CHAPTER

3
Right Triangle Trigonometry
Suppose you need to calculate the distance across a river for the construction
of a bridge or the height of a building or monument. Each of these distances
can be calculated using the properties of right triangles, similar triangles, and
trigonometry. Trigonometry is the branch of mathematics that studies the
relationships between angles and the lines that form them in triangles. It was
first developed for use in astronomy and geography. Today, trigonometry is
used in surveying, navigation, engineering, construction, and the sciences to
explore the relationships between the side lengths and angles of triangles.

Big Ideas
When you have completed this chapter, you will be able to …
• apply the Pythagorean theorem and primary trigonometric ratios to
solve problems involving right triangles
• solve problems involving indirect and direct measurement
• solve right triangles

Key Terms
hypotenuse
opposite side
adjacent side
tangent ratio
sine ratio
cosine ratio
primary
trigonometric
ratios

Your Measurement Organizer

Trigonometry

Imperial Measurement SI

Surface Area
& Volume

98 MHR • Chapter 3
 Astronomer

y Astronomers study matter in outer space

and the celestial bodies. They study
their compositions, motions, and origins.
Astronomers usually focus their work on
planetary science, solar astronomy, the
origin and evolution of stars, and the formation
of galaxies. Extragalactic astronomy is the study of
distant galaxies. It includes studying how distant
galaxies move, when they collide, and how they
transform as a result of this interaction.

Make the following Foldable™ to take notes on what you will learn in
Chapter 3.

1 Staple four 2 Make a mark ten 3 Cut through the top 4 Label the Foldable™
sheets of single- squares up from the two sheets up ten as shown. On the
sided grid paper bottom right edge more squares. As back of the Foldable™,
together, along of the top sheet. you do this, you will write the title What I
the left edge. Cut through the top form tabs along the Need to Work On.
Make sure the three sheets about five right side. Continue,
grid sides face squares in from this until you have four
down. mark as shown. tabs.

leave
Chapter 3 blank
3.1

3.2

10 squares 10 squares 3.3

5 squares 5 squares

Chapter 3 • MHR 99
 3.1 The Tangent Ratio

Focus on …
• explaining the
relationships between
similar triangles and
the definition of the
tangent ratio
• identifying the
hypotenuse, opposite
side, and adjacent side
for a given acute angle Vancouver, British Columbia
in a right triangle
• developing strategies
for solving right In addition to the Pacific Ocean, there are many lakes in Western
triangles Canada that are ideal for sailing. One important aspect of boating is
• solving problems using making sure you get where you want to go. Navigation is an area in
the tangent ratio which trigonometry has played a crucial role; and it was one of the
early reasons for developing this branch of mathematics.
People have used applications of trigonometry throughout history.
The Egyptians used features of similar triangles in land surveying
and when building the pyramids. The Greeks used trigonometry
to tell the time of day or period of the year by the position of the
various stars. Trigonometry allowed early engineers and builders
to measure angles and distances with greater precision. Today,
trigonometry has applications in navigating, surveying, designing
buildings, studying space, etc.

100 MHR • Chapter 3

Investigate the Tangent Ratio Materials
wind • grid paper
Sailing is a very popular activity. One of the • protractor
limitations of sailing is that a boat cannot sail • ruler
directly into the wind. Using a technique called
tacking, it is possible to sail in almost any
direction, regardless of the wind direction. When
sailing on a tack, you are forced to sail slightly
off course and then compensate for the distance
sailed when you change direction. You can use
trigonometry to determine the distance a boat is
off course before changing direction. tacking

1. a) On a sheet of grid paper draw a horizontal line 10 cm in

length to represent the intended direction.
b) Draw a tacking angle, θ, of 30°.
c) Every two centimetres, along your horizontal line, draw a
vertical line to indicate the off course distance. Label the
five triangles you created, ABC, ADE, AFG, AHI,
and AJK.

direction sailed off course

distance

θ
A
intended direction

2. Measure the base and the height for each triangle. Complete
the following table to compare the off course distance to
the intended direction. In the last column, express the
ratio, ____
off course distance , to four decimal places.
intended direction

Intended Off Course ____

Off Course Distance
Triangle Direction Distance Intended Direction
ABC
ADE
AFG
AHI
AJK

3.1 The Tangent Ratio • MHR 101

 3. a) The diagram you drew in step 1c) forms a series of nested
Did You Know?
similar triangles. How do you know the triangles are
• vertices of a triangle similar?
are commonly labelled
with uppercase letters, b) Use your knowledge of similar triangles to help describe
for example ABC how changing the side lengths of the triangle affects the
ratio ____
off course distance .
• angles of a triangle are
commonly labelled with intended direction
Greek letter variables
• some common Greek 4. a) Use your calculator to determine the tangent ratio of 30o.
letters used are To calculate the tangent ratio of 30o, make sure your
theta, θ, alpha, α, calculator is in the degree mode.
and beta, β.
Press C TAN 30 = .
b) How does the value on your calculator relate to the data
in step 2?

hypotenuse 5. In the two right triangles shown, the hypotenuse is labelled

• the side opposite the and an angle is labelled with a variable. Copy each triangle.
right angle in a right Use the words opposite and adjacent to label the side
triangle opposite the angle and the side adjacent to the angle.
opposite side
α
• the side across from
the acute angle being hypotenuse hypotenuse
 
considered in a right
triangle
• the side that does not θ
form one of the arms  
of the angle being
considered 6. Reflect and Respond
adjacent side a) Use your results from steps 1 to 4 and the terminology from
• the side that forms step 5 to describe a formula you could use to calculate the
one of the arms of tangent ratio of any angle.
the acute angle being
b) Use your formula to state the tangent ratios for ∠A and ∠B
considered in a right
triangle, but is not the in the following diagram.
hypotenuse A

c b

B a C

102 MHR • Chapter 3

Link the Ideas
A trigonometric ratio is a ratio of the A
measures of two sides of a right triangle.
adjacent hypotenuse
One trigonometric ratio is the
tangent ratio . tangent ratio

The short form for the tangent ratio of B C • for an acute angle in a
opposite
right triangle, the ratio
angle A is tan A.
of the length of the
tangent A = ______
length of side opposite ∠A opposite side to the
length of side adjacent to ∠A length of the adjacent
side adjacent

• tan A =
__
opposite
adjacent
Example 1 Write a Tangent Ratio
Write each trigonometric ratio. B
a) tan A
b) tan B
20 12

A 16 C

Solution

a) tan A =
__
opposite
b) tan B =
__
opposite
adjacent adjacent
tan A = _
BC tan B = _
AC
AC BC
tan A = _
12 tan B = _
16
16 12
tan A = _
3 _
tan B = 4
4 3

Your Turn
Calculate each trigonometric ratio. N 5
M
a) tan L
b) tan N
13 12

3.1 The Tangent Ratio • MHR 103

 Example 2 Calculate a Tangent and an Angle
a) Calculate tan 25° to four decimal places.
b) Draw a triangle to represent tan θ =
_5 . Calculate the angle θ to
4
the nearest tenth of a degree.

Solution
a) tan 25° ≈ 0.4663

b) Since tan θ =
_5 , the side opposite the A
4
angle θ is labelled 5 and the side adjacent θ
to the angle θ is labelled 4. 4
The inverse function on a calculator allows
you to apply the tangent ratio in reverse. If
you know the ratio, you can calculate the B 5 C
angle whose tangent this ratio represents.
tan θ = _5
4
θ = tan-1 _5
( )
4
θ = 51.340…°
The angle θ is 51.3°, to the
nearest tenth of a degree.

Your Turn
Explore your particular calculator to determine the sequence of
keys required. Then, calculate each tangent ratio and angle.
θ Tan θ θ Tan θ
27° 0.5095
45° 0.5543
57° 1.4653

104 MHR • Chapter 3

Example 3 Determine a Distance Using the Tangent Ratio
A surveyor wants to determine the width of a river for a proposed
bridge. The distance from the surveyor to the proposed bridge site
is 400 m. The surveyor uses a theodolite to measure angles. The
surveyor measures a 31° angle to the bridge site across the river.
What is the width of the river, to the nearest metre?

river
proposed
bridge

31°
400 m

Solution
Let x represent the distance across
the river.
Identify the sides of the triangle opposite
in reference to the given angle x
of 31°.
31°
400 m
adjacent

__
opposite
tan θ =
adjacent
_
tan 31° = x
400
400(tan 31°) = x
240.344… = x
To the nearest metre, the
width of the river is 240 m.

Your Turn
A ladder leaning against a wall forms an angle
of 63° with the ground. How far up the wall will
the ladder reach if the foot of the ladder is 2 m
from the wall?
63°
2m

3.1 The Tangent Ratio • MHR 105

 Example 4 Determine an Angle Using the Tangent Ratio
A small boat is 95 m from the base of a lighthouse that has a height
of 36 m above sea level. Calculate the angle from the boat to the
top of the lighthouse. Express your answer to the nearest degree.

36 m
θ
95 m

Solution
Identify the sides of the triangle in reference to the angle of θ.

opposite
36 m
θ
95 m
adjacent

tan θ = __
opposite
adjacent
tan θ = _
36
95
θ = 20.754…

The angle from the boat to the top of the lighthouse is

approximately 21°.

Your Turn
A radio transmission tower is to be supported by a guy wire. The
wire reaches 30 m up the tower and is attached to the ground a
horizontal distance of 14 m from the base of the tower. What angle
does the guy wire form with the ground, to the nearest degree?

106 MHR • Chapter 3

Key Ideas
• In similar triangles, corresponding angles are equal, and
corresponding sides are in proportion. Therefore, the ratios
of the lengths of corresponding sides are equal.
• The sides of a right triangle are labelled according to a
reference angle.
B B
β
hypotenuse hypotenuse
opposite adjacent

θ
A C A opposite C
adjacent
• The tangent ratio compares the length of the side opposite the
reference angle to the length of the side adjacent to the angle in a
right triangle.

tan θ = _____
length of side opposite θ
length of side adjacent to θ
• You can use the tangent ratio to
 determine the measure of one of the acute angles when the
lengths of both legs in a right triangle are known
 determine a side length if the measure of one acute angle and
the length of one leg of a right triangle are known

Check Your Understanding

Practise
1. Identify the hypotenuse, opposite, and adjacent sides
associated with each specified angle.
a) ∠X b) ∠T c) ∠L
Z S M

X Y

R T L N

2. Draw right DEF in which ∠F is the right angle.

a) Label the leg opposite ∠D and the leg adjacent to ∠D.
b) State the tangent ratio of ∠D.

3.1 The Tangent Ratio • MHR 107

 3. Determine each tangent ratio to four decimal places using
a calculator.
a) tan 74° b) tan 45°
c) tan 60° d) tan 89°
e) tan 37° f) tan 18°

4. Determine the measure of each angle, to the nearest degree.

a) tan A = 0.7 b) tan θ = 1.75
c) tan β = 0.5543 d) tan C = 1.1504

5. Draw and label a right triangle to illustrate each tangent

ratio. Then, calculate the measure of each angle, to the
nearest degree.
a) tan α =
_2 b) tan B =
_5
3 2
6. Determine the value of each variable. Express your answer to
the nearest tenth of a unit.
a)

33°
30.5 m

b)

airport 1.25 km
θ
20 km

7. Kyle Shewfelt, from Calgary, AB, was the Olympic floor

exercise champion in Athens in 2004. Gymnasts perform their
routines on a 40-ft by 40-ft mat. They use the diagonal of the
mat because it gives them greater distance to complete their
routine.
a) Use the tangent ratio to determine the angle of the
gymnastics run relative to the sides of the mat.
b) To the nearest foot, how much longer is the diagonal of the
mat than one of its sides?
Did You Know?
The Franco-Albertan Apply
flag was created by
8. Claudette wants to calculate the angles of
Jean-Pierre Grenier. The
flag was adopted by the the triangle containing the fleur-de-lys on
Association canadienne- the Franco-Albertan flag. She measures the
française de l’Alberta in legs of the triangle to be 154 cm and
March 1982. 103 cm. What are the angle measures?

108 MHR • Chapter 3

 9. A ramp enables wheelchair users and people pushing wheeled
objects to more easily access a building.
6°
3 ft
x
a) Determine the horizontal length, x, of the ramp shown.
State your answer to the nearest foot.
b) For a safe ramp, the ratio of vertical distance : horizontal
distance needs to be less than 1 : 12. Would the ramp
shown be considered a safe ramp? Explain.

10. Unit Project A satellite radio cell tower provides signals to

three substations, T1, T2, and T3. The three substations are
each located along a stretch of the main road. The cell tower is
located 24 km down a road perpendicular to the main road. A
surveyor calculates the angle from T1 to the cell tower to be 64°,
from T2 to the cell tower to be 33°, and from T3 to the cell tower
to be 26°. Calculate the distance of each substation from the
intersection of the two roads. Express your answers to the nearest
tenth of a kilometre.
cell tower

24 km

64° 33° 26°

A T1 T2 T3
main road

11. In the construction of a guitar, it is important to consider the

Did You Know?
tapering of the strings and neck. The tapering affects the tone
that the strings make. For the Six String Nation Guitar shown, The Six String Nation
Guitar, nicknamed
suppose the width of the neck is tapered from 56 mm to 44 mm
Voyageur, is made from
over a length of 650 mm. What is the angle of the taper for one 63 pieces of history and
side of the guitar strings? heritage, from every part
of Canada. It represents
many different cultures,
communities, and
characters. The guitar
is made from pieces of
wood, bone, steel, shell,
and stone from every
θ province and territory.
It literally embodies
Canadian history.

3.1 The Tangent Ratio • MHR 109

 12. When approaching a runway, a pilot needs to maneuver the
aircraft, so that it can approach the runway at a constant angle
of 3°. A pilot landing at Edmonton International Airport begins
the final approach 30 380 ft from the end of the runway. At
what altitude should the aircraft be when beginning the final
approach? State your answer to the nearest foot.

13. The Idaà Trail is a traditional route of the Dogrib, an Athapaskan-

speaking group of Dene. It stretches from Great Bear Lake to Great
Slave Lake, in the Northwest Territories. Suppose a hill on the
trail climbs 148 ft vertically over a horizontal distance of 214 ft.
a) Calculate the angle of steepness of the hill.
b) How far would you have to climb to get to the top of the hill?

Extend
14. One of the Ekati mine’s pipes,
Did You Know?
called the Panda pipe, has
Ekati mine is Canada’s northern and southern gates.
first diamond mine. It is
A communications tower
located 200 km south of
the Arctic Circle in the
stands 100 m outside the north
Northwest Territories. gate. The tower can be seen from
Diamond mines contain a point 300 m east of the south
pipes, which are gate at camp A.
cylindrical pits where
a) The distance between
diamonds are founds.
camp A and camp B is 600 m.
Calculate the diameter
of the Panda pipe.
b) Calculate the distance from
camp B to the tower.
Panda Pipe

tower
camp B
100 m 40°

Panda
pipe
open 600 m
pit
mine
300 m camp A

110 MHR • Chapter 3

15. Habitat for Humanity Saskatoon has designed a home that
provides passive solar features. The idea is to keep the sun off
the outside south wall during the summer months and to have
the wall exposed to the sun as much as possible during the
winter months. The highest angle of the sun during the summer
months is 73°.
a) Suppose the wall of the house is 20 ft tall. How much
overhang on the roof trusses should be provided so that the
shadow of the noonday sun reaches the bottom of the wall
during the summer months?
b) The lowest angle of summer
sun
the sun during the
winter months is 28°.
What height of the
winter
wall will be in direct sun
sunlight during the
winter months?
20 ft

73°
28°

16. Nistowiak Falls, located in Lac LaRonge Provincial Park is one of

the highest waterfalls in Saskatchewan. Delana, a surveyor, needs
to measure the distance across the falls. She sighted two points, C
and D, from the baseline AB. The length of baseline AB is 30 m.
Delana recorded these angle measures: ∠ACD = 90°, ∠CAB = 90°,
∠ACB = 31.3°, and ∠CDA = 44.6°
a) Determine the distance AC across the falls. Express your
answer to the nearest tenth of a metre.
b) Determine the distance CD. Express your answer
to the nearest tenth of a metre.

A
3.1 The Tangent Ratio • MHR 111
 17. Unit Project The first sound recordings were done on wax
cylinders that were 5 cm in diameter and 10 cm long. Wax
cylinders were capable of recording about 2 min of sound.
Modern music storage devices can have tremendous memory and
store thousands of songs. Janine calculated the number of wax
cylinders needed to match a 32 GB storage capacity. Imagine that
these cylinders are stacked one on top of another. From a distance
of 10 m, the angle of elevation to the top of the stack would
be 89.5°.
a) Draw and label a diagram to represent the situation.
b) Determine the height of the stack of cylinders, to the nearest
hundredth of a metre.
c) How many cylinders would need to be stacked to match
32 GB of storage?

Create Connections
18. Copy the following graphic organizer. For each item, describe its
meaning and how it relates to the tangent ratio.

ratio

adjacent side θ = 63°

tangent
opposite side tan 42°

tan θ = –3 tan θ = 1.428

4

19. Draw a right triangle in which the tangent ratio of one of the
acute angles is 1. Describe the triangle.

20. Devin stores grain in a cylindrical

granary. Suppose Devin places a 2-m-
tall board 9 m from the granary and
1.1 m away from a point on the ground.
Describe how Devin could use
trigonometry to calculate the angle
formed with the ground and the top of 2m
the granary. Then, determine this angle.
9m
1.1 m

112 MHR • Chapter 3

21. MINI LAB When measuring inaccessible distances, a surveyor Materials
can take direct measurements using a transit. A transit can • piece of cardboard
measure both horizontal and vertical angles. • large protractor
• drinking straw
Step 1 Construct a transit as shown in the diagram. Pin the
• tape
straw at the centre of the protractor.
• pin
• measuring tape
straw

tape pin tape

Step 2 Explain how a transit could be used to assess the

distance to an object. Hint: You will need to draw and
measure a baseline. This is the line from A to B in the
diagram.

A B

Step 3 To calculate the distance to some objects in your

schoolyard, use your transit to measure the
required angles.
Length of Measure Distance to the
Object Baseline AB of ∠A Object

3.1 The Tangent Ratio • MHR 113

 3.2 The Sine and Cosine Ratios

Focus on …
The first suspension bridge in Vancouver was built in 1889 by George
• using the sine ratio and
cosine ratio to solve Mackay. He had built a cabin along the canyon wall and needed a
problems involving right bridge to conveniently access his cabin. Mathematical tools, such
triangles trigonometry, can enable you to calculate distances that cannot be
• solving problems that measured directly, such as the distance across a river canyon.
involve direct and
indirect measurement In section 3.1, you learned about the tangent ratio. This ratio
compares the opposite and adjacent side lengths in reference to an
acute angle in a right triangle. There are two other trigonometric
ratios that compare the lengths of the sides of a right triangle. These
ratios, called the sine ratio and cosine ratio, involve the hypotenuse.

Materials Investigate Trigonometric Ratios

• protractor
• ruler 1. Choose an angle between 10° and 80°. This will be your
reference angle.

2. a) Draw right triangle ABC, using your reference angle.

b) Draw three right triangles similar to ABC using the
same reference angle.

114 MHR • Chapter 3

 3. Write the equivalency statements that show the similarity
of each triangle to ABC.

4. Label the sides of each triangle. Use the terms hypotenuse,

opposite, and adjacent according to the reference angle.

5. Measure the sides of each triangle. You may wish to record

the measurements in a table similar to this one or using
spreadsheet software. Express each ratio to four decimal places.
Length Length Ratio of
of of Opposite Ratio of Ratio of
Length of Opposite Adjacent to Opposite to Adjacent to
Triangle Hypotenuse Side Side Adjacent Hypotenuse Hypotenuse

6. Complete a similar table using the other acute angle in each

triangle as your reference angle.

7. Reflect and Respond Discuss with a partner the results of the

calculations of the ratios. Describe any similarities or patterns
that you notice.

8. What relationships do you observe among the ratios for the

angles between the two tables?

9. What conclusions can you make about how the ratios relate to
your reference angle?

Link the Ideas

The short form for the sine ratio of angle A is sin A. The short sine ratio
form for the cosine ratio of angle A is cos A. • for an acute angle in
B a right triangle, the
ratio of the length
of the opposite side
hypotenuse to the length of the
opposite
hypotenuse

• sin A =
___
opposite
reference angle hypotenuse
A adjacent C
cosine ratio

sin A = _____
length of side opposite ∠A • for an acute angle in
a right triangle, the
length of hypotenuse
ratio of the length
of the adjacent side
cos A = ______
length of side adjacent to ∠A to the length of the
length of hypotenuse hypotenuse

• cos A =
___
adjacent
hypotenuse

3.2 The Sine and Cosine Ratios • MHR 115

 Example 1 Write Trigonometric Ratios
Write each trigonometric ratio. B
a) sin A b) cos A
c) sin B d) cos B
5 4

A C
3

Solution

a) sin A =
___
opposite
b) cos A =
___
adjacent
hypotenuse hypotenuse
sin A = _
BC cos A = _
AC
AB AB
sin A = _
4 _
cos A = 3
5 5

c) sin B =
___
opposite
d) cos B =
___
adjacent
hypotenuse hypotenuse
sin B = _
AC cos B = _
BC
AB AB
sin B = _
3 cos B = _
4
5 5

Your Turn
Write each trigonometric ratio.
M 5
N

12 13

L
a) sin L b) cos N
c) cos L d) sin N

116 MHR • Chapter 3

Example 2 Evaluate Trigonometric Ratios
The primary trigonometric ratios and their inverses can be primary
evaluated using technology. trigonometric
a) Evaluate each ratio, to four decimal places. ratios
sin 42° cos 68° • the three ratios, sine,
b) Determine each angle measure, to the nearest degree. cosine, and tangent,
sin θ = 0.4771 cos β = 0.7225 defined in a right
triangle
Solution
a) sin 42° ≈ 0.6691 cos 68° ≈ 0.3746

b) sin θ = 0.4771 cos β = 0.7225

θ = sin-1 (0.4771) β = cos-1 (0.7225)
θ ≈ 28° β ≈ 44°

Your Turn
a) Evaluate each trigonometric ratio, to four decimal places.
sin 60° sin 30° cos 45°
b) What is the measure of each angle, to the nearest degree?
sin β = 0.4384 cos θ = 0.2079

3.2 The Sine and Cosine Ratios • MHR 117

 Example 3 Determine an Angle Using a Trigonometric Ratio
In the World Cup Downhill held at Panorama Mountain Village in
British Columbia, the skiers raced 3514 m down the mountain. If
the vertical height of the course was 984 m, determine the average
angle of the ski course with the ground. Express your answer to
the nearest tenth of a degree.

Solution
Visualize the problem by sketching
a diagram to organize the information. 3514 m
984 m
θ

sin θ = ___
opposite
For the unknown angle, the lengths of the opposite side
hypotenuse and hypotenuse are known. So, use the sine ratio.

sin θ = _984
3514
θ = sin-1 _
984
( )
3514
θ = 16.2615…°
The average angle of
the ski course is 16.3°,
to the nearest tenth
of a degree.

Your Turn
A guy wire supporting a cell tower is 24 m long. If the wire is
attached at a height of 17 m up the tower, determine the angle
that the guy wire forms with the ground.

Example 4 Determine a Distance Using a Trigonometric Ratio

A pilot starts his takeoff and climbs steadily at an angle of 12.2°.
Determine the horizontal distance the plane has travelled when it
has climbed 5.4 km along its flight path. Express your answer to
the nearest tenth of a kilometre.

Solution
Organize the information by sketching a diagram to illustrate the
problem.

5.4 km

12.2°
x

118 MHR • Chapter 3

 cos θ = ___
adjacent
How do you decide which trigonometric
hypotenuse ratio to use?
_
cos 12.2° = x
5.4
5.4(cos 12.2°) = x
5.278… = x

The horizontal distance travelled by the airplane is approximately

5.3 km.

Your Turn
Determine the height of a kite above the ground if the kite string
extends 480 m from the ground and makes an angle of 62° with
the ground. Express your answer to the nearest tenth of a metre.

Key Ideas
• The sine ratio and cosine ratio compare the lengths of the legs
of a right triangle to the hypotenuse.
sin θ = ___ cos θ = ___
opposite adjacent
hypotenuse hypotenuse
• The sine and cosine ratios can be used to calculate side lengths
and angle measures of right triangles.
• Visualizing the information that you are given and that you need
to find is important. It helps you determine which trigonometric
ratio to use and whether to use the inverse trigonometric ratio.
Determine the value of θ, to the nearest degree.

8
18
θ

cos θ = ___
adjacent
hypotenuse
cos θ = _
8
18
θ = cos-1 (_
8
18 )
θ = 63.6122…°
Angle θ is approximately 64°.

3.2 The Sine and Cosine Ratios • MHR 119

 Check Your Understanding
Practise
1. Evaluate each trigonometric ratio to four decimal places.
a) cos 34° b) cos 56.4°
c) sin 62.9° d) sin 19.6°
e) sin 90° f) cos 80°

2. Write each trigonometric ratio in lowest terms.

A 21
M G
26
10
20
C T 29
24

P
a) sin A b) sin C
c) cos C d) cos G
e) sin P f) cos P

3. Calculate the measure of each angle, to the nearest degree.

a) cos A = 0.4621 b) cos θ = 0.6779
c) sin β = 0.5543 d) sin C = 1.232

e) sin α =
_1 f) cos B =
_3
2 4

4. Determine each length of x. Express your answer to the nearest

tenth of a unit.
a) b)
x
40° 7
18°
20 x

5. Determine the measure of each angle θ. Express your answer to

the nearest tenth of a degree.
a) b)
12.8 θ
7
θ
20
16

120 MHR • Chapter 3

6. Determine the value of each variable. Express each answer to the
nearest tenth of a unit.
a)
10 m
7m
θ

b) c)

132 m

x
θ
100 m
65°
6 ft

d) vertical cliff
18°
x
70 m

Apply
7. Some farms use a hay elevator to move bales of
hay to the second storey of a barn loft. Suppose
the bottom of the elevator is 8.5 m from the
barn and the loft opening is 5.5 m above
the ground. What distance does a bale
of hay travel along the elevator?
Express your answer to the
nearest tenth of a metre.

8. A 30-m-long line is used to hold a helium weather balloon.

Due to a breeze, the line makes a 75° angle with the ground.
a) Draw a right triangle to model the problem. Label the
measurements you know. Use variables to represent the
unknown measurements.
b) Use trigonometry to determine the height of the balloon.
Express your answer to the nearest tenth of a metre.

3.2 The Sine and Cosine Ratios • MHR 121

 9. Oil rigs are found throughout Alberta. They play a crucial role in
the search for crude oil and natural gas products. Determine the
height of a rig if a 52-m-long guy wire is attached to the top of
the rig and forms an angle of 50° with the ground. Express your
answer to the nearest tenth of a metre.

10. Gerry is windsurfing at Squamish Wind

Pit, just north of Vancouver, BC. In
order to get upwind 6000 m, Gerry
sails at a 45° angle to the wind and
then turns 90° and heads toward 6000 m
his original destination. How far
would he have to sail to get directly
upwind the 6000 m? Express your
answer to the nearest tenth of
a metre. 45°

11. Toonik Tyme is Nunavut’s biggest spring festival, celebrating

the return of spring. To set up one of the holes for ice golf,
the organizers cleared a track in the form of a right angle. The
distance from the teeing area to the vertex of the right angle
is 180 yd.
a) The angle from the teeing area to the flag at the other end
of the track is 34°. Draw a diagram of the ice golf hole.
b) Determine the direct distance from the teeing area to the flag,
to the nearest yard.
c) How much shorter would the direct distance be than following
the track?

12. The PEAK 2 PEAK Gondola connects two mountain ski resorts,
Whistler Mountain and Blackcomb Mountain, near Vancouver,
BC. The straight-line distance between the two peaks is 4400 m.
The gondola travels 4600 m along a cable that sags in the centre.
Determine the approximate angle that the cable makes with the
horizontal, to the nearest degree.
4400 m

Blackcomb Whistler

122 MHR • Chapter 3

13. Dream Maker is a dolomite sculpture by
y Saskatoon,
SK, artist Floyd Wanner. In the sculpture,re, a
line can be drawn that passes through the he
centre of the two upper circles. Suppose e
this line is 129 1
___ in. long and the base
10
line is 45 in. Describe how you might
calculate
a) the height of Dream Maker
1 in.
129 __
b) the angle between the baseline 10
and the line through the two
upper circles

Did You Know?

Dolomite is a rock consisting mainly of calcium
and magnesium carbonate. It is mined around the
world, including in western Canada. Dolomite is θ
used to improve garden soil. It is also used as an 45 in.
ornamental stone, and in construction materials.

14. At Wapiti Valley Ski Area in Saskatchewan, the beginner slope

is inclined at an angle of 11.6° from the horizontal and the
advanced slope at an angle of 26.9° from the horizontal.
a) Suppose Francis skis 1200 m down the advanced slope while
Barbara skis the same distance down the beginner slope.
Predict who will cover a greater horizontal distance. Justify
your prediction.
b) Calculate the difference between the horizontal distances for
the two skiers, to the nearest tenth of a metre.

Extend
15. Michael is building a cabin at Cold Lake, AB. He has drawn a
diagram to design his roof truss. Determine the values of x, y,
and θ.

y
x
3.50 m

θ 20°
14.50 m

16. An equilateral triangle is inscribed in a circle. Determine the side

length of the triangle if the diameter of the circle is 200 cm.

3.2 The Sine and Cosine Ratios • MHR 123

 Create Connections
Materials 17. MINI LAB Work with a partner or in small groups to explore
• 1 m of foam pipe how varying the angle of a ramp in ski jumping changes the
insulation, cut launch angle and duration of flight.
lengthwise
• marble or small steel
ball
• eight to ten thick books
or bricks or a chair
• masking tape
• measuring tape
• table

Step 1 Build a ramp similar to the one shown. Place the edge
of the ramp at the end of the table. Make a sketch of the
right triangle formed by the pipe insulation, books, and
table. Include measurements of the length of each leg
of the triangle. Determine the angle formed between the
pipe insulation and the table.
Step 2 Place a marble at the top of the ramp. Without pushing,
let it roll. Observe the flight path. Mark the place where
the marble first lands on the floor, using masking tape.
Repeat this step two more times and record the horizontal
distance the marble lands from the edge of the table. You
may wish to complete a chart similar to this one.
Distance Measured
Sketch of the Measure of
Triangle the Angle (°) Trial 1 Trial 2 Trial 3

Step 3 Adjust the ramp so that it curves downward to the table

and runs flat along the table for about 20 cm before it
reaches the end. Roll the marble down the track and
record the distances.
Step 4 Add a book to the end of the ramp, so that the ramp
curves upward as it nears the end. Roll the marble and
record your measurements.
a) Describe how changing the launch angle of the ramp affects the
distance travelled by the marble. Explain why.
b) Would changing the angle of the ramp with the table affect the
distance the marble travels? Explain.

124 MHR • Chapter 3

 3.3 Solving Right Triangles

Focus on …
• explaining the
relationships between
similar right triangles
and the definitions of
the trigonometric ratios
• solving right triangles,
with or without
technology Aurora borealis above Churchill, Manitoba

• solving problems
involving one or more
right triangles The polar aurora is one of the most beautiful and impressive
displays of nature. There have been various attempts to explain
the phenomenon of these northern lights. Carl Stormer, a
Norwegian scientist, used a network of cameras that simultaneously
photographed the aurora. He used the photos to measure the parallax
angle shifts and then calculate the height of the aurora.

Materials Investigate Estimation of Distance

• metre stick or
measuring tape In this investigation you will use the method of parallax to help you
estimate the distance to an object.
1. Have a partner stand a distance away from you. Then, mark the
floor where each of you is standing using a small piece of paper
or other identifying item, such as masking tape. Stretch out your
arm with your thumb pointed upward and close your right eye.
Line your thumb up with your partner.

3.3 Solving Right Triangles • MHR 125

 2. Open your right eye and close your left eye. Do not move your
Did You Know?
outstretched arm. Have your partner move to his or her right
If you stretch your arm until he or she is in line with your thumb again. Then, mark
out in front of your face
the new location where your partner is standing.
with your thumb pointing
upward, and then close
3. Use a metre stick to measure the distance from you to your
one eye, your thumb
appears to shift slightly.
partner and the distance between your partner’s locations.
This shift is known as
4. Reflect and Respond
parallax. Your brain uses
this information to figure a) What is the relationship between the distance to your
out how far away from partner and the distance between your partner’s locations?
you objects are.
Hint: You may wish to repeat your measurements to help
you examine the pattern.
b) Explain how this relationship can help you estimate your
distance to an object.

Link the Ideas

The line of sight is the invisible line from one person or object
to another person or object. Some applications of trigonometry
involve an angle of elevation and an angle of depression.
• An angle of elevation is the angle formed by the horizontal
and a line of sight above the horizontal.
• An angle of depression refers to the angle formed by the
horizontal and a line of sight below the horizontal.
Measure the angle of elevation and the angle of depression in
the diagram. How are the measures of two angles related?
horizontal

angle of depression

line of sight

angle of elevation

horizontal

126 MHR • Chapter 3

Example 1 Use Angle of Elevation to Calculate a Height
Sean wants to calculate the height of the First Nations Native
Did You Know?
Totem Pole. He positions his transit 19.0 m to the side of the
totem pole and records an angle of elevation of 63° to the top of The First Nations Native
Totem Pole is in Beacon
the totem pole. If the height of Sean’s transit is 1.7 m, what is the
Hill Park, in Victoria, BC.
height of the totem pole, to the nearest tenth of a metre? The totem pole was
erected in 1956 and is
Solution one of the world’s tallest
Let x represent the height from the transit to the top totem poles.
of the totem pole.

tan θ = __
opposite
adjacent
tan 63° = _
x
19.0
x = 19.0(tan 63°)
x = 37.289…
Height of totem pole
= height of transit + height from transit to top of pole
= 1.7 + 37.289…
= 38.989…

The height of the First Nations Native Totem Pole

is 39.0 m, to the nearest tenth of a metre.

Your Turn
A surveyor needs to determine the height of a large
grain silo. He positions his transit 65 m from the
silo and records an angle of elevation of 52°. If the
height of the transit is 1.7 m, determine the height
of the silo, to the nearest metre.

19.0 m 63°
63

1.7 m

3.3 Solving Right Triangles • MHR 127

 Example 2 Calculate a Distance Using Angle of Depression
Natalie is rock climbing and Aaron is
Did You Know?
belaying. When Aaron pulls the
A belayer is the person rope taut to the ground, the
on the ground who
angle of depression is 73°. If
secures a climber who
is rock climbing. The
Aaron is standing 8 ft from
belayer and climber the wall, what length of
each wear a harness rope is off the ground?
that attaches to a rope.
The belayer controls Solution
how much slack is in the
Visualize the information by
rope. It takes skill and
concentration to be a sketching and labelling
successful belayer. a diagram.
Let h represent the
length of rope that
is off the ground.
73°

8 ft
Use the properties of angles to determine
θ = 90° - 73°
the angle measure of one of the acute angles θ = 17°
inside the right triangle.
The angle that the rope makes at the top with the vertical is 17°.

sin 17° = ___

opposite
hypotenuse
_
sin 17° = 8
h
h= __ 8
sin 17°
h = 27.362…
The rope off the ground is approximately 27 ft long.

Your Turn
A balloonist decides to use an empty football field for his landing
area. When the balloon is directly over the goal post, he measures
the angle of depression to the base of the other goal post to be
53.8°. Given that the distance between goal posts in a Canadian
football field is 110 yd, determine the height of the balloon.

128 MHR • Chapter 3

Example 3 Solve a Right Triangle A

Solve the triangle shown. Express each 22 cm

measurement to the nearest whole unit.
42°
Solution C B

To solve a triangle means to determine the lengths of all unknown

What information
sides and the measures of all unknown angles. To solve this are you given? Use
triangle, you need to determine the lengths of sides AC and CB the given information as
and the measure of ∠A. much as possible in your
∠A = 180° - (90° + 42°) What is the sum of the angles in a triangle? calculations.
∠A = 48°
Using ∠B as the reference angle and knowing the length of the
hypotenuse, apply the cosine ratio to calculate the length of side CB.

cos B = ___
adjacent
hypotenuse
cos 42° = _
CB
22
CB = 22(cos 42°)
CB = 16.349…

Calculate the length of side AC.

Method 1: Apply a Trigonometric Ratio
Since all angles are known, any of the primary trigonometric ratios
could be applied.
sin B = ___
opposite
How will you decide which ratio to use?
hypotenuse
sin 42° = _
AC
22
AC = 22(sin 42°)
AC = 14.720…
Method 2: Apply the Pythagorean Theorem
A
AB2 = AC2 + CB2
222 = AC2 + (16.349…)2 48°
22 cm
484 = AC2 + 267.295…
216.704…
__________
= AC2
√216.704… = AC 42°
14.720… = AC C (16.349…) cm B

Angle A measures 48°. Side CB is about 16 cm long and side AC is

about 15 cm long.

Your Turn F
Solve the triangle shown. Express each
measurement to the nearest whole unit. 42 m

D 31 m E

3.3 Solving Right Triangles • MHR 129

 Example 4 Solve a Problem Using Trigonometry
From a height of 50 m in his fire tower near Francois Lake, BC, a
ranger observes the beginnings of two fires. One fire is due west at
an angle of depression of 9°. The other fire is due east at an angle
of depression of 7°. What is the distance between the two fires, to
the nearest metre?

9° 7°
50 m

Solution
Model the problem using right triangles.
Let x and y represent the lengths of the bases of the triangles.

9° 7°
50 m
9° 7°
x y

tan 9° = __
opposite
adjacent
tan 9° = _
50
x
__
x = 50 Use the given angles to find the measure
tan 9° of one acute angle in each right triangle.
x = 315.687…

tan 7° = __
opposite
adjacent
tan 7° = _
50
y
__
y = 50
tan 7°
y = 407.217…

Add to determine the distance between the fires.

315.687… + 407.217… = 722.904…
The distance between the fires, to the nearest metre, is 723 m.

Your Turn
From his hotel window overlooking Saskatchewan Drive in
Regina, Ken observes a bus moving away from the hotel. The angle
of depression of the bus changes from 46° to 22°. Determine the
distance the bus travels, if Ken’s window is 100 m above street
level. Express your answer to the nearest metre.

130 MHR • Chapter 3

Key Ideas
• An angle of elevation is the angle between the line of sight and
the horizontal when an observer looks upward.

angle of
elevation
horizontal

• An angle of depression is the angle between the line of sight and

the horizontal when the observer looks downward.
horizontal
angle of
depression

• To solve a triangle means to calculate all unknown angle

measures and side lengths.

Check Your Understanding

Practise
1. Solve each triangle, to the nearest tenth of a unit.
a) b)
10 45°
y y x
30°
x
7
c) C d) J

7
3

U 12 B
61°
M D

2. Calculate the length of BC, to the nearest tenth of a centimetre.

a) C b) A

20°
A 40°
10 cm 8 cm
30°
D 40°
B C
D

3.3 Solving Right Triangles • MHR 131

 3. Determine the measure of ∠CAB, to the nearest degree.
C

16 cm

12 cm
A
8 cm

4. Describe each angle as it relates

to the diagram. 1
a) ∠1 2
b) ∠2 3
c) ∠3 4
d) ∠4

5. The heights of several tourist attractions are given in the table.

Determine the angle of elevation from a point 100 ft from the base
of each attraction to its top.
Attraction Location Height
a) World’s largest fire hydrant Elm Creek, MB 29 ft
b) World’s largest dinosaur Drumheller, AB 80 ft
c) Saamis Tipi Medicine Hat, AB 215 ft
d) World’s largest tomahawk Cut Knife, SK 40 ft
e) Igloo church Inuvik, NT 78 ft

Apply
6. An airplane is observed by an air traffic
controller at an angle of elevation of 52°.
The airplane is 850 m above the
observation deck of the tower. What is 850 m
the distance from the airplane to
control
the tower? Express your answer to the tower 52°
nearest metre.

7. Cape Beale Lighthouse, BC, is on a cliff that is 51 m above sea

level. The lighthouse is there to warn boats of the danger of
shallow waters and the possibility of rocks close to the shore. The
safe distance for boats from this cliff is 75 m. If the lighthouse
keeper is 10 m above ground and observes a boat at an angle of
depression of 50°, is the boat a safe distance from the cliff? Justify
your conclusion.

132 MHR • Chapter 3

 8. At night, it is possible to make precise measurements of cloud
height using a search light. An alidade is set 720 ft away from the
search light. It measures the angle of elevation to the place where
the light strikes the cloud to be 35°. What is the altitude of the
cloud? Express your answer to the nearest foot.

alidade
35°
720 ft alidade

9. The working arm of a tower crane is 192 m high and has a length
Did You Know?
of 71.6 m. Suppose the hook reaches the ground and moves along
the arm on a trolley. For the 2010 Olympic
Games in Vancouver,
the Millennium Water
Project involved building
1100 condominiums.
This project made
use of eight tower
cranes that lifted steel,
71.6 m concrete, large tools, and
generators. The cranes
often rise hundreds of
192 m feet into the air and can
reach out just as far.

a) Determine the maximum distance from the hook to the

operator when the trolley is fully extended at 71.6 m and the
minimum distance when the trolley is closest to the operator at
8.1 m. Hint: The operator is located at the vertex of the crane.
b) Determine the maximum and minimum angles of
depression from the operator to the hook on the
ground. State your answer to the nearest tenth
of a degree.

10. Arctic Wisdom involved children, parents, and

Elders gathering on Baffin Island, NU, to send a
message. To achieve the best picture of the human
image on the sea ice, an aerial photograph
was taken. The angle of depression from the
helicopter was 58° and the height of the
helicopter was 140 m. How far away from
the image was the helicopter?

3.3 Solving Right Triangles • MHR 133

 11. Unit Project A cell phone can be used to send music, but as
your location changes, you move in and out of range from one
cell to the next. Three or more cellular towers may pick up a cell
phone’s signal. A cell phone signal has been located 5 mi from
tower 1.

tower 3

7 mi

tower 1
62° tower 2

5 mi caller

a) What is the distance from the caller to tower 3?

b) How far is tower 1 from tower 3?

12. The Disabled Sailing Association had its

first sessions at the Jericho Sailing Centre in
Vancouver, BC. At a recent regatta, a television
n
news team tracked two sailboats from a
helicopter 800 m above the water. The team
observed the sailboats on the left and right
sides of the helicopter at angles of depression
of 58° and 36°, respectively.
a) Which boat is located closer to the
helicopter? Explain.
b) Determine the distance between the
two boats. Express your answer to the
nearest metre.

13. Two tourists stand on either side of the

Veterans Pole, honouring Canadian Aboriginall
war veterans, in Victoria, BC. One tourist
measures the angle of elevation of the top of
the pole to be 21°. To the other tourist, the
angle of elevation is 17°. If the height of the
pole is 5.5 m, how far apart are the tourists?
Express your answer to the nearest tenth of
a metre.

134 MHR • Chapter 3

Extend

14. From the top of a 35-m-tall building, an observer sees a truck

heading toward the building at an angle of depression of 10°.
Ten seconds later, the angle of depression to the truck is 25°.
a) Determine the distance that the truck has travelled. Express
your answer to the nearest metre.
b) If the speed limit for the area is 40 km/h, is the truck driver
following the speed limit? Explain.

15. A rectangular prism has base dimensions of 24 cm by 7 cm. A

metal rod is run from the bottom corner diagonally to the top
corner of the prism. If the rod forms an angle of 40° with the
bottom of the box, calculate the volume of the box.

24 cm 7 cm

Create Connections
16. From her apartment, Jennie measures the angle of depression
to Mike’s house. At the same time, Mike measures the angle of
elevation to Jennie’s apartment.

line of sight

a) Mike’s brother Richard observes Mike and states that Mike

made an error, because the angle of elevation must be greater
than the angle of depression. Is Richard correct? Explain
your reasoning.
b) In order to calculate the measure of angle θ, you can be given
any of the following measurements:
• the height of Jennie’s window
• the horizontal distance between buildings
• the length of line of sight
• the measure of angle α
Which measurement(s) would you prefer to be given? Explain
how you would use these measurements to calculate θ.

3.3 Solving Right Triangles • MHR 135

3 Review

3.1 The Tangent Ratio, pages 100—113

Where necessary, express your answers to the nearest tenth of a unit.
1. Triangles ABC and XYZ are similar. Calculate the lengths
of the unknown sides.
X 9 Y

12
B 3 A

2. Determine the value of the variable in each triangle.

a) b) 11

8
x
θ
60°
10

c)

37°
3

3. A group of conservationists
needs to calculate the
angle of elevation of the
river bank of the North
Saskatchewan River. They
set up a right triangle using
two measuring poles. If
they measure the vertical
height to be 64 cm and the
horizontal distance to be
50 cm, what is the angle of
elevation of the river bank?

136 MHR • Chapter 3

3.2 The Sine and Cosine Ratios, pages 114—124
Where necessary, express your answers to the nearest tenth of a unit.
4. Determine the value of the variable in each triangle.
a) b)
x 4 13
24°
32°
x
c)
12.5

15
θ

5. Augers are used to move grain into storage bins. Suppose an

auger is 67 ft long and the granary is 44 ft high. Determine the
angle formed by this auger and the ground.

6. A 14-ft ladder leans against the bottom of a window and makes

an angle of 64° with the ground. What is the height to the bottom
of the window?

3.3 Solving Right Triangles, pages 125—135

Where necessary, express your answers to the nearest tenth of a unit.
7. In ABC, BC = 7.4 km, ∠B = 90°, and ∠A = 38°.
a) Draw and label the triangle.
b) Solve ABC.

8. The angle of depression from the top of an 80-m-high cliff to a

sailboat is 21°. Determine the distance from the base of the cliff to
the sailboat.
21°
80 m

9. A lifeguard sitting on a platform that is 14 ft high observes

someone swimming. The first sighting of the swimmer is at an
angle of depression of 60°. The angle of depression becomes
30° the next time the lifeguard looks at the swimmer. Explain
whether the swimmer is moving toward or away from the
lifeguard. Use a diagram to support your answer. Then, determine
the distance that the swimmer has travelled.

Chapter 3 Review • MHR 137

3 Practice Test

Multiple Choice
For #1 to #4, choose the best answer.
1. For the similar triangles shown, R
which expression is true?

A
_
FG = _RG B
_
PQ
=_ RG
QP PG GR QF F G

C
_
RF = _GR D
_
GF = _ QP
QR RP RF RP
Q P

2. Madeleine’s dad is designing a rafter

garage to build beside their house. 0.9 m
He wants a 30-cm overhang on
each side. How long should each 1.8 m
rafter be?
A 2.0 m B 2.3 m
C 3.8 m D 4.4 m

3. A gardener uses topsoil to improve garden soil for his Regina

customers. He purchased a special pickup truck that acts like a
dump truck. If the 80-in. truck bed is raised to a 40° angle, how
high is the upper end of the truck box above the wheels?
A 51 in. B 61 in. C 67 in. D 80 in.

4. The 17th hole at the Rivershore Golf

Course near Kamloops, BC, is 197 yd
9° 197 yd
from the teeing area to the centre of the 9°
green. Suppose the largest angle at
which you can drive the golf ball to the
left or right and still land on the green is 9°.
What is the width of the green, to the nearest yard?
A 15 yd B 31 yd C 47 yd D 62 yd

138 MHR • Chapter 3

Short Answer
5. During the annual Windscape Kite Festival in Swift Current, SK,
Yves and Lucian’s kite got caught in the top of a tree. Yves wants
to use similar triangles to calculate the height of the tree. A
nearby 9-m flagpole casts a shadow that is 6 m long. Yves and
Lucian estimate the shadow of the tree to be 3.5 m long. What is
the height of the tree, to the nearest tenth of a metre? Include a
diagram of the situation.

6. Evaluate each trigonometric ratio, to four decimal places.

a) tan 17° b) sin 68° c) cos 23°

7. Calculate the measure of each angle, to the nearest degree.

a) sin θ = 0.2588 b) tan α = 5.6713 c) cos θ = 0.7431

8. Zachary was calculating the length of side CD in the figure.

His partial solution is shown.
cos 23° =
_
80
BD D

BD =
__ 80 18°
C
(cos 23°)
sin 18° =
_
BD 23°
CD A 80 cm B

CD =
_ BD
sin 18°
Before Zachary completed his work, he realized that he had made
an error. Identify Zachary’s error. Explain a strategy to help him
avoid making this error again.

Extended Response
9. The Quikcard Edmonton Minor Hockey Week is one of the largest
hockey tournaments in North America. The tournament has
grown to include more than 480 teams from Alberta.
a) Suppose the goalie’s shoulder rises to 40 in., and a
player takes a shot 20 ft from the net. Through what
angle of elevation of the puck’s flight will the goalie
make the save? Give your answer to the nearest tenth
of a degree.
b) The height of the net is 48 in. A player takes a shot over the
right shoulder of the same goalie from part a) at an angle of
elevation of 8.5°. If the puck travels a distance of 29 ft, will the
player score a goal? Explain why.

Chapter 3 Practice Test • MHR 139

 1 Unit Connections

Unit 1 Project
Use your answers to the unit project questions throughout chapters
1, 2, and 3, as well as your own research, to prepare a presentation on
music distribution. Your presentation should include the following:
• research on the history of music recording
• a comparison of various storage devices
• a description of the impact technology has had on music distribution

To complete your presentation, predict what the next technological

advance in music distribution might be. Include answers to the
following questions:
• Describe what you think the next advance in music distribution
might look like. Provide measurements for length, area, and volume
of the new equipment in both SI and imperial units.
• How might this equipment work?
• What impact might the advance have on how you access music?
• How might this equipment distribute music to people around
the world?

Unit Review
Chapter 1 Measurement Systems
1. Identify referents that could be used for the following linear
measurements.
millimetre centimetre metre
inch foot yard

2. For each total length, choose a comparable unit of measurement

in the SI system. Determine the length to the nearest tenth of a
unit. Justify your choice of units.
a) A table-tennis ball has a diameter of the width of two fingers.
Each finger is half an inch wide.
b) Samantha often wears her hair in a ponytail. Her ponytail is
5 hand-widths long. The width of her hand is 3 in.
c) It takes Everrett 14 steps to leave the classroom. One of his
paces measures half a yard.

140 MHR • Unit 1 Connections

 3. Convert each measurement to the indicated unit.
a) 3500 mm =  cm b) 3.5 ft =  in.
c) 8723 m =  km d) 4.25 ft =  m
e) 67 cm =  in. f) 14 km =  mi

4. A rectangular oak table measures 5 ft 10 in. by 3 ft 9 in. What is

the perimeter of the table, in feet and inches?
_1
5. An artist sculpts a 10 -in. tall clay model of a horse. If the scale
4
used for the sculpture is 1 : 6, how tall would the actual horse be,
in feet and inches?

Chapter 2 Surface Area and Volume

6. Calculate the area of each figure, as indicated.
a) a rectangle with dimensions 250 cm by 180 cm, in square
metres
b) a square of side length 4 mi in square yards

7. Melody is helping prepare a cake for a banquet in Nanaimo, BC.

She needs to know the amount of icing needed for the initial
covering of the cake before she adds the final decorations.
Assume that Melody does not ice the bottom of any layer.
a) What surface area does Melody need to ice for the top three
layers, in square centimetres, if the top three layers are square
and have the following dimensions:
Top layer: side length of 10 cm and a height of 7 cm
Second layer: side length of 14 cm and a height of 8.5 cm
Third layer: side length of 18 cm and a height of 9 cm
b) The volume of cake used for the bottom layer is 4000 cm3.
The bottom square layer has a height of 10 cm. What surface
area, in square centimetres, does Melody need to ice on the
bottom layer?

8. Nalze went on a field trip with his class

to the RCMP detachment in Yellowknife,
NT. In the Henry Larson Building, there
was a display on crime investigation.
Nalze used a magnifying glass to look at a
fingerprint.
a) From the photo, estimate the diameter
of the magnifying glass Nalze used.
b) If the glass is approximately 0.3 cm
thick, what is the volume of
glass used?

Unit 1 Connections • MHR 141

 9. Saskatchewan artist Jacqueline Berting created The Glass
Wheatfield - A Salute to Canadian Farmers. It is made up
of 11 000 individually crafted waist-high stalks of glass
wheat mounted in a steel base. The average cylindrical stem
is 40 in. tall with a diameter of _
1 in. Each head of
8
wheat contains the equivalent amount of glass as a cone that
is 4 in. long with a base diameter of _3 in. Approximately how
4
much glass did Jacqueline use for the sculpture?

The Glass Wheatfield

Regina Plains Museum

Chapter 3 Right Triangle Trigonometry

10. Determine the measurements of each unknown side and
unknown angle. State side lengths to the nearest tenth of
a unit and angles to the nearest degree.
a) b) 22 cm
θ
26 cm 14 cm
x x

θ
19 cm

142 MHR • Unit 1 Connections

11. A tree casts a shadow 12 m long. The
angle measured to the top of the tree
from the end of the shadow is 68°.
What is the height of the tree?
Express your answer to the nearest
tenth of a metre.

68º
12 m

12. An oil rig is held vertical by two guy wires of unequal lengths on
opposite sides of the oil rig. One of the wires makes an angle of
45° with the platform. The other wire is 90 ft long and makes an
angle of 55° with the platform. Both wires are attached 8 ft down
from the top of the rig.
a) Sketch and label a diagram of this situation.
b) Calculate the height of the oil rig, to the nearest foot.
c) Do you think the length of the unknown wire is greater than
the 90-ft wire? Justify your prediction. Then, determine the
measurement, to the nearest half of a foot.
d) Determine the distance on the platform between the two guy
wires, to the nearest half of a foot.

13. A 15-m-long ladder is placed in a driveway between two

buildings. The ladder leans against one building and reaches
12 m up the side. If the ladder is rotated to lean on the other
building, it reaches 8 m up the side. How wide is the driveway
between the two buildings?

14. Neighbourhoods A and B are situated on opposite sides of a

mountain that stands 780 m high. The angles of elevation from
each neighbourhood to the top of the mountain are 67° and 54°.
What would be the length of a tunnel from neighbourhood A to
neighbourhood B? Express your answer to the nearest tenth
of a metre.

Unit 1 Connections • MHR 143

 1 Unit Test

Multiple Choice
For #1 to #4, choose the best answer.
1. What is the distance measured between the two arrows on this
imperial ruler?

0 1 2 3 4 5 6

A
_7 in. B
_
16 in. C
_
7
in. D
_
15
in.
8 14 16 16
2. Elijah is helping install baseboards in a bedroom in the basement.
He knows that one of his paces is approximately equal to 1 yd.
If he walks 15 paces along the width of the room and 18 paces
along the length, what is the approximate perimeter of the room,
in feet?
A 99 ft B 198 ft C 270 ft D 792 ft

3. Carrie was asked to calculate the slant height of a right cone. She
is given that the surface area is 251.3 cm2 and the diameter is
10 cm. Her work is shown below.
Step 1 SA = πr2 + πrs
Step 2 251.3 = π(52) + π(5)s
Step 3
__ 251.3
=s
(25π + 5π)
Step 4 2.7 = s
When Carrie examined her work, she realized that she made her
first error in
A Step 1 B Step 2 C Step 3 D Step 4

4. The equation that could be used to calculate

the value of h in the diagram is
A cos 58° =
_ h
7.8
B tan 58° =
_ h 14.2 cm
h
7.8
C cos 61° =
_ h
14.2 61º 58º
D sin 61° =
_h
6.9 6.9 cm 7.8 cm

144 MHR • Unit 1 Test

Numerical Response
Complete the statements in #5 to #7.
5. Jett measures the diagonal of the television screen in his family
room to be 117 cm. Laura measures the diagonal of her television
screen to be 54 in. Laura’s television is  in. larger than Jett’s
television, expressed to the nearest inch.

6. A glass paperweight is in the shape of a sphere and has a volume

of 356 818 mm3. The radius of the paperweight is  mm.

7. Your school is installing a wheelchair ramp outside the front

doors. The current stairs reach a height of 0.7 m. If the ramp is
8 m long, the horizontal distance to the end of the ramp, to the
nearest tenth of a metre, is  m.

Written Response
8. Alicia found a unique gift for her friend’s birthday. She bought a
purse that is in the shape of a right pyramid with a square base.
The dimensions of the base are 12.0 cm by 12.0 cm, and the slant
height is 16.16 cm.
a) Determine the height of the purse.
b) How much space is inside the purse?
c) Alicia wants to place the purse in a gift box with a lid. She has
gift boxes of the following volumes:
• 2100 cm3
• 2200 cm3
For each size of gift box, explain whether the purse will fit
inside.

9. Given ACD is adjacent to ABC.

C
a) Write an equation that could be B
40º
used to calculate the length of AC.
5 cm
b) Calculate the length of AC.
D
c) Calculate the length of DC, to 30º
the nearest centimetre. A

10. As a spectator at a hockey game, Brennan is sitting 40 m

horizontally from the goal net. His seat is 10 m above ice level.
a) At what angle of depression is Brennan watching the goalie
make a save?
b) A seat becomes available directly below Brennan, so he moves
3 m down. Will the angle of depression from Brennan to the
goalie increase or decrease? Justify your answer.

Unit 1 Test • MHR 145