerd the length of timber that must be

ordered if the treads and stringers are constructed from 250-mm by 50-mm hardwood, and the stairs are 1 metre wide.

2 Draw a diagram for Mrs Kelly, showing the structure of her staircase (with the correct number of treads). Mark on your diagram all known measurements and indicate the total length of timber required for the job.

Constructing a paper boxResources: Paper, scissors.

How is it possible to determine the starting dimensions for constructing a box of a particular size? Pythagoras’ theorem has the answer.

1 Take a square piece of paper of a reasonable size.a Valley fold along all lines of symmetry.

Remember to crease all folds sharply.Open the square out flat.

b Fold each of the corners in a valley fold in towards the centre.

Treads

Stringers × 2

Footings

level Ground

170 mm

250 mm

250 mm

50 mm

inve


n



c Valley fold the top, bottom, left and right sides of the resulting shape in towards the centre, opening each fold before making the next. These folds will form the base of the box.

d Open two of the corner folds opposite each other as indicated.

e Fold up the edges opposite each other (those which already have centre folds) to make one pair of the box’s sides.

f Fold in both other ends as shown, completing the other two sides of the box.

2 With the base of the box completed, a lid could then be fashioned from another piece of square paper about 1 cm larger.



3 Having completed our box, the question now arises: ‘How is it possible to determine the dimensions of square paper required to construct a box of a particular size?’

a Open your box and look at all the creases formed. It should appear as below.

b The base of the box is outlined with respect to the size of the original square paper. Note that the box’s base is 2 ‘units’ in length, while the whole diagonal of the paper is 8 ‘units’ in length. This means that the box’s base is one-quarter the length of the diagonal of the original paper.

c What is the height of the box compared with the length of the diagonal of the original square?

4 We now have the knowledge to make a box with a side length of, say, 5 cm.

a The diagonal of our starting square must be4 × 5 cm = 20 cm

b Using Pythagoras’ theorem, we can now determine the length of the side of the square (S).


202 = S2 + S2

400 = 2S2

= S2

200 = S2

So S = = 14.1 cm

c So, starting with a piece of paper 14.1 cm square, after folding, the resulting box would be 2.5 cm high, with a side length of 5 cm.

d Verify these measurements for yourself by folding such a piece of paper.

e Try this exercise with other paper sizes.

This is yet another example of the importance of geometry in many facets of our daily lives.

S

S20 cm

4002

---------

200



Similar triangles and shadow sticksSometimes it is not possible to measure heights of objects such as trees, tall buildingsand mountains. It is, however, often possible to measure the length of a shadow cast bythese objects. If we compare the length of these shadows with the length of a shadowcast by a stick of known height under the same conditions, it is possible to calculateheights which are difficult to measure.

Consider the situation below.

We need to determine the height of the building. Take a stick whose height we know(or can measure) and place it, vertically, in the sunshine, near the building. The raysfrom the sun are parallel, so we have two similar right-angled triangles. (The buildingand the stick are at right angles to the ground.) We can measure both shadows.

The larger right-angled triangle formed by the building and its shadow is some scalefactor of the smaller right-angled triangle formed by the stick and the stick’s shadow.

Scale factor =

We can then apply the same scale factor to the height of the stick to determine theheight of the building. So,

Height of building = height of stick × scale factor

Using shadows to determine the height of a tall object

Resources: Tall object, shadow stick, measuring tape, sunshine!

We shall apply the theory just developed, to determine the height of a tall object in your school grounds.

1 Choose a tall, accessible object in your school grounds (tree, building, etc.) as the object whose height you wish to determine. Let its height be B metres.

BuildingStick

Stick’sshadow

Building’s shadow

Building

StickStick’sshadow

Building’s shadow

length of building shadowlength of stick shadow

--------------------------------------------------------------

inve


n

MQ Maths A Yr 11 - 05 3 Wednesday, July 4, 2001 4:39 PM


2 Measure the height of your shadow stick as S metres.

3 Place your shadow stick vertically in the sunshine near your tall object. Measure the length of its shadow on the ground. Let this be s metres.

4 Measure the length of the building’s shadow on the ground. Let this be b metres.

5 Draw a diagram representing this situation. Mark all known measurements on the diagram.

6 Calculate the scale factor F (tall-object shadow length divided by stick shadow

length; that is, F = ).

7 The height of your tall object can then be determined by multiplying the height of the stick by the scale factor; that is, B = S × F

8 Write up your activity, explaining all steps involved.

bs---

At the same time as a tree cast a shadow of 14 m, a 168-cm-tall girl cast a shadow of 140 cm. Find the height of the tree. Give the answer to 1 decimal place.

THINK WRITE

Identify the two similar triangles and draw them separately. (We assume that both the girl and the tree are perpendicular to the ground.)

Identify the side of the triangle whose length is required.

Calculate the scale factor. Note: Measurements must be in the same units.

Scale factor =

=

= 10Apply the scale factor to the girl’s height.

Height of tree = girl’s height × scale factor= 1.68 m × 10= 16.8 m

Write the final answer, specifying units. Height of tree is 16.8 m

Shadow(140 cm)

Sun’s

rays

Girl(168 cm)

14 metres

1

x

168 cm

140 cm 14 m

2

3length of tree shadow

length of girl’s shadow-------------------------------------------------------

14 m1.4 m-------------

4

5

7WORKEDExample



Shadow sticks

1 The following diagrams represent the measurements taken from shadows of sticks andobjects. Use the figures to determine the heights of the objects.a b

c d

2 At the same time as a building cast a shadow of 14.3 metres, a 2-metre stick cast ashadow of 5.3 metres. What is the height of the building?

3 A boulder on the shoreline cast a shadow of 15.8 m on the beach at the same time as a1.5-metre stick cast a shadow of 3.7 m. Determine the height of the boulder.

4 Find the height of the flagpole shown in the diagram at right.

5 The shadow of a tree is 4 metres and at the same time the shadow of a 1-metre stick is 25 cm. Assuming both the tree and stick are perpendicular to the horizontal ground, what is the height of the tree?

remember1. To determine the height of objects too difficult to measure, we can use the

‘shadow stick’ method.

2. Scale factor = .

3. Height of object = height of stick × scale factor.

length of shadow of objectlength of shadow of stick

----------------------------------------------------------------

remember

5BWORKEDExample

7

a

3.6 m

8.2 m

2.5 m

b

4.5 m

20 m

3.5 m

c

1.2 m

4.2 m

1.5 m

d

5.8 m

15 m

3 m

0.9 m

9 m1 m

Guy wire



Questions 6 and 7 refer to the followinginformation.A young tennis player’s serve is shown inthe diagram.

Assume the ball travels in a straight line.

6

The height of the ball just as it is hit, x, isclosest to:A 3.6 m B 2.7 m C 2.5 mD 1.8 m E 1.6 m

7

The height of the player, y, as shown isclosest to:A 190 cm B 180 cm C 170

cmD 160 cm E 150 cm

Calculating trigonometric ratiosWe have already looked at Pythagoras’ theorem, which enabled us to find the length ofone side of a right-angled triangle given the lengths of the other two. However, to dealwith other relationships in right-angled triangles, we need to turn to trigonometry.

Trigonometry allows us to work with the angles also; that is, deal with relationshipsbetween angles and sides of right-angled triangles. For example, trigonometry enablesus to find the length of a side, given the length of another side and the magnitude of anangle.

So that we are clear about which lines and angles we are describing, we need to identify the given angle, and name the shorter sides with reference to it. For this reason, we label the sides opposite and adjacent — that is, the sides opposite and adjacent to the given angle. The diagram shows this relationship betweenthe sides and the angle, θ.

The trigonometric ratios are constant for a particular angle, and this is the reason theshadow-stick method worked.

x

y1.

1 m

5 m 10 m

0.9 m



WorkS

HEET 5.1

hypotenuseopposite

adjacentθ



Trigonometry uses the ratio of side lengths to calculate the lengths of sides and the sizeof angles. The ratio of the opposite side to the adjacent side is called the tangent ratio.This ratio is fixed for any particular angle.

The tangent ratio for any angle, θ, can be found using the result:

In the investigation above, we found that for a 30° angle the ratio was approximately0.58. We can find a more accurate value for the tangent ratio on a calculator by usingthe button.

Calculators require a particular sequence of button presses in order to perform thiscalculation. Investigate the sequence required for your particular calculator.

For all calculations in trigonometry you will need to make sure that your calculatoris in DEGREES MODE. For most calculators you can check this by looking for a DEGin the display.

Looking at the tangent ratioResources: Ruler, spreadsheet.

The tangent ratio for a given angle is a ratio of sides in similar right-angled triangles, such as those in the diagram. ∠BAC is common to each triangle and is equal to 30°. We are going to look at the ratio of the opposite side length to the adjacent side length in each triangle.

1 Complete each of the following measurements.a BC = mm AB = mmb DE = mm AD = mmc FG = mm AF = mmd HI = mm AH = mm

2 Set up the spreadsheet below:

3 Enter values in columns 2 and 3 from the measurements above.

4 Provide a formula in column 4 to calculate the ratio of the opposite side length to the adjacent side length. This is the tangent ratio.

5 The ∠BAC is common to each triangle. Consider the calculated values in column 4. What is your conclusion?

inve


n

AB

C

E

G

I

DFH

TriangleOpposite side

lengthAdjacent side

length

ABC

ADE

AFG

AHI

OppositeAdjacent-----------------------

tan θ opposite sideadjacent side-------------------------------=

tan



When measuring angles:1 degree = 60 minutes1 minute = 60 secondsYou need to be able to enter angles using both degrees and minutes into your calcu-

lator. Most calculators use a (Degrees, Minutes, Seconds) button or a button. Check with your teacher to see how to do this.

The tangent ratio is used to solve problems involving the opposite side and the adjacentside of a right-angled triangle. The tangent ratio does not allow us to solve problemsthat involve the hypotenuse.

The sine ratio (abbreviated to sin; pronounced sine) is the name given to the ratio ofthe opposite side and the hypotenuse.

Looking at the sine ratioResources: Ruler, spreadsheet.

The tangent ratio for a given angle is a ratio of the opposite side and the adjacent side in a right-angled triangle. The sine ratio is the ratio of the opposite side and the hypotenuse.

1 Complete each of the following measurements as before. (As we saw earlier, ∠BAC is common to all these similar triangles and so in this exercise, we look at the ratio of the side opposite ∠BAC to the hypotenuse of each triangle.)

a BC = mm AC = mm

b DE = mm AE = mm

c FG = mm AG = mm

d HI = mm AI = mm

DMS ° ’ ”

Using your calculator, find the following, correct to 3 decimal places.

a tan 60° b 15 tan 75° c d tan 49°32′

THINK WRITE/DISPLAY

a Press and enter 60, then press . a tan 60° = 1.732

b Enter 15, press and , enter 75, then press .

b 15 tan 75° = 55.981

c Enter 8, press and , enter 69, then press .

c = 3.071

d Press , enter 49, press , enter 32, press , then press .

d tan 49°32′ = 1.172

Note: Some calculators require that the angle size be entered before the trigonometric functions.

8tan 69°------------------

tan =× tan

=

÷ tan=

8tan 69°-----------------

tan DMSDMS =

8WORKEDExample

inve


n

AB

C

E

G

I

DFH



In any right-angled triangle with equal angles, the ratio of the length of the oppositeside to the length of the hypotenuse will remain the same, regardless of the size of thetriangle. The formula for the sine ratio is:

The value of the sine ratio for any angle is found using the sin function on the calcu-lator.

sin 30° = 0.5Check this on your calculator.

2 Set up the spreadsheet below:

3 Enter values in columns 2 and 3 from the measurements above.

4 Provide a formula in column 4 to calculate the ratio of the length of the opposite side to the length of the hypotenuse. This is the sine ratio.

5 The ∠BAC is common to each triangle. You should notice that the values in column 4 are the same (or very close, allowing for measurement error).

TriangleOpposite side

lengthHypotenuse

length

ABC

ADE

AFG

AHI

OppositeHypotenuse------------------------------

sin θ opposite sidehypotenuse

-------------------------------=

Find, correct to 3 decimal places:

a sin 57° b 9 sin 45° c d 9.6 sin 26°12′.

THINK WRITE/DISPLAY

a Press and enter 57, then press . a sin 57° = 0.839


b 9 sin 45° = 6.364

c Enter 18, press and , enter 44, then press .

c = 25.912

d Enter 9.6, press and , enter 26, press , enter 12, press , then press .

d 9.6 sin 26°12′ = 4.238

Note: Check the sequence of button presses required by your calculator.

18sin 44°-----------------

sin =× sin

=

÷ sin=

18sin 44°-----------------

× sinDMS DMS=

9WORKEDExample



A third trigonometric ratio is the cosine ratio. This ratio compares the length of theadjacent side and the hypotenuse.

The cosine ratio is found using the formula:

To calculate the cosine ratio for a given angle on your calculator, use the cos func-tion. On your calculator check the calculation:

cos 30° = 0.866

Similarly, if we are given the sine, cosine or tangent of an angle, we are able to calcu-late the size of that angle using the calculator. We do this using the inverse functions.On most calculators these are the second function of the sin, cos and tan functions andare denoted sin−1, cos−1 and tan−1.

Looking at the cosine ratioResources: Ruler, spreadsheet.

1 Complete each of the following measurements.a AB = mm AC = mmb AD = mm AE = mmc AF = mm AG = mmd AH = mm AI = mm

2 Set up a spreadsheet of similar format to the two previous spreadsheets to calculate the ratio of the length of the adjacent side to that of the hypotenuse.

3 You should again find this ratio constant (or nearly so). This is the cosine ratio.

inve


n

AB

C

E

G

I

DFH

cos θ adjacent sidehypotenuse

-------------------------------=

Find, correct to 3 decimal places:

a cos 27° b 6 cos 55° c d .

THINK WRITE/DISPLAY

a Press and enter 27, then press . a cos 27° = 0.891


b 6 cos 55° = 3.441

c Enter 21.3, press and , enter 74, then press .

c = 77.275

d Enter 4.5, press and , enter 82, press , enter 46, press , then press .

d = 35.740

Note: Check the sequence requirements for your calculator.

21.3cos 74°------------------

4.5cos 82°46′--------------------------

cos =× cos

=

÷ cos=

21.3cos 74°------------------

× cosDMS DMS=

4.5cos 82°46′--------------------------

10WORKEDExample



Problems sometimes supply angles in degrees, minutes and seconds, or requireanswers to be written in the form of degrees, minutes and seconds. On most calcu-lators, you will use the DMS (Degrees, Minutes, Seconds) function or the function.If you are using a graphics calculator, the ‘angle’ function provides this facility. Checkwith your teacher for the requirements of your particular calculator.

Find θ, correct to the nearest degree, given that sin θ = 0.738.

THINK WRITE/DISPLAY

Press [sin–1] and enter .738, then press .Round your answer to the nearest degree.

θ = 48°

Note: Check the sequence requirements for your calculator.

1 2nd F=

2

11WORKEDExample

° ’ ”

Given that tan θ = 1.647, calculate θ to the nearest minute.

THINK WRITE/DISPLAY

Press [tan–1] and enter 1.647, then press .Convert your answer to degrees and minutes by pressing .

θ = 58°44′

1 2nd F=

2DMS

12WORKEDExample

remember1. The tangent ratio is the ratio of the opposite side and the adjacent side.

tan θ =

2. The sine ratio is the ratio of the opposite side and the hypotenuse.

sin θ =

3. The cosine ratio is the ratio of the adjacent side and the hypotenuse.

cos θ =

4. The value of the trigonometric ratios can be found using the sin, cos and tan functions on your calculator.

5. The angle can be found when given the trigonometric ratio using the sin−1, cos−1 and tan−1 functions on your calculator.

opposite sideadjacent side-------------------------------

opposite sidehypotenuse

-------------------------------

adjacent sidehypotenuse

-------------------------------

remember



Calculating trigonometric ratios

1 Calculate the value of each of the following, correct to 3 decimal places.



4 Calculate the value of each of the following, correct to 3 decimal places, if necessary.


6 Find θ, correct to the nearest degree, given that sin θ = 0.167.

7 Find θ, correct to the nearest degree, given that:

8 Find θ, correct to the nearest minute, given that cos θ = 0.058.

9 Find θ, correct to the nearest minute, given that:

Finding an unknown sideWe can use the trigonometric ratios to find the length of one side of a right-angled triangle if we know the length of another side and an angle. Consider the triangle at right.

In this triangle we are asked to find the length of the opposite side and have been given the length of the adjacent side.

a tan 57° b 9 tan 63° c d tan 33°19′

a sin 37° b 9.3 sin 13° c d

a cos 45° b 0.25 cos 9° c d 5.9 cos 2°3′

a sin 30° b cos 15° c tan 45°d 48 tan 85° e 128 cos 60° f 9.35 sin 8°

g h i

a sin 24°38′ b tan 57°21′ c cos 84°40′d 9 cos 55°30′ e 4.9 sin 35°50′ f 2.39 tan 8°59′

g h i

a sin θ = 0.698 b cos θ = 0.173 c tan θ = 1.517.

a tan θ = 0.931 b cos θ = 0.854 c sin θ = 0.277.

5CWORKEDExample

8 8.6tan 12°-----------------

WORKEDExample

9 14.5sin 72°----------------- 48

sin 67°40′-------------------------

WORKEDExample

10 6cos 24°------------------

4.5cos 32°------------------ 0.5

tan 20°----------------- 15

sin 72°-----------------

19tan 67°45′------------------------- 49.6

cos 47°25′-------------------------- 0.84

sin 75°5′----------------------

WORKEDExample

11

WORKEDExample

12

hypopp

adj14 cm

30°x



We know from the formula that: tan θ = . In this example, tan 30° = . From

our calculator we know that tan 30° = 0.577. We can set up an equation that will allowus to find the value of x.

tan θ =

tan 30° =

x = 14 tan 30°= 8.083 cm

In the example above, we were told to use the tangent ratio. In practice, we need to beable to look at a problem and then decide if the solution is found using the sin, cos ortan ratio. To do this we need to examine the three formulas.

tan θ =

We use this formula when we are finding either the opposite or adjacent side and aregiven the length of the other.

sin θ =

The sin ratio is used when finding the opposite side or the hypotenuse when giventhe length of the other.

cos θ =

The cos ratio is for problems where we are finding the adjacent side or the hypot-enuse and are given the length of the other.

To make the decision we need to label the sides of the triangle and make a decisionbased on these labels.

oppositeadjacent-------------------- x

14------

oppadj---------

x14------

Use the tangent ratio to find the value of h in the triangle at right, correct to 2 decimal places.

THINK WRITE

Label the sides of the triangle opp, adj and hyp.

Write the tangent formula. tan θ =

Substitute for θ (55°) and the adjacent side (17 cm).

tan 55° =

Make h the subject of the equation. h = 17 tan 55°Calculate. = 24.28 cm

17 cm55°

h

1

hypopp

adj17 cm

55°

h

2oppadj---------

3h

17------

45

13WORKEDExample

Cabri Geometry

Trigratiosopposite side

adjacent side-------------------------------

opposite sidehypotenuse

-------------------------------

adjacent sidehypotenuse

-------------------------------



To remember each of the formulas more easily, we can use this acronym:SOHCAHTOA

We pronounce this acronym as ‘Sock ca toe her’. The initials of the acronym represent the three trigonometric formulas.

sin θ = cos θ = tan θ =

Care needs to be taken at the substitution stage. In the previous two examples, theunknown side was the numerator in the fraction, hence we multiplied to find theanswer. If after substitution, the unknown side is in the denominator, the final step isdone by division.

Find the length of the side marked x, correct to 2 decimal places.

THINK WRITE

Label the sides of the triangle.

x is the opposite side and 24 m is the hypotenuse, therefore use the sin formula.

sin θ =

Substitute for θ and the hypotenuse. sin 50° =

Make x the subject of the equation. x = 24 sin 50°Calculate. = 18.39 m

24 m

50°

x

1hyp

opp

adj

24 m

50°

x

2opphyp---------

3x

24------

45

14WORKEDExample

SOH

opphyp---------

CA

H

adjhyp---------

TOA

oppadj---------

Find the length of the side marked z in the triangle at right.

THINK WRITE

Label the sides opp, adj and hyp.

Choose the cosine ratio because we are finding the hypotenuse and have been given the adjacent side.

Write the formula. cos θ =

12.5 m

23°15'z

1

opp

hyp

adj12.5 m

23°15'z

2

3adjhyp---------

15WORKEDExample



Trigonometry is used to solve many practical problems. In these cases, it is necessaryto draw a diagram to represent the problem and then use trigonometry to solve theproblem. With written problems that require you to draw the diagram, it is necessary togive the answer in words.

THINK WRITE

Substitute for θ and the adjacent side. cos 23°15′ =

Make z the subject of the equation. z cos 23°15′ = 12.5

z =

Calculate. = 13.60 m

Note: In calculating the length of the hypotenuse, the process will always involve division by the trigonometric function.

412.5

z----------

5

12.5cos 23°15′--------------------------

6

A flying fox is used in an army training camp. The flying fox is supported by a cable that runs from the top of a cliff face to a point 100 m from the base of the cliff. The cable makes a 15° angle with the horizontal. Find the length of the cable used to support the flying fox.

THINK WRITE

Draw a diagram and show information.


Choose the cosine ratio because we are finding the hypotenuse and have been given the adjacent side.

Write the formula. cos θ =

Substitute for θ and the adjacent side. cos 15° =

Make f the subject of the equation. f cos 15° = 100

f =

Calculate. = 103.5 m

Give a written answer. The cable is approximately 103.5 m long.

1

15°100 m

opp

hyp

adj

f

2

3

4adjhyp---------

5100

f---------

6

100cos 15°------------------

7

8

16WORKEDExample



Finding an unknown side

1 Label the sides of each of the following triangles, with respect to the angle markedwith the pronumeral.a b c

2 Use the tangent ratio to find the length of the side marked x (correct to 1 decimal place).

3 Use the sine ratio to find the length of the side marked a (correct to 2 decimal places).

4 Use the cosine ratio to find the length of the side marked d (correct to nearest whole number).

remember1. Trigonometry can be used to find a side in a right-angled triangle when we are

given the length of one side and the size of an angle.

2. The trig formulas are:

sin θ = cos θ = tan θ =

3. Take care to choose the correct trigonometric ratio for each question.

4. Substitute carefully and note the change in the calculation, depending upon whether the unknown side is in the numerator or denominator.

5. Before using your calculator, check that it is in degrees mode.

6. Be sure that you know how to enter degrees and minutes into your calculator.

7. Written problems will require you to draw a diagram and give a written answer.

opphyp--------- adj

hyp--------- opp

adj---------

remember

5D

θα

γ

WORKEDExample

13

51 mm71°

x

13 m

23°a

35 cm31°

d


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 1875 The following questions use the tan, sin or cos ratios in their solution. Find the size of

the side marked with the pronumeral, correct to 1 decimal place.

a b c

6 Find the length of the side marked with the pronumeral in each of the following(correct to 1 decimal place).

a b c

7 Find the length of the side marked with the pronumeral in each of the following(correct to 1 decimal place).

a b c

d e f

g h i

j k l

WORKEDExample

14

68°

13 cmx

49°48 m

y41°

12.5 km

z

WORKEDExample

15

21°

4.8 m

t

77°

87 mm

p

36°8.2 m

q

23°

2.3 m

a

39°

0.85 kmb76°

8.5 km

x

116 mm

9°

m

16.75 cm11°

d

13°

64.75 m

x

83°

44.3 m

x

20°15.75 km

g

2.34 m

84°9'

m

84.6 km

60°32'q

21.4 m75°19't

26.8 cm

29°32'

r



8

Look at the diagram at right and state which of the following is correct.

9

Study the triangle at right and state which of the following is correct.

10

Which of the statements below is not correct?A The value of tan θ can never be greater than 1.B The value of sin θ can never be greater than 1.C The value of cos θ can never be greater than 1.D tan 45° = 1

11

Study the diagram at right and state which of the statements is correct.

12 A tree casts a 3.6 m shadow when the sun’s angle of elevation is 59°. Calculate theheight of the tree, correct to the nearest metre.

13 A 10-m ladder just reaches to the top of a wall when it is leaning at 65° to the ground.How far from the foot of the wall is the ladder (correct to 1 decimal place)?

14 The diagram at right shows the paths of two ships, A and B, after they have left port.If ship B sends a distress signal, how far must ship A sail to give assistance (to the nearest kilometre).

15 A rectangular sign 13.5 m wide has a diagonal brace that makes a 24° angle with the horizontal.a Draw a diagram of this situation.b Calculate the height of the sign, correct to the nearest metre.

16 A wooden gate has a diagonal brace built in for support. The gate stands 1.4 m highand the diagonal makes a 60° angle with the horizontal. a Draw a diagram of the gate.b Calculate the length that the diagonal brace needs to be.

17 The wire support for a flagpole makes a 70° angle with the ground. If the support is3.3 m from the base of the flagpole, calculate the length of the wire support (correctto 2 decimal places).

A x = 9.2 sin 69° B

C x = 9.2 cos 69° D

A tan φ = B tan φ = C sin φ = D cos φ =

A w = 22 cos 36° B

C w = 22 cos 54° D w = 22 sin 54°


69°9.2

xx

9.2sin 69°-----------------=

x9.2

cos 69°------------------=


1715

8

φ

815------ 15

8------ 15

17------ 8

17------



36°

22 mmw

w22

sin 36°-----------------=

WORKEDExample

16

60°Port

23 km

A

B


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 18918 A ship drops anchor vertically with an

anchor line 60 m long. After one hour the anchor line makes a 15° angle with the vertical.a Draw a diagram of this situation.b Calculate the depth of water, correct

to the nearest metre.c Calculate the distance that the ship

has drifted, correct to 1 decimal place.

Find the size of the side marked with the pronumeral in each of the following. Wherenecessary, give your answer correct to 1 decimal place.

1 2

3 4

5 6

7 8

9 10

1

15 cm

8 cm

a

10 m 20 m

b

30 km

40 km

c 21°23 md

40°39.2 cm

e

37°42.1 m

f

50°

14.9 mm

g42°

119 mm

h

17°93.2 m i 45°

17 mj



Finding anglesSo far, we have concerned ourselves with finding side lengths. We are also able to usetrigonometry to find the sizes of angles when we have been given side lengths. We needto reverse our previous processes.

Consider the triangle at right.We want to find the size of the angle marked θ.

Using the formula sin θ = we know that in this triangle

sin θ =

=

= 0.5

We then calculate sin−1 (0.5) to find that θ = 30°. As with all trigonometry it is important that you have your calculator set to degrees

mode for this work.

In many cases we will need to calculate the size of an angle, correct to the nearestminute. The same method for finding the solution is used; however, you will need touse your calculator to convert to degrees and minutes.

10 cm5 cm

θopphyp---------

510------

12---

Find the size of angle θ, correct to the nearest degree, in the triangle at right.

THINK WRITE

Label the sides of the triangle and choose the tan ratio.

tan θ =

Substitute for the opposite and adjacent sides in the triangle.

=

Make θ the subject of the equation. θ = tan−1

Calculate. = 33°

6.5 m

4.3 mθ

1

4.3 m

6.5 madj

hypopp

θ

oppadj---------

24.36.5-------

34.36.5-------

4

17WORKEDExample



The same methods can be used to solve problems. As with finding sides, we set thequestion up by drawing a diagram of the situation.

Find the size of the angle θ at right, correct to the nearest minute.

THINK WRITE

Label the sides of the triangle and choose the sin ratio.

sin θ =

Substitute for the opposite side and the hypotenuse in the triangle.

=

Make θ the subject of the equation. θ = sin−1

Calculate and convert your answer to degrees and minutes.

= 40°23′

4.6 cm

7.1 cmθ1 opp

hypadj

4.6 cm

7.1 cmθ

opphyp---------

24.67.1-------

34.67.1-------

4

18WORKEDExample

A ladder is leant against a wall. The foot of the ladder is 4 m from the base of the wall and the ladder reaches 10 m up the wall. Calculate the angle that the ladder makes with the ground.

THINK WRITE

Draw a diagram and label the sides.

Choose the tangent ratio and write the formula.

tan θ =

Substitute for the opposite and adjacent sides. =


Calculate. = 68°12′Give a written answer. The ladder makes an angle of 68°12′ with

the ground.

1

4 m

10 m

adj

opphyp

θ

2oppadj---------

3104

------

4104

------

56

19WORKEDExample



Finding angles

1 Use the tangent ratio to find the size of the angle marked with the pronumeral in eachof the following, correct to the nearest degree.

a b c

2 Use the sine ratio to find the size of the angle marked with the pronumeral in each ofthe following, correct to the nearest minute.

a b c

3 Use the cosine ratio to find the size of the angle marked with the pronumeral in eachof the following, correct to the nearest minute.

a b c

remember1. Make sure that the calculator is in degrees mode.2. To find an angle given the trig ratio, press or and then the

appropriate ratio button.3. Be sure to know how to get your calculator to display an answer in degrees and

minutes. When rounding off minutes, check if the number of seconds is greater than 30.

4. When solving triangles remember the SOHCAHTOA rule to choose the correct formula.

5. In written problems draw a diagram and give an answer in words.

2nd F SHIFT

remember

5EWORKEDExample

17

7 m

12 mθ 11 m

3 mφ

25 mm

162 mmγ

WORKEDExample

18

24 m13 m

θ

4.6 m

6.5 mθ

9.7 km

5.6 km

α

15 cm

9 cmθ

2.6 m4.6 mα

27.8 cm

19.5 cmβ


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 1934 In the following triangles, you will need to use all three trig ratios. Find the size of the

angle marked θ, correct to the nearest degree.

a b c

d e f

5 In each of the following find the size of the angle marked θ, correct to the nearestminute.

a b c

d e f

6

Look at the triangle drawn at right.Which of the statements below is correct?

7

Consider the triangle drawn at right.θ is closest to:

8

The exact value of sin . The angle θ =

A ∠ABC = 30° B ∠ABC = 60°C ∠CAB = 30° D ∠ABC = 45°

A 41°55′ B 41°56′ C 48°4′ D 48°5′

A 30° B 45° C 60° D 90°

7 cm

11 cm

θ

15 cm

8 cm

θ14 cm

9 cm

θ

3.6 m

9.2 m

θ

196 mm32 mm

θ 14.9 m26.8 m

θ

30 m

19.2 m

θ

10 cm

63 cm

θ

2.5 m0.6 m

θ

3.5 m

18.5 m

θ16.3 m

8.3 m

θ

6.3 m

18.9 m

θ


10 cm5 cm

A

BCθ


12.5 m9.3 m

θ


θ 32

-------=



9 A 10-m ladder leans against a wall 6 m high. Find the angle that the ladder makeswith the horizontal, correct to the nearest degree.

10 A kite is flying on a 40-m string. The kite is flying 10 m away from the vertical as shown in the figure at right.Find the angle the string makes with the horizontal, correct to the nearest minute.

11 A ship’s compass shows a course due east of the port from which it sails. After sailing 10 nautical miles, it is found that the ship is 1.5 nautical miles off course as shown in the figure below.

Find the error in the compass reading, correct to the nearest minute.

12 The diagram below shows a footballer’s shot at goal.

By dividing the isosceles triangle in half in order to form a right-angled triangle, calculate, to the nearest degree, the angle within which the footballer must kick to get the ball to go between the posts.

13 A golfer hits the ball 250 m, but 20 m off centre. Calculate the angle at which the ball deviated from a straight line, correct to the nearest minute.

WORKEDExample

19

40 m

10 m

kite

10 nmPort

1.5 nm

7 m

30 m



1 Find the length of the hypotenuse in the triangle at right.

2 Find x, correct to 1 decimal place.

3 Find y, correct to 1 decimal place.

4 Find z, correct to 3 decimal places.

5 Calculate sin 56°, correct to 4 decimal places.

6 Calculate 9.2 tan 50°, correct to 2 decimal places.

7 Calculate ,

correct to 2 decimal places.

8 Find θ, given that sin θ = 0.5.

9 Find θ to the nearest degree, given that cos θ = 0.299.

10 Find θ to the nearest minute, given that tan θ = 2.

2

12 cm

5 cm

12 cm

12 cmx

30 cm

20 cmy

3.7 m

7.4 m z

132cos 8°45′-----------------------



Angles of elevation and depressionThe angle of elevation is measured upwards from a horizontal and refers to the angleat which we need to look up to see an object. Similarly, the angle of depression is theangle at which we need to look down from the horizontal to see an object.

We are able to use the angles of elevation and depression to calculate the heights anddistances of objects that would otherwise be difficult to measure.

In practical situations, the angle of elevation is measured using a clinometer. Therefore,the angle of elevation is measured from a person’s height at eye level. For this reason,the height at eye level must be added to the calculated answer.

Angle of elevation

Horizontal

Angle of depression

Horizontal

From a point 50 m from the foot of a building, the angle of elevation to the top of the building is measured as 40°. Calculate the height, h, of the building, correct to the nearest metre.

THINK WRITE

Label the sides of the triangle opp, adj and hyp.Choose the tangent ratio because we are finding the opposite side and have been given the adjacent side.

Write the formula. tan θ =

Substitute for θ and the adjacent side.

tan 40° =

Make h the subject of the equation. h = 50 tan 40°Calculate. = 42 mGive a written answer. The height of the building is approximately 42 m.

50 m40°

h

1

50 m40°

hypopp

h

adj

2

3oppadj---------

4 h50------

56

7

20WORKEDExample

Bryan measures the angle of elevation to the top of a tree as 64°, from a point 10 m from the foot of the tree. If the height of Bryan’s eyes is 1.6 m, calculate the height of the tree, correct to 1 decimal place.

10 m 1.6 m

64°

21WORKEDExample



A similar method for finding the solution is used for problems that involve an angle ofdepression.

THINK WRITE

Label the sides opp, adj and hyp.Choose the tangent ratio because we are finding the opposite side and have been given the adjacent side.


Substitute for θ and the adjacent side. tan 64° =

Make h the subject of the equation. h = 10 tan 64°Calculate h. = 20.5 mAdd the eye height. Height of tree = 20.5 + 1.6

Height of tree = 22.1Give a written answer. The height of the tree is approximately 22.1 m.

1

10 madj

hypopp

h

64°

2

3oppadj---------

4h

10------

56

7

8

When an aeroplane in flight is 2 km from a runway, the angle of depression to the runway is 10°. Calculate the altitude of the aeroplane, correct to the nearest metre.

THINK WRITE


Choose the tan ratio, because we are finding the opposite side given the adjacent side.


Substitute for θ and the adjacent side, converting 2 km to metres.

tan 10° =

Make h the subject of the equation. h = 2000 tan 10°Calculate. = 353 m

Give a written answer. The altitude of the aeroplane is approximately 353 m.

10°2 km

h

1

10°2 km

h

adj

hypopp

2

3oppadj---------

4h

2000------------

5

6

7

22WORKEDExample



Angles of elevation and depression can also be calculated by using known measure-ments. This is done by drawing a right-angled triangle to represent a situation.

Angles of elevation and depression

1 From a point 100 m from the foot of a building, the angle of elevation to the top of the building is 15°. Calculate the height of the building, correct to 1 decimal place.

2 From a ship the angle of elevation to an aeroplane is 60°. The aeroplane is 2300 m due north of the ship.Calculate the altitude of the aeroplane, correct to the nearest metre.

A 5.2-m building casts a 3.6-m shadow. Calculate the angle of elevation of the sun, correct to the nearest degree.

THINK WRITE

Label the sides opp, adj and hyp.Choose the tan ratio because we are given the opposite and adjacent sides.


Substitute for opposite and adjacent. tan θ =


Calculate. = 55°Give a written answer. The angle of elevation of the sun is approximately 55°.

θ3.6 m

5.2 m

1

θ3.6 madj

hypopp

5.2 m2

3oppadj---------

45.23.6-------

55.23.6-------

6

7

23WORKEDExample

remember1. The angle of elevation is the angle at which you look up to see an object. 2. The angle of depression is the angle at which you look down to see an object.3. Problems can be solved by using angles of elevation and depression with the

aid of a diagram.4. Written problems should be given an answer in words.

remember

5FWORKEDExample

20

100 m15°

60°2300 m


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 1993 From a point out to sea, a ship sights the top of a lighthouse

at an angle of elevation of 12°. It is known that the top of the lighthouse is 40 m above sea level. Calculate the distance of the ship from the lighthouse, correct to the nearest 10 m.

4 From a point 50 m from the foot of a building, Rod sights the top of a building at an angle of elevation of 37°.Given that Rod’s eyes are at a height of 1.5 m, calculate the height of the building, correct to 1 decimal place.

5 Richard is flying a kite and sights the kite at an angle of elevation of 65°. The altitude of the kite is 40 m and Richard’s eyes are at a height of 1.8 m.Calculate the length of string the kite is flying on, correct to 1 decimal place.

6 Bettina is standing on top of a cliff, 70 m above sea level. She looks directly out to sea and sights a ship at an angle of depression of 35°. Calculate the distance of the ship from shore.

7 From an aeroplane flying at an altitude of 4000 m, the runway is sighted at an angle of depression of 15°. Calculate the distance of the aeroplane from the runway, correct to the nearest kilometre.

8 There is a fire on the fifth floor of a building. The closest a fire truck can get to the building is 10 m. The angle of elevation from this point to where people need to be rescued is 69°. If the fire truck has a 30-m ladder, can the ladder be used to make the rescue?

9 From a navy vessel, a beacon which is 80 m above sea level is sighted at an angle of elevation of 5°. The vessel sailed towards the beacon and thirty minutes later the beacon is at an angle of elevation of 60°. Use the diagram at right to complete the following.a Calculate the distance that the vessel was from the beacon, when the angle of

elevation to the beacon was 5° (the distance AC).b Calculate the distance that the vessel sailed in the 30 minutes between the two

readings.

10 A 12 m high building casts a shadow 15 m long. Calculate the angle of elevation of the sun, to the nearest degree.

40 m12°

x

WORKEDExample

21

1.5 m 50 m37°

40 m

1.8 m

65°

x

WORKEDExample

2235°

70 m

15°4000 m

69°10 m

5° 60°

80 m

A

D

CB

WORKEDExample

23θ

12 m

15 m



11 An aeroplane which is at an altitude of 1500 m is 4000 m from a ship in a horizontal direction, as shown at right.Calculate the angle of depression from the aeroplane to the ship.

12 The angle of elevation to the top of a tower is 12° from a point 400 m from the foot of the tower.a Draw a diagram of this situation.b Calculate the height of the tower,

correct to 1 decimal place.c Calculate the angle of elevation

to the top of the tower, from a point 100 m from the foot of the tower.

Alternative method to determine the height of a tall object

Resources: Measuring tape, clinometer or protractor, calculator.

In a previous investigation, you determined the height of a tall object using a shadow stick, observing the lengths of shadows. An alternative method could be employed using a clinometer. This is a device which measures angles. Your school may have one. If not, a protractor could be used as a substitute (although the accuracy of the reading would not be as great).

1 Locate a tall building or tree whose height you wish to determine.

2 Take a position some distance from its base, and measure your horizontal distance from the base.

3 Use the measuring tape to measure your height at eye level.

4 Take the clinometer (or protractor) and measure the angle of elevation from your eye level to the top of the tall object.

5 Draw a diagram representing the situation, labelling known lengths and angles.

6 Use a trigonometric ratio to determine the height of your tall building (or tree). (Remember to allow for your height at eye level in your calculations.)

7 Write up this activity, showing detailed diagrams and procedures used to obtain the height of your tall object.

θ1500 m

4000 m

WorkS

HEET 5.2

inve


n



201

Revisiting staircases

Resources: Pen, paper (graph paper if possible), calculator.

The mathematics used in constructing a staircase is more complex than you may at first suspect. Earlier in this chapter we looked at a simple staircase. Let us now examine the mathematics in more detail. The Home Building Code lays down construction standards for staircases. As a starting point, it is necessary to understand some key terms.

Rise and going

Rise

and

going

refer to the staircase treads, and their arrangement and construction.

Rise

The

rise

is the vertical measurement from the upper surface of one tread to the upper surface of the next tread.

Rise of flight

The

rise of flight

is the vertical measurement taken from a floor or landing to the top of the next floor or landing; that is, the height of the staircase.

Going

The term

going

is the horizontal measurement from the front edge of one tread to the front edge of the next.

Going of flight

The

going of flight

is the horizontal measurement for one flight, taken from the face of the first or bottom step to the face of the last or top step in that flight.

The figure below shows rise (

R

) and going (

G

) labelled in a staircase.

The home building code sets the following maximums and minimums:

It also states that the sum of two times the rise plus the going must be a maximum of 630 and a minimum of 585. Using

R

to represent the rise and

G

to represent going, this could be written as:

2

R

+

G

=

585 to 630 mm

The building code also states that rise and going must be constant for any one staircase. (Imagine walking up a staircase that had stair heights of varying sizes!)

inve


n

Rise R

Rise of flight

Going of flight

R

Going G

GR G

R GR

Max. (mm) Min. (mm)

Rise 190 115

Going 395 250

MQ Maths A Yr 11 - 05 Page 201 Friday, July 6, 2001 8:50 AM


The following figure shows a staircase with 9 rises. The number of treads used is 8, one less than the number of rises, because the top floor acts as the final step. This staircase has 8 equal goings.

Task 1A staircase is to have a rise of flight of 2844 mm. The builder wants to use a rise of approximately 160 mm.

1 Within the building code regulations, determine:a the number of risesb an appropriate rise (R) measurementc a suitable going (G) measurement.

2 Draw a diagram showing the design of the staircase. Include all measurements. Provide calculations to confirm that your staircase conforms to the home building code.

Task 2Often, there are restrictions on the space available for the location of a staircase, and the going must be kept to a minimum.

Tony and Mary are renovating their home and wish to install a staircase. The rise of flight is 2608 mm and the restricted going of flight is 3945 mm.

1 Find a suitable rise, if a rise of approximately 160 mm is desired.

2 State the number of treads.

3 Determine the going.

4 Calculate the number of rises.

5 Check that the dimensions are in accordance with the building code.

6 Draw a diagram of your completed staircase, indicating all measurements.

Going

Going of flight

Rise of flight

Existingtop floor

Rise



Pythagoras’ theorem• When using Pythagoras’ theorem to calculate the length of the hypotenuse of a

right-angled triangle, the formula is:hypotenuse2 = base2 + height2

• To find one of the shorter sides of a right-angled triangle, the formula is:base2 = hypotenuse2 − height2

or height2 = hypotenuse2 − base2

• Pythagoras’ theorem can be used to test whether a triangle is right-angled.If hypotenuse2 = base2 + height2, the triangle is right-angled.If hypotenuse2 > base2 + height2, the triangle is obtuse-angled.If hypotenuse2 < base2 + height2, the triangle is acute-angled.

• Pythagorean triads or Pythagorean triples are sets of three numbers that satisfy Pythagoras’ theorem.

Shadow sticks• Shadow sticks can be used to determine the height of objects too difficult to measure.• Height of the object = height of shadow stick × scale factor

Trigonometry formulas for right-angled triangles

• tan θ =

• sin θ =

• cos θ =

• SOHCAHTOA — this acronym will help you remember trig formulas.

Steps to find a side of a right-angled triangle• Label the sides of the triangle opposite, adjacent and hypotenuse.• Choose the correct ratio.• Substitute given information.• Make the unknown side the subject of the equation.• Calculate.

Steps to find an angle in a right-angled triangle• Label the sides of the triangle opposite, adjacent and hypotenuse.• Choose the correct ratio.• Substitute given information.• Make the unknown angle the subject of the equation.• Calculate by using an inverse trig function.

Angles of elevation and depression• The angle of elevation is the angle we look up from the horizontal to see an object.• The angle of depression is the angle we look down from the horizontal to see an

object.• Problems are solved using angles of elevation and depression by the same methods

as for all right-angled triangles.

summary

oppadj---------

opphyp---------

adjhyp---------



1 Find the length of the side marked with a pronumeral, in each case writing your answer correct to 2 decimal places.a b c

d e f

2 To travel between the towns of Bolong and Molong, you need to travel west along a road for 45 km, then north along another road for another 87 km. Calculate the straight-line distance between the two towns.

3 A rope is 80 m long and runs from a cliff top to the ground, 45 m from the base of the cliff. Calculate the height of the cliff, to the nearest metre.

4 Classify the following triangles as acute, obtuse or right-angled.a b

c d

5 Which of the following are Pythagorean triads?a 9, 39, 40 b 3, 4, 5c 0.3, 0.4, 0.5 d 6, 7, 10

6 A flagpole casts a shadow of 15 metres at the same time as a 1.5-metre stick casts a shadow of 2 metres. How high is the flagpole?

7 Calculate each of the following, correct to 4 decimal places.

8 Calculate θ, correct to the nearest degree, given that:

a sin 46° b tan 76°42′ c 4.9 cos 56°

d 8.9 sin 67°3′ e f

a cos θ = 0.5874 b tan θ = 1.23 c sin θ = 0.8.

CHAPTERreview

5A

9.2 m

9.2 mm

32 cm

26 cm

n

p4.8 m

3.2 m

q

17.25 cm7.25 cm

r

1.9 km

1.3 km t

0.6 m

2.4 m

5A

5A

5A5 cm

13 cm

12 cm

3 cm5 cm

4 cm

7 m19 m

9 m

8 cm16 cm

15 cm

5A

5B

5C5.69

cos 75°------------------ 2.5

tan 9°55′----------------------

5C


C h a p t e r 5 R i g h t - a n g l e d t r i a n g l e s a n d t r i g o n o m e t r y 2059 Calculate θ, correct to the nearest minute, given that:

10 Find the length of each side marked with a pronumeral, correct to 1 decimal place.a b c

d e f

g h i

j k l

11 A rope that is used to support a flagpole makes an angle of 70° with the ground. If the rope is tied down 3.1 m from the foot of the flagpole, find the height of the flagpole, correct to 1 decimal place.

a cos θ = 0.199 b tan θ = 0.5 c sin θ = 0.257. 5C

5D

q9°

6 cm

x78°

3.9 m m

22°12.6 cm

n22°

12.6 cm

q

7.8 cm

32°t

6.8 m65°

g

26°42'

2.9 mh

77°18'

4.8 cm

z 83°30' 138 mm

j

4.32 m

29°51'

k38.5 m

16°8'

m

63 km

85°12'

5D



12 A dirt track runs off a road at an angle of 34° to the road. If I travel for 4.5 km along the dirt track, what is the shortest direct distance back to the road (correct to 1 decimal place)?

13 A fire is burning in a building and people need to be rescued. The fire brigade’s ladder must reach a height of 60 m and must be angled at 70° to the horizontal. How long must the ladder be to complete the rescue?

14 Find the size of the angle marked θ in each of the following, giving your answer correct to the nearest degree.a b c

15 Find the size of the angle marked θ in each of the following, giving your answer correct to the nearest minute.a b c

16 A kite on an 80-m string reaches a height of 50 m in a strong wind. Calculate the angle the string makes with the horizontal.

17 There is 50 m of line on a fishing reel. When all the line is out, the bait sits on the bed of a lake and has drifted 20 m from the boat. Calculate the angle that the fishing line makes with the vertical.

18 The top of a building is sighted at an angle of elevation of 40°, when an observer is 27 m back from the base.Calculate the height of the building, correct to the nearest metre.

19 Hakam stands 50 m back from the foot of an 80-m telephone tower. Hakam’s eyes are at a height of 1.57 m. Calculate the angle of elevation that Hakam must look to see the top of the tower.

5D

5D

5E

θ

19 m16 m

θ

4.6 m

2.3 m θ 116 cm43 cm

5Eθ

10.8 m

4.6 m

θ2.9 m

6.1 m

θ11.9 cm

13.8 cm

5E

5E

5F

40°27 m

h

testtest

CHAPTERyyourselfourself

testyyourselfourself

5

5F

θ1.57 m 50 m

80 m


maths a - chapter 5

Technology

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t h s q u e s t

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p t e r

f o r q u e e n s

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