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NSM Enh 10_5.1-5.2 HB 02 4th.fm Page 11 Tuesday, March 10, 2009 11:21 AM
Trigonometry 1
Chapter
2
2:01
Student Name
Class
Parent Signature
Date
2:02
Right-angled Triangles
Score
Right-angled Triangles:
The Ratio of Sides
Outcome MS5.1.2
Outcome MS5.1.2
When the size of one of the other two angles in a
right-angled triangle is known, we give special
names to all three sides:
• the hypotenuse (h) is the side opposite the
right-angle
• the adjacent side (a) joins (or is next to) the
known angle to the right-angle
• the opposite side (o) is opposite (not joined)
to the known angle.
Here are some right-angled triangles. In each one there
is an angle of 40°, so the triangles are similar (same
shape), but the lengths of the sides are different.
41
Triangle 1
Triangle 3
40°
known
angle
32
hy
p
es
55°
ote
nu
s
adjacent
e
40°
opposite
Here are three labelled triangles.
B
28°
E
H
41°
62°
C
F
G
40°
I
pl
A
D
e
1
pa
g
Triangle 2
right-angle
Sa
m
Complete this table to show the labels of the three
types of side.
Triangle 4
Triangle Hypotenuse Opposite Adjacent
side
side
ΔABC
ΔDEF
AB
1
ΔGHI
2
The sides of these triangles have been labelled h, o
and a. Place a small arc with the special angle label
θ in the correct position to match the labels.
a
b
h
o
a
40°
For each one, measure in mm the opposite side,
adjacent side and hypotenuse, and then complete
the table below. Give the ratios as decimals correct
to 2 dp.
Part of the first one has been done for you.
Triangle
o
a
h
1
26
32
41
h
a
o
o
--- =
h
a
--- =
h
o
--- =
a
32
-----41
= 0.78
2
3
✂
4
CHAPTER 2 TRIGONOMETRY
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2
3
Explain what the ratio of the opposite side (o) to
the hypotenuse (h) is equal to in each of these 40°
right-angled triangles.
Explain what the ratio of the adjacent side (a) to
the hypotenuse (h) is equal to in each of these 40°
right-angled triangles.
These conversions can also be done on a
calculator that has a Degrees/Minutes/Seconds
button.
A calculator gives the value of the tan ratio for
any given angle. For example, on a calculator,
tan70° = 2.747 (3 dp).
1
Write down the tangent ratio, as a fraction and as
a decimal, for angle θ in each diagram.
a
b
13
7
θ
θ
8
Explain what the ratio of the opposite side (o) to
the adjacent side (a) is equal to in each of these
40° right-angled triangles.
2
Write down the values of tanP and tanQ in this
triangle. Give your answers as fractions and
also decimals.
es
4
10
pa
g
R
21
The Tangent Ratio
P
Outcome MS5.1.2
e
tanP:
3
pl
The tangent in a right-angled triangle is:
side opposite angle θ = o
tan θ = --------------------------------------------------------side adjacent to angle θ
a
Sa
m
5
In the example, tan θ = --- = 0.625 .
8
4
5 cm (o)
θ
8 cm (a)
At this level, angles are expressed in degrees.
Parts of degrees can either be written in decimal
form or by using units called ‘minutes’. There are
60 minutes in a degree. We use a dash, like ′, to
show minutes.
5
6
Degrees and minutes to decimals:
Example: 30°15′ = 30 + 15
------ = 30.25°
60
(Note: 15 ÷ 60 = 0.25)
tanQ:
Write these angles in decimal form.
a 42°30′
b 5°45′
c 85°3′
d 135°51′
Write these angles in degrees and minutes.
a 60.2°
b 18.8°
c 9.25°
d 112.35°
Calculate the value of these trigonometric
expressions, correct to 3 decimal places.
a tan35°
b tan49.6°
c tan53°15′
d tan86°33′
Use Pythagoras’ theorem to calculate the length
of QR, and then state the value of tanP correct to
4 decimal places.
Q
12 cm
Decimals to degrees and minutes:
Example: 71.3° = 71 + 0.3 × 60′
= 71° + 18′
= 71°18′
12
NEW SIGNPOST MATHEMATICS ENHANCED 10 STAGE 5.1, 5.2 HOMEWORK BOOK
Q
29
R
17 cm
P
✂
2:03
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2
2:04
Trigonometry 2
Student Name
Class
Parent Signature
Date
Finding an Unknown Side
2
Outcomes MS5.1.2, MS5.2.3
We can use the tan ratio equation
side opposite angle θ = o
tan θ = --------------------------------------------------------side adjacent to angle θ
a
to work out the length of the opposite side in a
right-angled triangle when the given information
is a (length of the adjacent side) and θ (one of
the angles).
20 cm
42°
x
b
52°
x
Solution: --- = tan52°
6
x = 6 × tan52°
x = 7.68 (2 dp)
A flagpole casts a shadow that is 2.7 m long when
the sun is at an angle of 58º in the sky. Estimate
the height of the flagpole to the nearest 0.1 m.
pl
e
24 cm
x
pa
g
3
Example 2: Calculate the length marked y.
75°
22°
16 m
x
y
Calculate the length of the side marked x in each
right-angled triangle. Give your answers rounded
to 2 dp.
a
Example 1: Calculate the length marked x.
6 cm
Score
es
Chapter
58°
2.7 m
1
Sa
m
Solution: Find the angle opposite y first:
y
------ = tan15°
24
y = 24 × tan15°
y = 6.43 (2 dp)
Calculate the length of the side marked x in each
right-angled triangle. Give your answers rounded
to 2 dp.
a
x
2m
4
A pencil only just fits into an empty glass that is
11 cm tall inside. The pencil makes an angle of
66o with the bottom of the glass. Calculate the
width of the inside of the glass.
60°
11 cm
66°
b
27°
✂
12 cm
x
CHAPTER 2 TRIGONOMETRY
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2:05
Finding an Unknown Angle
3
Outcomes MS5.1.2, MS5.2.3
The tan ratio can be used to calculate the size of
an angle in a right-angled triangle given the length
of the side opposite the angle and the length of
the side adjacent to the angle, as follows:
• Substitute the numbers into the tan ratio
o
formula tanθ = --, then
a
• Change the fraction to a decimal and use the
‘inverse’ tan key on your calculator.
Work out the sizes of the marked angles in these
right-angled triangles. Give your answers to the
nearest minute.
a
17 cm
θ
7 cm
b
53.6 m
θ
Example 1: Work out angle A if tanA = 0.94.
Solution: Use the tan–1 key: This gives 43.2o
(1 dp), or 43°13′ in degrees and minutes.
39.2 m
Example 2: Calculate the size of angle A:
θ
4
pl
a tanA = 0.7137
1
8
c tanA = ---
2
60 cm
Use a calculator to work out the angles that make
these trigonometric equations true. Round your
answers to 1 dp.
Sa
m
1
e
o 13.2
Solution: tanθ = -- = ---------- = 0.6168 (4dp)
a 21.4
or 31°40′ in degrees and minutes.
Check that you can get the same result on
your calculator.
A suitcase is 60 cm high. It is placed as close as
possible to the sloping roof in an attic. In this
position the suitcase is 40 cm from where the
sloping roof meets the floor. Calculate the angle
of slope of the roof.
pa
g
13.2
es
21.4
b tanA = 1.777
40 cm
5
5
19
d tanA = ------
Calculate to the nearest degree
the sizes of the angles marked
p and q in this rhombus.
p
q
Work out the sizes of the marked angles in these
right-angled triangles. Give your answers in
degrees rounded to 1 dp.
11 cm
8 cm
Fun Spot
a
3 cm
θ
A circular cake has been decorated around
the outside with 10 numerals as shown.
4 cm
4
b
24 m
18 m
3
θ
2
5
6
7
8
1
0
9
14
NEW SIGNPOST MATHEMATICS ENHANCED 10 STAGE 5.1, 5.2 HOMEWORK BOOK
✂
Show where it could be sliced with two cuts only,
so that the numerals on each resulting piece add to the
same total. Each cut is a chord of the circular cake.
NSM Enh 10_5.1-5.2 HB 02 4th.fm Page 15 Tuesday, March 10, 2009 11:21 AM
Chapter
2
2:06
Trigonometry 3
Student Name
Class
Parent Signature
Date
Sine and Cosine Ratios
3
Outcomes MS5.1.2, MS5.2.3
Score
Use Pythagoras’ theorem to calculate the length of
the unknown side in this triangle, and then state
the value of sinθ correct to 3 decimal places.
In any right-angled triangle, with a marked angle θ,
the sine and cosine ratios shown below are
constant. See the exercise in Section 2:02 for
the measurements and calculations you made
to confirm this.
45 cm
θ
53 cm
sinθ =
o
4
θ
a
side opposite θ
o
sin θ = ------------------------------------- = -hypotenuse
h
Example: Calculate the value of sinθ and cosθ
in this triangle.
θ
25
1
pl
20
cosθ = ------ = 0.8
25
Sa
m
15
Solution: sinθ = ------ = 0.6
25
e
20
15
Evaluate sinθ and cosθ for this triangle. Give each
answer as a fraction.
40
9
5
a cos60°
b sin47°
c sin54.3°
d cos9.8°
e sin74°18′
f
There is an angle A between 0° and 90° that has
the same sine value as its cosine value⎯that is,
sinA = cosA. What is the value of angle A?
2:07A
Finding Unknown Sides
with Sine and Cosine:
Finding a Short Side
Outcomes MS5.1.2, MS5.2.3
We can use the sin and cos ratios to work out the
length of the opposite side and the adjacent side in
a right-angled triangle when the given information
is h (length of hypotenuse) and θ (one of the angles).
Example 1: Calculate the length marked p.
36°
41
θ
cos5°53′
pa
g
side adjacent to θ
a
cos θ = ------------------------------------------- = -hypotenuse
h
Use your calculator to find the value of the
following, correct to 4 decimal places.
es
h
p
12 cm
2
Use Pythagoras’ theorem to calculate the length
of PR, and then state the value of cosθ correct to
4 decimal places.
P
16
Q
✂
PR =
θ
63
R
Solution: p is the opposite side to the marked
angle, 36°, so use the sin ratio equation:
o
sinθ = -h
p
sin36° = -----12
p = 12 × sin36°
p = 7.0 (to one decimal place)
cosθ =
CHAPTER 2 TRIGONOMETRY
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2
Example 2: An awning is constructed over a bus
stop to provide shelter. It is supported by a steel
rod 2.8 m in length, attached to the back of the
shelter at an angle of 50°. Calculate how far below
the shelter roof the rod is attached.
For each triangle, calculate the length of the
unknown marked side. Give answers to 2 dp.
a
c
51°
18 cm
b
Solution: We can draw a simplified right-angled
triangle and transfer the information in the
question to it.
42°
2.8 m
3
x is the adjacent side to the marked angle, 50°,
so use the cos ratio equation:
a
cosθ = -h
The top of a tent pole is fastened by a tight piece
of rope measuring 2.3 m to a point on the ground.
The angle between the rope and the ground is 70°.
pa
g
50°
es
x
d
26 m
2.3 m
rope
tent pole
1
Sa
m
pl
e
x
cos50° = ------2.8
x = 2.8 × cos50°
x = 1.8 m (to one decimal place)
70°
Calculate the height of the tent pole.
For each triangle, calculate the length of the
unknown marked side. Give answers to 2 dp.
a
5 cm
4
a
37°
When roads go around bends, engineers design
them with a ‘camber’. This makes it easier for cars
to stay on the road when cornering.
At one place on a curve a road has a camber of
4° and is 6.6 m across. Calculate the difference
in height between the sides of the road.
x
b
42°
6.6 m
4°
b
16
NEW SIGNPOST MATHEMATICS ENHANCED 10 STAGE 5.1, 5.2 HOMEWORK BOOK
✂
8m
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Chapter
Trigonometry 4
2
Student Name
Class
Parent Signature
Date
2:07B
Finding Unknown Sides
with Sine and Cosine:
Finding the Hypotenuse
b
Score
14 m
x
57°
Outcomes MS5.1.2, MS5.2.3
Trigonometric problems where the hypotenuse
has to be worked out involve solving equations.
Choose the correct trigonometric ratio, then
substitute the values from the diagram.
c
x
49°
6 km
Example: Find x, correct to 2 dp.
d
x
6
83 m
es
37°
28.6°
x
pa
g
o
Solution: sinθ = -h
6
sin37° = --x
Then invert:
x
1
--- = ---------------6 sin 37°
e
3
6
x = ---------------- = 9.97 (to one decimal place)
sin 37°
x
pl
3 km
25°
Solve these equations by inverting. Give each
answer correct to 2 dp.
Sa
m
1
10
h
a 0.4018 = ------
Calculate the distance flown by the aeroplane
through the air to a point above the coastline.
Give your answer to the nearest km.
8
h
b cos27° = ---
4
2
Calculate the length of the side marked x in each
right-angled triangle. Choose carefully whether
you should use sin or cos. Give your answers
rounded to 2 dp.
An aeroplane takes off at an angle of 25°. It
follows a straight-line course and when it reaches
the coastline it is at a height of 3 km.
A nail protrudes from a wall at an
angle of 68°. The head of the nail is
56 mm from the wall. Calculate the
length of the nail that protrudes from
the wall.
68°
56 mm
a
x
✂
32°
20 cm
CHAPTER 2 TRIGONOMETRY
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2:08
Using Sine and Cosine to
Find an Unknown Angle
3
Work out the sizes of the marked angles in these
right-angled triangles. Give your answers in
degrees and minutes.
Outcomes MS5.1.2, MS5.2.3
a
To calculate the size of an angle in a right-angled
triangle given the length of the hypotenuse and
one other side:
• Decide which pair of two sides is given (choose
from opposite/hypotenuse and adjacent/
hypotenuse). This pair tells you whether to use
the sin or cos formula.
• Substitute the numbers into the formula.
• Change the fraction to a decimal and use an
‘inverse’ trig key on your calculator.
15 m
A
20 m
b
8.9 m
A
12.6 m
Example 1: Work out angle A if sinA = 0.76.
Solution: Use the cos–1 key: This gives 49.5°
(1 dp) or 49°28′.
4
A
3 cm
Lee placed a 5-metre-long ladder against a window
ledge that was 4.6 metres above the ground.
a 3
cosA = -- = --- = 0.4286
h 7
A = 64.6° (1 dp)
a Add the measurements 5 and 4.6 to the diagram.
a sinθ = 0.3412
b Make some calculations (show your working)
pl
Find the size of θ in degrees correct to 1 decimal
place when:
Sa
m
1
window
e
Solution: The sides that measure 3 and 7 are the
adjacent and hypotenuse⎯use the cos formula:
pa
g
7 cm
es
Example 2: Calculate the size of angle A.
to decide whether the ladder was placed at a
safe angle.
b cosθ = 0.8933
4
9
c cosθ = ---
5.2
7.3
d sinθ = ------2
WARNING!!
Angle between ladder and
ground must not exceed 75°.
5
Work out the sizes of the marked angles in these
right-angled triangles. Give your answers rounded
to 1 dp.
A ski slope is exactly 150 m long and has a vertical
drop of 70 m from the top down to where the
slope ends. Calculate the angle that the ski slope
makes with the horizontal.
a
4 cm
150 m
A
b
A
15.1 m
18
70 m
6.3 m
NEW SIGNPOST MATHEMATICS ENHANCED 10 STAGE 5.1, 5.2 HOMEWORK BOOK
✂
10 cm
NSM Enh 10_5.1-5.2 HB 02 4th.fm Page 19 Tuesday, March 10, 2009 11:21 AM
Chapter
2
Trigonometry 5
Student Name
Class
Parent Signature
Date
2:09A
Miscellaneous
Exercises: Angles of
Elevation and Depression
Outcomes MS5.1.2, MS5.2.3
Surveying often involves calculations with angles
measured in relation to the horizontal.
Angles of elevation are measured upwards from
the horizontal.
Angles of depression are measured downwards
from the horizontal.
angle of elevation
2
Score
Fiona is sitting in an IMAX theatre looking at the
giant screen on the opposite wall. She is 12 m
from the wall. The apparent angle of elevation
from Fiona to the top of the screen is 55° and
the angle of elevation to the bottom of the screen
is 21°.
screen
θ
es
a Add the measurements 12 and 21° to the
angle of depression
diagram.
pa
g
b What is the size of the angle marked θ?
c Calculate the height of the screen.
Peter has climbed a stepladder and is looking
across to a flagpole, which is 6 m away from his
eyes. The angle of depression to the bottom of the
flagpole is 65° and the angle of elevation to the
top of the flagpole is 47°.
Sa
m
pl
e
1
2:09B
Miscellaneous Exercises:
Compass Bearings
Outcomes MS5.1.2, MS5.2.3
6m
a Add lines to the diagram and then label the
angles of elevation and depression.
b Use the given information to calculate the
height of the flagpole correct to the nearest
metre.
Bearings use angles together with a starting
direction of north to give directions.
True bearings are measured clockwise from
north.
Magnetic bearings give the starting direction in
which you face, and then give the angle needed
to turn.
N
N
N50°W
W
E
S
310°
✂
The diagram shows the bearing 310° and its
magnetic equivalent N 50° W.
CHAPTER 2 TRIGONOMETRY
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1
2:09C
A jet-ski rider travels for 600 m on a bearing of
025°. How far does this take her north of her
starting point?
Miscellaneous
Exercises: Other Topics
Outcomes MS5.1.2, MS5.2.3
N
N
W
1
E
025°
A car drives for 600 m up a straight road that rises
at an angle of 9°. Calculate the change in the car’s
height above sea level.
S
600 m
9°
2
A yacht sails on a bearing of 070° before it tacks
in another direction. How far has it travelled if
it tacks at a point which is 550 m north of its
starting point?
A hard-covered book on a shelf is leaning against
a bookend. The book is 232 mm high, and the
angle between the book and the shelf is 80°.
es
2
pa
g
N
070°
80°
A helicopter travels 200 km in a straight line until
it is 160 km north of its starting point.
N
3
Sa
m
3
pl
e
Calculate the distance between the bottom of the
book and the bookend.
The diagram shows a side view of a shelf
supported by an 18-cm-long bracket. The angle
between the bracket and the wall is 48°. Calculate
the width of the shelf.
N
W
shelf
E
wall 48°
bracket
18 cm
S
a Add a line to the diagram to form a right-
angled triangle and place the measurements
200 and 160 on two of the sides.
4
b Calculate the bearing that the helicopter
The roof of a warehouse slopes at an angle of
22.5°. The ‘roof run’ from the top to the side is
20 metres. Calculate the length of the rafter.
travelled on.
20 m
height
22.5°
20 m
rafter
20
NEW SIGNPOST MATHEMATICS ENHANCED 10 STAGE 5.1, 5.2 HOMEWORK BOOK
✂
length