Year 10A Mathematics

Module 2 Measurement and Trigonometry

Name: _____________________________________ Home Group: _________________

Contents:

Set 1 Rounding to a given number of decimal places.

Set 2 Pythagorean Theorem

Set 3 Pythagorean Triads (Triples)

Set 4 Labelling sides of Triangles

Set 5 Rounding the size of an angles to the nearest minute

Set 6 Trig Ratios Assignment (You will need a protractor and a mm ruler)

Set 7 Trigonometric Ratios

Set 8 Using Trigonometry to calculate side length

Set 9 Using Trigonometry to calculate angle size

SAC Calculator active. (Set 1 to Set 9 inclusive)

Set 10 Angles of Elevation and Depression

Set 11 Bearings

Set 12 Drawing 3-D Shapes

Set 13 Applications

Set 14 Area

Set 15 Volume

SAC Calculator active. (Set 10 to Set 15 inclusive)

Extension, Revision and Video support via Maths Online.

(search using lesson code)

Pythagorean Theorem 5130, 5132, 3321, 3320

Trigonometric Ratios and Applications: 8268, 8269, 8270, 4258, 8271 4271, 4273

Bearings 4247, 4248, 4272

Area & Volume 8274, 8280, 8281, 4301, 4303

III b 077 III.. c 1.204 111

121 e 0.903 121 f 7.235 121

13j h 0.2978 131 i 23004 131

1 Round each ot the following numbers to the number of decimal places shown in brackets,

a 8.47

d 0.569

g 03502

Rounding to a given number of decimal places To round a decimal to a given number of decimal places, follow these steps. Step 1 Count out the required number of decimal places and consider the next digit. That is, if required to round to I decimal

place, consider the digit in the second decimal place; if required to round to 2 decimal places, consider the digit in the third decimal place and so on.

Step 2 11 the next digit is less than 5, simply omit this digit and all digits that follow. Step 3 If the next digit is 5 or larger, add 1 to the preceding digit and omit all digits that follow.

WORKED EXAMPLE 1

Round each of the following numbers to the number or decimal planes shove n in brackets. a 2.371 111 b 8.7234 121 CID WP I TF a Count out 1 decimal place. The next digit (the digit in the second decimal place) a 7 371 2.4

is 7, which is greater than 5. So add I to the preceding digit (that is, to 3) and omit all digits that follow.

b Count out 2 decimal places. The next digit (the digit in the third decimal place) b 8.7234 8.72 is 3, which is less than 5. So simply omit this digit and all digits that follow (that is, omit 3 and 4).

Evaluate VT/ correct to 1 decimal place.

CID 1 Evaluate J3 using a calculator and record the number shown on the display.

2 Count out I decimal place. The next digit (the digit in the second decimal place) is 5. so add I to the preceding digit and omit all digits after the first decimal place.

CID a= 5.65(1854 249

R. 5.7

Set 1

j 0.594 55 [41

2 Evaluate each of the following correct to I decimal place.

• sfit = b 4729 = c 91I

d = e 4F41

ANSWERS Rounding to a given number of decimal places

1 a g.5 b 0.8 c 1.2 d 0.57 e 0.90 f 7.24 g 0.550 h 0.298 i 2.3(1) j 0.5946

2 a 2_8 b 5.4 c 5.5 d 7.2 e 8.2

Pythagorean Theorem The Pythagorean Theorem describes the relationship between the lengths of the legs and the hypotenuse of a right triangle.

a 2 b 2 = C 2

side A

Side B right angle

Note: a + b < c

1 a 2 is with

ir—le

Pythagorean Theorem The relationship a 2 + b 2 = C 2 can be shown visually.

2 The areas of a 2 and b fit i nto C 2

2

2

Pythagorean Theorem Given the length of legs a and b, the length of the hypotenuse can be found using the formula a 2 + b 2 = C 2.

2

2

Pythagorean Theorem Given the length of legs a and b, the length of the hypotenuse can be found using the formula a 2 + b 2 = C 2.

Pythagorean Theorem The Pythagorean Theorem will work for any right triangle.

c2= a Z 4. b2

c 2 = 52 + 72

C 2 = 25 + 49

c 2 = 74

C =47-4-

c z: 8.6023

4.5 a2 + 1.2 2

20.25 + 1.4.4 20.25 — 1.44

- 18.81

,

— tf— 4- el"

= 4.3 m

Finding the hypotenuse • Ti, calculate the length of the hypotenuse when given the length of the two shorter sides,

substitute the known values into the formula c = a? + 1,1 .

For the triangle at right, calculate the length of the hypotenuse, z, correct to 1 decimal place.

4 y

=

= =

42 16

+ 1,2

+ 72 + 49

—65

7 x -47)75. x = 8.1

Finding a shorter side • Sometimes a question will givv you the length of the hypotenuse and ask you to find one of

the shorter sides. In such examples. we need to rearrange Pythagoras' formula.

Calculate the length. correct to 1 decimal place, of the unmarked side of the triangle at right.

= a2 + fr 142 = + 82

196 = a 2 + 64 a2 = 196 — 64

132

a = = 11.5cm

• In many cases we are able to use Pythagoras' theorem to solve practical problems. First model the problem by drawing a diagram.

A ladder that is 4.5 m long leans up against a vertical wall. The foot of the ladder is 1.2 m from the wall. How far up the wall does the ladder reach? Give your answer correct to 1 decimal place.

1.2 iii

The ladder will reach u height 014.3 m up the wall.

I. The hypotenuse is the longest side of the triangle and is opposite the right angle.

2. On your diagram, check whether you are finding the length of the hypotenuse or one of the shorter sides.

3. The length of a side can be found if we are given the length of the other sides by using the formula c 2 a2 +

4. When using Pythagoras' theorem, always check the units given for each measurement.

5. If necessary, convert all mea.surements to the same units before using the rule.

6. Worded problems can be solved by drawing a diagram and using Pythagoras' theorem to solve the problem.

7. Worded problems should be answered in a sentence.

Set 2 Pythagoras' theorem

For each of the following triangles. calculate the length of the hypotenuse, giving

answers correct to 2 decimal places.

a 4.7 b 19.3 c

6.3 7 804 N Find t he value of the pronumeral. correct to 2 decimal places.

1.98

"1411111111111k47.1 2.56 rtX -1

l 7 .

3 The diagonal of the rectangular sign at right is 34cm. lithe height of this sign is 25 cm. find the width.

4 A right-angled triangle has a base of 4 cm and a height of 12cm. Calculate the length of the hypotenuse to 2 decimal places.

5 Calculate the lengths of the diagonals Ito 2 decimal places) of squares that have side lengths of:

a 10011

8 An isosceles triangle has a base of 30cm and a height of 10cm. Calculate the length of the two equal sides.

9 An equilateral triangle has sides of length 20cm. Find the height of the triangle.

10 A right-angled triangle has a height of 17.2cm, and a base that is half the height. Calculate the length of the hypotenuse. correct to 2 decimal places.

11 The road sign shown below is in the form of an equilateral triangle. Find the height of the sign and, hence. find its area.

\ GIVE wAY

76 cm

12 A flagpole, 12 m high, is supported by three wires, attached from the top of the pole to the ground Each wire is pegged into the ground 5m from the pole. How much wire is needed to support the pole?

13 Ben's dog 'Macca' has wandered onto a frozen pond. and is too frightened to walk hack. Ken estimates that the dog is 3.5 m from the edge of the pond. He finds a plank, 4 m long, and thinks he can use it to rescue Macca. The pond is surrounded by a bank that is I in high. Ben uses the plank to make a rump for Macca to walk up. Will he be able to rescue his dog?

14 Sarah goes canoeing in a large lake. She paddles 2.1 km to the 3.8 km north. then 3.8 km to the west. Use the triangle at right to find out how far she must then paddle to get back to her starting point in the shortest possible way. 2.1 km

Starting point

16 Penny. a carpenter, is building a roof for a new house. The roof has a gable end in the form of an isosceles triangle, with a base of 6m and sloping sides of 7.5 m. She decides to put 5 evenly spaced vertical strips of wood as decoration on the gable as shown at right. How many metres of this decorative wood does she need?

Answers - Pythagoras' theorerr 1 a 7.86 b 33.27 c 980.95

d 12.68 e 2.85 f 175.14 2 a 36.36 b 1.62 C 15.37

d 0.61 e 2133.19 f 453.90 3 23.04 cm 4 12.65 cm 5 a 14.14 cm b 24.04 cm C 4.53 cm 6 a 97.47 cm b 334.94cm c 6822.90 cm2 7 a 6.06 b 4.24 C 4.74 8 18.03 cm 9 17.32 cm

10 19.23 cm

11 65.82 cm; 2501.16 cm2 12 39m 13 Yes

14 4.34 km

16 20.61m

Set 3 Pythagorean Triads (or ttinlcs1

A Pythagorean triad (a, b, c) where a b < c is a set of numbers which obey the Theorem of Pythagoras a 2 + b2 = C2 for right angled triangles.

b

1. Use the Theorem of Pythagoras to find the unknown pronumeral in each of the following Pythagorean Triads.

Note that the parts of this question will represent key integer triads, which occur regularly in mathematical questions where Pythagoras is used. Triads (a) and (b) are particularly important.

(a) (3, 4, x) (b) (5, 12,x) (c) (6, 8, x)

(d) (8, x, 17) (e) (x, 24, 25) (f) (9, x, 41)

2. Use the Theorem of Pythagoras to find the unknown pronumeral in each of the Following Pythagorean Triads.

Where necessary leave your answer in simplest exact (ie surd) form, as each of these triads will have at least one irrational member.

(e) (3, 6, x)

(f) (151, x, 5)

(i) (2, 2, x)

(a) (2, 5, x)

(d) (Irf, 3, x)

(g) x, 6)

Answers 1. (a) x 5 ie (3, 4, 5)

(d) x = 15 ie (8, 15, 17)

(b) (2, 6, x)

(e) (x, 3, 4)

(h) (x, 3,VT.7)

(b) x =13 ie (5, 12,13)

(e) x = 7 ie (7, 24, 25)

(c) x = 10 ie (6, 8, 10)

(f) x = 40 ie (9, 40, 41)

Parts (a) ie (3, 4, 5) and (b) ie (5, 12, 13) are important triads which occur regularly in mathematical problems.

(Scalar) multiples of these triads, particularly of (3, 4, 5), such as part (c) ie (6, 8, 10) .... also (9, 12, 15), (12, 16, 20) (30, 40, 50) etc are also very common.

2. (a) x = A/F3

(d)x= tTi

(b) x = 21/17

(e) x = Nri

(g) x 5 (h) x = 2Ari (i) x =

40,416666. % 1

X

Labelling the sides of a right-angled triangle In a right-angled triangle, the longest side is called the hypotenuse. The hypotenuse is always opposite the right angle. The other two sides are named according to their position with respect to a specific angle. The side that is opposite to the angle is called opposite and the side next to that angle is called adjurent.

WORKED EXAMPLE

Label the 'ides of the following right-angled triangle using the letters H (for hypotenuse), 0 (for opposite) and A (for adjacent) with respect to angle a

1 The side opposite the right angle (side BC) is the hypotenuse. Label it H.

2 Look at the position of the other two sides with respect to angle 0. Side AB is opposite angle 0, so label it 0. Side AC is next to angle 0, so label it A.

Set 4

8 h.. A

Label the sides of the following right-angled triangles using the letters H (for hypotenuse), 0 (for opposiie) and A (for adjacent) with respect to angle a

3

A

4

— ANSWERS

Labelling the sides of a right-angled triangle hittab. 3 0

/AM a

A A

6R 0 S

I A T

\ 5 t 1 rpr II

7 I I 8 • V A

■ H 7 X

9 e A • 4‘11 1:

0

0 w

Rounding the size of an angle to the nearest minute When a calculator is used to convert an angle to degrees. minutes and seconds, the display usually shows eight digits. The first two digits represent the number of degrees. the next two digits represent the number of minutes and the last four digits (together with the decimal point) represent the number of seconds to the nearest hundredths. For example, if the number on the display is

2518'12.35", it means that the size of the angle is 25 degrees. 18 minutes and 32.35 seconds. The angle size often needs to be rounded to the nearest minute. or to the nearest second. To round to the neareNt minute, we need to consider the number of seconds. Since each minute contains 60 seconds, we need

to rnund as follows. • If the number of seconds is less than 30. simply omit the seconds. • If the number of seconds is 30 or above, round up; that is, add 1 to the number of minutes and then omit the seconds.

WORKED EXAMPLE 1

Round each of the following angle sizes to the nearest minute. a 13°48'42.65" b 75°10'22.36" CID o The number of seconds is 42.65. which is more than 30. Add 1 to the number of

minutes (that is, 48 + 1 = 49) and omit seconds.

b The number of seconds is 22.36. Since this is less than 30, simply omit the I seconds.

CID 13 04842.65" • 13°49'

b 7510'2136" 75°1(Y

Set 5 1 Round each of the following angle sizes to the nearest minute.

a 54°32'12.60"

C 16°59'12.61"

a 39°15'20.00"

g 40°37'32.36"

i 68°50'11.12" j 37°19'52.76" •

2 Round each of the angle sizes in question I to the nearest second.

a 54°32'12.60" • b 11°4120.08"

c 16°59'12.63" •

d 82°27'25.11"

e 39°15'20.00 •

I 78°2651.24"

g 40°37'32.3e

h 22°10'46.83"

i 68°50'11.12" --- j 37°19'52.76"=

ANSWERS

Rounding the size of an angle to the nearest minute and second 1 a 54°37 b war c 16°59' d 82°27' a 39°15'

I 78°27' g 40°38' h 22°11' i 68°50' j 37°20'

2 a 54°32'13" b 11°41'2(Y. c 16°59'13" d 82°27'25" e 390 iy2(r I 78°26'5 r g 40°3732" h 22°10'47" i 68°50'11" j 37°19'53"

= b I 1°4120.08"

d 82"2725.11" -

• f 78°2651.24" •

=, h 22°10'46.83"

Set 6

Trig Ratios Assignment

Introducing the side ratio Sin 0, Cos 0, and Tan 0

(You will need a mm ruler, scientific calculator and a protractor)

Method

• Measure each angle (labelled A to 0) using a protractor to the nearest Y2 degree.

• Lable each side of the triangle Opp, Adj. and Hyp

• Measure each side for every triangle and record on your value to the nearest mm on the table provided

• Using your values calculate side ratios (two decimal places)

For example if Opp = 16mm and Hyp = 120mm

Opp 16

Hyp 120

= 0.1333

= 013

This ratio is called Sin ()

• Now use your scientific calculator to compare your result. Record the difference Your value should be very close to the calculator value

• Repeat for Cos ii and Tan ii

• Now complete the table

Assessment: You will need to complete the table to the satisfaction of your teacher to gain a "Satisfactory completed grade". This grade will contribute to your effort assessment in your final semester report.

„--,--,-.- ,---'.-----' __,--- -,-

_r _.....------'---..-' 1-

C

Trigonometry Assignment

Name:

Elomegroup:

Trigonometric ratios Angles and the calculator • Last year you were shown that each angle has specific values for its sine, cosine and tangent. • These values are needed for practically every trigonometry problem and can be obtained with

the aid of a calculator.

Calculate the value of each of the following. correct to 4 d ecimal places. using a calculator a cos 65°57' b tan 56°45'30'

CID

C=1 a cos 65'57' =0.4075

b tan 56' -'45'3(r = 1.5157

Calculate the size of angle 0, correct to the nearest degree, given sin 0 — 0.6583.

CLIO CID sin 0=0.6583 sin 0= 0.6583

0= sin' (0.6583) 0= sin -1 (0.6583)

6=41'

I. When using the calculator to find values of sine, cosine and tangent. make sure the calculator is in Degree mode.

2. To find the size of an angle whose sine, cosine or tangent is given, perform an inverse operation; that is. sin -1 , ens' or tan 1 .

3. Use the calculator's conversion function to convert beim= decimal degrees and degrees, minutes and seconds.

4. There are 60 minutes in 1 degree and 60 seconds in I minute.

Set 7 Trigonometric ratios FLUENCY

1 Calculate each of the following, correct to 4 decimal places. a sin 30' b cos 450 c tan 25° d sin 570 e tan 83° f cos 44°

2 Cakulatc each of the following, correct to 4 decimal places. a sin 40°30' b cos 53'57' c tan 27°34' d tan 1230401 a sin 92°32' f sin 4298' g cos 35'42'35 h tan 27°42'50" i cos 143°25'23" j sin 23°58'21" k cos 8°54'2"' I sin 286° m tan 420' n cos 8450 o sin 367°35'

3 Find the size of angle 8 correct to the nearest degree, for each of the following. a sin 0 0.763 b cos 0 = 0.912 c tan 0 = 1.351 d cos 6=0.321 a tan 0 = 12.86 f cos 8=0.756

4 (22:231 Find the size of the angle 0 in each of the following, correct to the nearest minute. a sin 0=-0.814 b sin 0=-0.110 c tan 8=0.015 d cos 0 = 0.296 e tan 0.-- 0.993 f sin 0 = 0.450

5 en Find the site of the angk 9 in each of the following, correct to the nearest second. a tan 0 = 0.5 b cos 0 = 0.438 c sin 0 = 0.9047 d Ian 0 -= 1.1141 e cos 9 0.8 f tan 0 = 43.76

Answers - Trigonometric ratios

la 0.5000 b 0.7071 0.4663 0.8387 e 8.1443 0.7193

2a 0.6944 b 0.5885 0.5220 -1.5013 e 0.9990 0.6709 0.8120 h 0.5253 -0.8031

j 0.4063 0.9880 -0.9613 m 1.7321 -0.5736 0.1320

3 a 50° 24° 53° d 71° 86° 41°

4 a 54°29' 6°19' 0°52' d 72°47' 44°48' 26°45'

5 a 26°33'54" 64° I '25" 64046159" d 48°5'22" 36°52'12" 88"41'27"

Using trigonometry to calculate side lengths

Review • We are able to find a side length in a right-angled triangle if we are given one other side

length and the size of one of the acute angles. These sides and angle are related using one of the three trigonometric ratios.

• The sine ratio The sine ratio is defined as the ratio of the length of the side opposite angle 0(0) to the length of the hypotenuse (H). This is

0 written as sin 9=

The sine of an angle is not dependent on the size of the right-angled triangle as all these triangles are similar in shape.

• The cosine ratio The cosine ratio is defined as the ratio of the length of the adjacent side (A) to the length of the hypotenuse (H) and is written as

A cos =—.

The cosine of an angle also does not depend on the size of the right-angled triangle_

• The tangent ratio 0

The tangent ratio is defined as tun 0 --. where 0 is the length of A

the side opposite angle Band A is the length of the side adjacent to it. Again, the tangent ratio does not depend on the size of the right-angled triangle.

Adjacent

• The steps used in m)lving, the problem are as follows. Step I. Label the sides of the triangle, which are either given, or need to be found, with

respect to the given angle. Step 2. Consider the sides involved and determine which of the trigonometric ratios is

required. (a) Use the sine ratio if the hypotenuse (H) and the opposite side (0) are

involved. (b) Use the cosine ratio if the hypotenuse (H) and the adjacent side (A) are

involved. (c) Use the tangent ratio if the opposite (0) and the adjacent (A) sides are

involved. Step 3. Substitute the values of the pronumerals into the ratio. Step 4. Solve the resultant equation for the unknown side length.

Find the value of the pronumeral for each of the foikm ing. (Ave answers correct to 3 decimal places.

A

6 crn a

0.346 crii

WRITE/DRAW

a

• 0

sin 9-- H

A

0.3,16A HA:ni f

•

a sin

6

6 sin 35° = a a 6 sin 35'

a 3.441 cm

A cos 9

= Ti

cos Ar = 0.346

0.346 cos 32° =f

fz-- 0.346 cos 32'

0.293 cm

Find the value of the pronumeral in the triangle shown. Give the answer correct to 2 decimal places.

r20 m

A

tan (1=-0 A

120 tan 5° =—

P

Px tan 50= 120 120 p =

tan 5`

P.= 1371.61 m

75.23/Icm

11.7 ni

glgiligNo.9 cm 1

11111

a

The trigonometric ratios can he used to find a side length in a right-angled triangle when we are given one other side length and one of the acute angles.

Set 8 Using trigonometry to calculate side lengths Find the length of the unknown side in each of the following, correct to 3 decimal

places. a

.t

10 eni a

14

2 find the length of the unknown side in each of the following triangles, correct to 2 decimal places.

11111111111111111bm-23 in

4.e in 13'

3 Find the length of the unknown side in each of the following, correct to 2 decimal places.

4 Find the value of the pronumeral in each of the following, correct to 2 decimal places .

a 11

-.411111111111111111111143.9 cm

12.3 ni 15.3 m hiott

4).732 kir

47.385 km

5 Given that the angle Ois 42° and the length of the hypotenuse is 11.95m in a right -angled triangle. find the length of: a the opposite side b the adjacent side. Give each answer correct to I decimal point.

6 A ladder rests against a wall. If the angle between the ladder and the ground is 35 0 and the foot of the ladder is 1.5 rit from the wall, how high up the wall dots the ladder mach?

Answers using trigonometry to calculate side lengths

1 a 8.660 b 7.250 c 8.412 2 a 0.79 b 4.72 c 101.38 3 a 33.45 m b 74,89m c 44.82m

d 7.76mm e 80.82km f 9.04cm 4 a x = 31.58 cm y = 17.67 m

c z= 14.87 m p = 67.00m e p = 21.38km. q = 42.29 km f a = 0.70 km, b = 0.21 km

5 a 6.0m b 6.7m 6 1.05m

Using trigonometry to calculate angle sae

• To fund the size of an angle using the trigonometric nttios, we need to be given the length of any two sides.

WORKED EXAMPLE 14

For each of the following, find the size of the angle, 0, correct hi the nearest degree. a

m

5 cm • 3. 5 cm It m

•

„441111 0 5 m II at A

0 tan 0 =—

A

•

sin 0 .=— H

• 3.5 stn -7

=0.7

0= sin -1 0.7 = 44.427 004°

0= 44°

tan 0=5

0=-11m

= 24.443 954 711'•

0., 24°

Find the size of angle Cmn each of the triangles shown below.

a 3.1 m A

0

7.2 m

tan 0=-0

A

7.2 tan 0 =Ti.

= 2.322580 645

0= tan 2.322 580 645

(Answer correct to 0= 66.7D543675°

the nearest minute.) = 6(ia4219.572"

9 66"42'

a Al 4.8

I. The trigonometric ratios can be used to find the size of the acute angles in a right-angled triangle when we are given the length of two sides.

2. To find an angle size we need to use the inverse trigonometric functions. 3. Answers may be given correct to the nearest degree, minute or second, or as decimal

degrees.

Set 9 Using trigonometry to calculate angle size

FLUENCY

1 ME11 Find the site of the angle, 9 in each of the following. Give your answer correct to the nearest degree.

Find the size of the angle marked with the pronumeral in each of the following. Give your answer correct to the nearest minute. a b 1 A

17

• 12

Find the site of the angle marked with the pronumeral in each of the following. Give your answer motet to the nearest seeinxi_ 0 4

a rn

,ite41 2.7 3 5

4 Find the size of the angle marked with the pronumeral in each of the following. giving your answer correct to the nearest degree.

1 3.1111111101' 15.3

Niiivi C 106.4

v.' 7

43.7 1E1.7

•

• 1 1 .85 12--

18.56

7.3 ci

Find the size of each of the angles in the following, giving your answers correct to the nearest minute.

2.3

IIIIImIlIII111_ •

• 6 a Calculate the length of the sides r, 1 and h. Write

your answers correct to 2 decimal places. b Calculate the area of ABC, correct to the nearest

square centimetre. c Calculate ZBCA.

A

D 1:44111111111111h■ 20 cm B 30cm C

7 In the sport of air racing, small planes have to travel between two large towers (or pylons). The gap between a pair of pylons is smaller than the wing-span of the plane, so the plane has logo through on an angle with one wing `above' the other. The wing-span of a competition airplane is 8 metres.

a Determine the angle. correct to 1 decimal place, that the plane has to tilt if the gap between pylons is:

i 7 metres ii 6 metres iii 5 metres.

Answers - Using trigonometry to calculate

1 a 67° b47° c 69° 2 a 54'47' b 33045 1 c 33°33' 3 a 75°31'21" b 36°52'12" c 37°38'51" 4 a 41° b 30° c49°

d 65° e48° f 37° 5 a a=25°47',b=64°13' b d = 25°23', e = 64037'

C .v = 663 12', y = 23°48'

6 a r = 57.58, / = 34.87, h = 28.56 b 714 cm2 c 29.7*

7 a i 29.0° 41.4* ii 51.30

SAC Calculator active. (set 1 to Set 9 inclusive)

Angles of elevation and depression When we need to look up or down in order to see a certain object, our line of vision (that is, the straight line from the observer's eye to the object) is inclined. The angle of inclination of the line of vision to the horizontal when looking up is referred to as the

II II .9 -

I. To solve a problem involving trigonometric ratios, follow these steps: • Draw a diagram to represent the situation. • Label the diagram with respect to the angle involved (either given or that

needs to be found). • Identify what is given and what needs to be found. • Select an appropriate trigonometric ratio and use it to find the unknown side

or angle. • Interpret the result by writing a worded answer.

2. The angle of elevation is measured up and the angle of depression is measured . down from the horizontal line to the line of vision.

Horizontal 4 Angle of depression

Angle of 0 elevation

Horizontal

WORKED Example From an observer, the angle of elevation of the top of a tree is 50 0. If the observer is 8 metres from the tree, find the height of the tree.

WRITE

ri

0 tan 0=

A

h tan 50' = 8-

h = 8 tan 50° 9.53

The height of the tree is 9.53 m.

1--5.8 km - Li

---- -2-i-2: 58° •"' ,I

.•'. / „• .. .-'

. . rib( -

Set 10 Angles of elevation and depression 1 WE16 The angle of elevation from an observer to the top of a tree is 54°22'. lithe tree is

known to be 12.19m high. how far is the observer from it?

2 From the top of a cliff I 12m high, the angle of depression to a boat is 9 ° 15'. How far is the boat from the foot of the de?

3 A person on a ship observes a lighthouse on the cliff, which is 830 metres away from the ship. The angle of elevation of the top of the lighthouse is 12°. a How far above sea level is the top of the lighthouse? b lithe height of the lighthouse is 24m, how high is the cliff'?

4 Al a certain time of the day a post, 4 m tall, casts a shadow of 1.8 tn. What is the angle of elevation of the sun at that time?

5 An observer, who is standing 47m from a building, measures the angle of elevation of the top of the building as 17'. If the observer's eye is 167cm from the ground, what is the height of the building?

A lookout tower has been erected on top of a cliff. At a distance of 5.8km from the foot of the cliff, the angle of elevation to the base of the tower is 15.7° and to the observation deck at the top of the tower is 16° respectively as shown in the figure below. How high from the top of the cliff is the observation deck?

a Find the height of a telegraph pole in the photograph at right if the angle of elevation to the top of the pole is 8 from a point at the ground level 60m from the base of the pole.

b Find the height of the light pole in the figure below.

11 From a point on top of a cliff, two boats are observed. lithe angles of depression are 58' and 32' and the cliff is 46m above sea level, how far apart are the boats?

Answers — Angles of elevation and depression 1 8.74m 2 687.7m 3 a 176.42m b 152.42m 4 65°46' 5 16.04m

11 44.88 m

Bearings To determine our position or directions we need to use uni-form methods of navigation. There arc two ways this can be done. We can use a compass bearing or a true bearing.

Compass bearings Compass bearings (also known as con-ventional bearings) are measured from the north—south line in either a clock-wise or anticlockwise direction.

To identify the compass bearing of an object we need to state:

I. whether the angle is measured from north (N) or south (S),

2. the size of the angle, and 3. whether the angle is measured

in the direction of west (W) or east (E).

For example, the bearing of S20°E means

the bearing N40°W means

True bearings

true bearings are measured from north in a clockwise direction.

The diagrams below show the bearings of 025' true and 250' true respectively.

2

Drawing a diagram from given directions When drawing a diagram of a path. it is important to remember that the direction of a movement can be given either as a true bearing or a compass bearing. If the direction is given as a true bearing. the angle has to be measured from north in a clockwise direction. lithe direction is given using a compass hearing, the specified angle is measured from either north or south, in the

direction of either cast or west. In either case it is Unpin tam to mark the angle that is beim: given on a diagnun.

Draw a diagram to show each of the folkmin. cr An object moved 70 km from A to B on a hearing of 050°T and then 641km from It to Con a bearing of 1709: b An object moved 100 km from A to Hon a hearing of S20'W and then 95 km from B to t' on a bearing of N70°W.

DRAW

0 1St

Set 11 Bearings and compass directions I Change each of the following compass directions to true bearings.

a N20°E b N20°W c S35°W d S28°E e N34°E f S42°W

2 Change each of the following true bearings to compass directions.

a 049'T b 132'T c 2671' d 3301' e 086'T f 2341'

3 Describe the following paths using true bearings.

0

•

35' 2.5 kin 4 kin

4 Show each of the following by drawing the paths.

O A ship travels 040`T for 40km and then 1001' for 30 km. b A plane flies for 230 km in a direction 135'7 and a further 140 km in a direction 240°1 c A bushwalker travels in a direction 2601 for 0.8 km. then changes direction to 1201' for

1.3 km. and finally travels in a direction of 32° for 2.1 km. d A boat travels N4U°W for 8km, then changes direction to S30°W for 5 km and then

S50°E for 7 km. e A plane Ira. els N20°E for 32() km. N70`E. for 180 km and S30°E for 220 km.

5

0 You are planning a trip on your yacht. If you travel 20km from A to B on a bearing of 0427:

1 how far east of A is B? how far north of A is B?

iii what is the hearing of A from 11? b In the next part of the journey you decide to travel 80 km from B to C on a bearing of

130`T. Show the journey to be travelled using a diagram.

ii How far south of B is C? iii How far cast of B is C? lv What is the bearing of B from C?

4 a

6 215°T

10 a 600431 T b 69°27' T

[73

6 Ira farmhouse is situated nom N35°E from a shed, what is the true bearing of the shed from the house?

10 Find the final bearing ior each of the folknving. Express your answer in true bearings, correct to the nearest minute. a A boat travels due cast for 4 km and then travels

N200E for 3 km. What is the final bearing of the boat from the starting point?

b A hushwalker travels due north kw 3km. then due east for 8 km. What is the final bearing of the bushwalker from the starting point?

c A ear travels due south for 80 km, then travels due west for 50 km, and finally due south for a further 30km. What is the final bearing of the ear from the starting point?

Answers - Bearings and compass directions

1 a 020°T d I52°T

2 a N49°E d N30°W

3 a 3 km 325°T b 2.5 km 112aT c 8km 235°T d 4 km 090°T, then 2.5 kin 035"rT e 12 km 115°T, then 7 km 050°T f 300m 31001, then 500m 220°T

b 34001 C 215°T e 034°T f 222°T b S48°E c S87°W e N86°E f S54°W

hi 222°T ii 51.42km

iii 6 I .28 km iv 310°T

ii 38.97 km ii 22.5 km iv 030"T

Drawing 3-D shapes The worked example below shows step-by-step instructions that will help you to draw a rectangular prism, a right-angled wedge and a square-based pyramid.

Construct each of the following shapes. a A rectangular prism whose dimensions are 5 cm x 3 cm x 4 cm b A right-angled wedge with base 5 cm x 4 cm and height (at the tallest end) 3 cm c A square-based pyramid with the length of the base 5 cm and the height 6 cm

CID DRAW a 1 Draw a rectangle ABCD. 0 B

Now: Figures reduced to fit.

2 Draw a rectangle FFGH. It must be congruent (have the same shape and size) to rectangle ABCD and overlap it at the top right corner.

3 Join the corresponding vertices of the two rectangles with straight lines. That is, join A to E, B to F, C to G and D to H. Write in the given dimensions (length, width and height).

b 1 Draw a parallelogram ABCD.

A

2 On the side DC of the parallelogram draw a rectangle DCFE. (That is, DC is the shared side, belonging to both the parallelogram and the rectangle.)

3 Join A to E and B to F with straight lines. Write in the dimensions of the wedge that are given.

: 1 Draw a parallelogram ABCD. c 1) (

A

2 Draw the diagonals of the parallelogram (AC and BD). From the point of intersection of the diagonals draw a vertical line 6 cm long, which will represent the height of the pyramid. Call the end-point of the line E

3 Join point E to points A, B, C and D with straight lines. Write in the dimensions of the pyramid that are given.

A

5 cm

Set 12 Try these Construct each of the following shapes on a separate sheet of paper.

1 A rectangular prism whose dimensions are 4 cm x 3 cm x 2 cm. 2 A rectangular prism whose dimensions are 3 cm x 2 cm x 6 cm.

3 A cube with side length 3 cm.

4 A right-angled wedge with base 4 cm x 3 cm and height (at the tallest end) 5 cm.

5 A right-angled wedge with base 6 cm x 7 cm and height (at the tallest end) 35 cm.

6 A right-angled wedge with a square base 4 cm x 4 cm and height (at the tallest end) 2.4 cm.

7 A square-based pyramid with the length of the base 4 cm and the height 7 cm.

8 A square-based pyramid with the length of the base 3.5 cm and the height 5 cm.

9 A rectangular-based pyramid with base 4 cm x 6 cm and height 8 cm.

4

A 4 cm R A 6 cm

- ANSWERS Drawing 3-D shapes Note: Answers not to exact size.

2 F G 1 F G

B A .1Piril

P R Ai 1

2 cm

Milreill

6 crn 14

A 4 cm D

r2cmil A 3 cm D

6 L F

3.5 cm dD c A4 4 cm

A 4 cm

9 7 8

Aar .5 cm A 3.5 cm

I) AVM A 4 cm

\ 6 cm A 4 cm

Applications Many people use trigonometry at work. It is particu-larly important in careers such as the building trades, surveying, architecture and engineering. Trigonometric ratios have a variety of applications, some of which will be discussed in this section and the one that follows.

Trigonometric ratios can be used to solve prob-lems. When solving a problem, the following steps can be of assistance. 1. Sketch a diagram to represent the situation

described in the problem. 2. Label the sides of the right-angled triangle with

respect to the angle involved. 3. Identify what is given and what needs to be

found. 4. Select an appropriate trigonometric ratio and use

it to find the unknown measurement. 5. Interpret your result by writing a worded answer.

WORKED Example A ladder of length 3 m makes an angle of 32° with the wall. a How far is the foot of the ladder from the wall? b How far up the wall does the ladder reach?

What angle does the ladder make with the ground?

WRITE (wall)

A

0 a sin = —

sin 32° =

x = 3 sin 32° 1.59 m

The foot of the ladder is wall.

1.59 m from the

0

b cos 0 — —

cos 32° =

y= 3 cos 32° v 2.54 m

The ladder reaches 2.54 m up the wall.

c 90° + 32° = 180° a+ 122° 1800

a = 180°— 122° a= 58°

The ladder makes a 58' angle with the ground.

To solve a problem involving trigonometric ratios, follow these steps:

(a) Draw a diagram to represent the situation.

(b) Label the diagram with respect to the angle involved (either given or that needs to be found).

(c) Identify what is given and what needs to he found.

(d) Select an appropriate trigonometric ratio and use it to find the unknown side or angle.

(e) Interpret the result by writing a worded answer.

Set 13 Applications 1 A 3m-long ladder is placed against a wall so that it reaches 1.8m up the wall.

a What angle does the ladder make with the ground? b What angle does the ladder make with the wall? c How far from the wall is the foot of the ladder?

2 Jamie decides to build a wooden pencil box. He wants his ruler to be able to lie across the bottom of the box, so he allows 32cm along the diagonal. The width of the box is to be 8 cm.

9.'311 8 cm 0 r

Calculate: a the size of angle b the length of the box.

3 A carpenter wants to make a roof pitched at 29°30 1 . as shown in the diagram. How long should he cut the beam PR?

29°30' P Q

I Oh rn

4 The sloping sides of a gable roof are each 7.2m long. They rise to a height of 2.4m in the centre. What angle do the sloping sides make with the horizontal?

5 The ma.st of a boat is 7.7m high. A guy wire fmm the top of the mast is fixed to the deck 4 m from the base of the mast. Determine the angle the wire makes with the horizontal.

6 A desk top of length 1.2m and width 0.5 m rises to 10 cm.

E F i /0 cm 1

05m ,/c / ,

,/

. .

A 1.2m B

Calculate: a LDBF b LCBE.

7 A cuboid has a square end.

n cm

25 cm 13

a If the length of the cuboid is 45 cm and its height and width are 25 cm each, calculate: i the length of BD ii the length of BG

iii the length of BE iv the length of BH v ZFBG vi LEBH.

8 In a right square-based pyramid, the length of the side of the base is 12 cm and the height is 26 cm.

12 cm

Determine: a the angle the triangular face makes with the base b the angle the sloping edge makes with the base c the length of the sloping edge.

10 In a right square-based pyramid, the height is 47 cm. If the angle between a triangular face and the base is 73 0, calculate: a the length of the side of the square base b the length of the diagonal of the base c the angle the sloping edge makes with the base.

11 The height of a vertical cone is 24.5 cm.

If the angle at the apex is 48°37'10", determine: a the length of the slant edge of the cone b the radius of the cone.

12 Aldo the carpenter is lost in a rainforest. He comes across a large river and he knows that he can not swim across it. Aldo intends to build a bridge across the river. He draws some plans to calculate the distance across the river as shown in the diagram below.

Tree

a Aldo used a scale of 1 cm to represent 20 m. Find the real-life distance represented by 4.5 cm in Aldo's plans.

b Use the diagram below to write an equation for h in terms of d and the two angles.

• z92 I- d x ) 1 <

1< 1

c Use your equation from b to find the distance across the river, correct to the nearest metre.

Answers Applications 1 a 36°52' 2 a 14°29' 3 6.09m

6 a 11°32'

b 4°25' 7 a i 35.36 ern

ii 51.48 cm iii 51.48 ern

iv 57.23 cm v 29°3' vi 25°54'

b i 25.74 cm

ii 12.5 cm iii 25°54' iv 28.61 cm

8 a 77°

b 71°56' c 27.35 cm

b 53'8' b 31cm

4 19°28' 5 62°33'

10 a 28.74 cm 11 a 26.88 cm

12 a 90 m

C 250 m

2.4m

b 40.64 cm c 66°37' b 11.07 em d tan()

b h= x tan 612 tan 19, + tan (92

Shape

Formula

Area I. Square

2. Rectangle

3. Triangle

4. Trapezium

5. Circle

6. Parallelogram

7. Sector

8. Rhombus

AL

AW

.1*

A = 1 2, where 1 is a side length.

A = lw, where 1 is the length and w is the width.

A = where b is the base length and h the height.

A = (a + b)h, where a and b are lengths of parallel sides and h the height.

A = Kr 2, where r is the radius.

A = bh, where b is the base length and h the height.

0 A = — x x-r2, where B is the sector angle 360°

in degrees and r is the radius.

A = xy, where x and y are diagonals.

9. Ellipse A = ?tab, where a and b are the lengths b: of the semi-major and semi-minor axes

a respectively.

Alternative way to find the area of a triangle • If the lengths of all three sides of a triangle are known, its area,

b A, can be found by using Heron's formula: a

A =Vs(s–aXs–bXs–c) where a, b and c are the lengths of a+ b+c

the three sides and s is the semi-perimeter or s = —. 2

a A=Vs(s—aXs—bXs—c)

A = V7(7 —3X7 —5X7 —6)

=V7 x4x 2x1

=

a =3, b =5, c =6

s — a +b+ c

2 3+5+6

2 _ 14 — 2 =7

b A = gab

a = 5, b = 2

A=rx 5x 2 = 31.42 cm2

e c A = X gr2 360°

0= 40°, r= 15

40° A= — x 152 360°

= 78.54 cm 2

WORKED EXAMPLE

Find the areas of the following plane figures, correct to 2 decimal places. a

2 cmi 3 cm 5 cm

5 cm 15 cm

40* 6 cm

WORKED EXAMPLE

Find the area of each ot the totiowing composite shapes.

a a Area ACBD = Area LABC + Area AABD AB =8 cm E C = c n Fri c r

A - — - B E 'LT

1 Atriangle = 2 bh

LABC: b= AB = 8, h = EC = 6 Area of LABC = x AB x EC

x 8x 6

=24 cm2

LABD: b= AB = 8, h=FD = 2

Area of LABD = AB x FD

= x 8 x 2

= 8 cm2

Area of ACBD = 24 cm2 + 8 cm 2 = 32 cm2

Set 14 Area Where appropriate, give answers correct to 2 decimal places.

1 Find the areas of the following shapes.

a 1 c

4 cm

12 cm

—4 cm

1 0 cm

d e 12 cm

8 cm

i 18 cm

8 mm 13 mm

-4- 7 mm—>-

3

Use Heron's formula to find the area of the following triangles.

a

5

rn

8 m m I 3 '1m 1 •

-4-5 m

4 Find the area of the following

a

9 mm

4 mm:

Answer correct to 1 decimal place. A

Find the area of the following composite shapes.

20 cm 28m -4-40 tn 8 a 5 cm

8 cm

3 +2 cm cm 4 cm

— Area 1 a 16 crn2 b 48 cm2 C 75 cm

d 120 cm2 e 706.86 cm2 f 73.5 mm2 g 254.47 cm2 h 21 m2 i 75 cm2

3 a 203 cm2 b 7.64 cm2 4 a 113.1 mm2 b 37.70 cm2

8 a 123.29 cm2 b 1427.88m C 52 cm2 d 78 cm2 e 153.59m2 f 13.86m2

Volume of prisms and other shapes • The volume of a 3-dimensional figure is the amount of space it takes up. • The units for volume are mm 3 , cm 3 and m3 .

• The volume of any solid with a uniform cross-sectional area is given by the formula: V= AH, where A is the cross-sectional (or base) area and H is the height of the solid.

Cylinder Volume = Area X Height = area of a circle x height

Krzh

If the solid has a point then: Volume = 1 X Area X Height

3

Volume of a pyramid = AH

sea of base = A

base

• Since a cone is a pyramid with a circular cross-section, the volume of a cone is one-third the volume of a cylinder with the same base area and height.

Volume of a cone = AH

= zr 2h

Volume of spheres • Volume of a sphere of radius, r, can be calculated using the formula: V= irr.3.

WORKED EXAMPLE

Find the volume of each of the following solids. a

10icm

cm

Volume = I X Area X Height 3

V =- 7tr2h

r = 8, h = 10 1

17 = -3 X7rX82 X10

= 670.21 cm 3

Set 15 Volume 1 Find the volumes of the following prisms.

a

0 3 cm

Volume = I X Area X Height 3

V = AH 3

A = 12 where 1 = 8 A =82

= 64 cm2

H= 12

V+64x 12

=256=3

C 12 cm 0

15 cm

20 cm

2 Calculate the volume of each of these solids. a r It8 mm

[Base area: 25 mm 2]

[Base area: 24 cm 2 ]

3 Find the volume of each of the following. Give each answer correct to 1 decimal place wnere appropriate. a C

411110

IS 14 1 10 cm

pi t

cm

I

7 cm #

8 cm

t 15 cm

4 Find the volume of a sphere (correct to 1 decimal place) with a radius of: a I .2 m b 15 cm

5 Find the volume of each of the following cones, correct to 1 decimal place. a

[ 0 cm 20 mml 61)100 _ _ _ _

6 cm

6 Find the volume of each of the following pyramids.

a 12 cmj, b r 4)11 -4- , 1.1 -- . 24 cm ,-''-- 10 cm

30 cm

7 CID Calculate the volume of each of the following composite solids correct to 2 decimal places where appropriate.

a b l 0 cm

8 cm

5 cm 12 cm

5 cm

20 cm 20

12 cm .4'

—Volume 1 a 27cm3

C 3600 cm3 2 a 450 mm3 b 360 cm2 3 a 6333.5 crn3

C 280crn3 4 a 7.2m3 b 14 137.2 cm3

5 a 377.0 cm3 b 2303.8 mm3 6 a 400cm3 b 10080cm3 7 a 1400 crn3 b 10379.20 cm3 c 41.31cm3

e 218.08 cm3

SAC Calculator active. (Set 10 to Set 15 inclusive)