Download Oblique Triangle Trigonometry

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Chapter
4
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Oblique
Triangle
Trigonometry
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LEARNING GOALS
You will be able to develop your spatial
sense by
Using the sine law to determine side
lengths and angle measures in obtuse
triangles
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•
NEL
•
Using the cosine law to determine side
lengths and angle measures in obtuse
triangles
•
Solving problems that can be modelled
using obtuse triangles
? This scenic hole at Furry Creek golf
course near Vancouver has a dogleg
left. On a dogleg hole, golfers have
a choice between playing it safe
and making the green in two shots
or taking a chance and trying for
the green in one shot. Jay can hit a
ball between 170 and 190 yd from
the tee with a 3-iron. Is it possible
for Jay to make the green at this
hole in one shot with a 3-iron?
Explain.
159
4
Getting Started
GPS Trigonometry
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Global Positioning System (GPS) satellites were first used by the U.S. Navy
in the 1960s. At first, the U.S. Navy had only five satellites and was able
to receive precise navigational locations only once every hour. As well, the
signal was intentionally altered for civilian use, so that only the military
would have access to the full precision of the system. In 2000, civilians
also gained access, bringing the available precision of the GPS system from
1000 ft to 65 ft.
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By 2010, the GPS system in the United States had expanded to include
30 GPS satellites. Also in 2010, Russia had almost completed its own
system of 24 satellites, and both the European Community and China
were beginning to deploy GPS systems. Europe and China say their
systems will be precise to 10 m.
160
Chapter 4 Oblique Triangle Trigonometry
NEL
?
What is the altitude of the GPS satellite?
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Two of the 10 GPS tracking stations in Canada are located in Victoria,
British Columbia, and Prince Albert, Saskatchewan, 1340 km apart.
Suppose that these stations locate a GPS satellite at the same time, when
the satellite is vertically above the line segment connecting them. The
angles of elevation from the tracking stations to the satellite measure
87.7° and 88.5°.
Draw a triangle that models the two tracking stations and the satellite.
What type of triangle did you draw?
B.
Solve your triangle.
C.
Create a plan that will allow you to calculate the altitude of the satellite.
D.
Carry out your plan to determine the altitude of the satellite.
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A.
WHAT DO You Think?
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Decide whether you agree or disagree with each statement. Explain your
decision.
1. Each value of a primary trigonometric ratio corresponds to one unique
angle.
2. The sine law and cosine law relationships apply only to angles and
sides in acute or right triangles.
3. When modelling a problem that can be solved using trigonometry, the
information provided will lead to only one possible triangle.
NEL
Getting Started
161
4.1
YOU WILL NEED
Exploring the Primary
Trigonometric Ratios
of Obtuse Angles
GOAL
• calculator
EXPLORE the Math
Until now, you have used the primary
trigonometric ratios only with acute
angles. For example, you have used
these ratios to determine the side
lengths and angle measures in right
triangles, and you have used the sine
and cosine laws to determine the side
lengths and angle measures in acute
oblique triangles.
C
a
A
c
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a
B
sine law
a
b
c
5
5
sin A
sin B
sin C
cosine law
a25b21c222bccosA
oblique triangle
100°
Joe investigated the values of the primary trigonometric ratios for obtuse
angles. Using a calculator, he determined that the value of sin 100° is
0.9848… .
He knew that he could not create a right triangle with a 100° angle.
However, he knew that he could create a triangle using the supplement of
100°, which is 80°. Out of curiosity, he evaluated sin 80° and determined
that it has the same value, 0.9848… .
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Atrianglethatdoesnotcontain
a90°angle.
80°
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b
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Determine the relationships between the primary trigonometric
ratios of acute and obtuse angles.
u
100°
sin u
cos u
tan u
0.9848
(180° 2 u)
sin (180° 2 u)
80°
0.9848
cos (180° 2 u)
tan (180° 2 u)
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110°
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Joe decided to broaden his investigation. He created a table like the one
below.
120°
130°
180°
?
162
What relationships do you observe when comparing the
trigonometric ratios for obtuse angles with the trigonometric
ratios for the related supplementary acute angles?
Chapter 4 Oblique Triangle Trigonometry
NEL
Reflecting
A.
Compare your observations with a classmate’s observations. How are
they different? How are they alike?
B.
Describe any patterns you observed as the measure of the obtuse angle
increased.
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In Summary
Key Idea
• There are relationships between the value of a primary trigonometric
ratio for an acute angle and the value of the same primary trigonometric
ratio for the supplement of the acute angle.
Need to Know
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• For any angle u,
sin u 5 sin (180° 2 u)
cos u 5 2cos (180° 2 u)
tan u 5 2tan (180° 2 u)
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FURTHER Your Understanding
A
80°
B
C
1. Which of the following equations are valid? Give reasons for your
D
d) sin 122° 5 sin 58°
e) cos 135° 5 cos 45°
f ) tan 175° 5 2tan 5°
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have an equal or opposite trigonometric ratio. Check your prediction.
a) sin 15°
c) tan 35°
b) cos 62°
d) sin 170°
K
3. Determine two angles between 0° and 180° that have each sine ratio.
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F
20°
2. Calculate each ratio to four decimal places. Predict another angle that will
a) 0.64
c) 0.95
1
7
b)
d)
3
23
4. a) Identify pairs of angles with equal sine ratios in the five triangles to
the right.
b) What do you know about the cosine and tangent ratios for these
pairs of angles?
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55°
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choices.
a) sin 25° 5 sin 65°
b) cos 70° 5 2cos 110°
c) tan 46° 5 tan 134°
110°
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N
G
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O
40°
H
20°
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4.1 Exploring the Primary Trigonometric Ratios of Obtuse Angles
163
4.2
YOU WILL NEED
Proving and Applying
the Sine and Cosine Laws
for Obtuse Triangles
GOAL
• calculator
• ruler
EXPLORE…
• Anisoscelesobtusetriangle
hasoneanglethatmeasures
120°andonesidelengththat
is5m.Whatcouldtheother
sidelengthsbe?
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Explain steps in the proof of the sine and cosine laws for obtuse
triangles, and apply these laws to situations that involve obtuse
triangles.
INVESTIGATE the Math
In Lesson 3.2, you analyzed Ben’s proof of the
A
sine law for acute triangles. Ben wanted to adjust
his proof to show that the sine law also applies to
obtuse triangles. Consider Ben’s new proof:
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Step 1 I drew obtuse triangle ABC with height AD.
b
c
B
a
C
Step 2 I wrote equations for sin (180° 2 / ABC ) and sin C using the two
right triangles.
In ^ ABD,
In ^ ACD,
opposite
sin (180° 2 / ABC) 5
hypotenuse
AD
sin (180° 2 / ABC) 5
c
c sin (180° 2 / ABC) 5 AD
c sin / ABC 5 AD
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opposite
hypotenuse
AD
sin C 5
b
b sin C 5 AD
sin C 5
Step 3 Both expressions for AD equal each other (transitive property), so:
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c sin / ABC 5 b sin C
1 c sin /ABC 2
5 b
sin C
c
b
5
sin C
sin /ABC
Step 4 I drew a new height, h, from B to base b in the triangle.
In ^ ABE,
h
sin A 5
c
c sin A 5 h
In ^ CBE,
h
sin C 5
a
a sin C 5 h
A
b
E
c
D
h
B
a
C
Step 5 Both expressions for h equal each other, so:
c sin A 5 a sin C
c
a
5
sin C
sin A
164
Chapter 4 Oblique Triangle Trigonometry
NEL
I have already shown that
c
b
5
, so
sin C
sin /ABC
c
b
a
5
5
sin C
sin /ABC
sin A
How can you explain what Ben did to prove the sine law for
obtuse triangles?
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?
Why did Ben choose to write expressions for the sin (180° 2 / ABC)
and sin C ?
B.
In step 3, Ben mentions the transitive property. What is this property,
and how did he use it in this step?
C.
In step 4, Ben drew a new height in ^ ABC. Why was this necessary?
D.
Why was Ben able to equate all three side angle ratios in step 5?
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A.
Reflecting
Compare the proof above to Ben’s original proof in Lesson 3.2, pages
118 to 119. How is the proof of the sine law for obtuse triangles the
same as that for acute triangles? How is it different?
F.
If Ben started his proof by writing expressions for sin (180° 2 / CBA)
and sin A, where would he have drawn the height in step 1?
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E.
APPLY the Math
example
1
Use reasoning and the sine law to determine
the measure of an obtuse angle
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In an obtuse triangle, /B measures 23.0° and its opposite side, b, has a
length of 40.0 cm. Side a is the longest side of the triangle, with a length of
65.0 cm. Determine the measure of / A to the nearest tenth of a degree.
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Bijan’s Solution
C
b  40.0 cm
a  65.0 cm
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I drew an obtuse triangle to represent ^ ABC.
23°
B
I knew that the longest side is always opposite the
largest angle, so the 65.0 cm side must be opposite
the obtuse angle, / A.
Since ^ ABC is not a right triangle, I knew that I
could not use the primary trigonometric ratios to
determine the measure of /A.
4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles
165
Inoticedthatthediagramhastwoside-anglepairswith
onlyoneunknown,/A.Idecidedtousethesinelaw.
sin A
sin 23°
5
65.0
40.0
Themeasureofanangleistheunknown,soIused
theformofthesinelawthathastheanglesinthe
numerator.
sin A
sin 23°
b5a
b65.0
65.0
40.0
sin A 5 0.6349…
IisolatedsinA.
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65.0a
sin A
sin B
5
a
b
/ A 5 sin−1(0.6349…)
/ A 5 39.4153…°
Iusedtheinversesinetodeterminethemeasureof/ A.
Myreasoningsuggeststhat/ Amustbetheobtuse
angle.IusedtherelationshipsinA5sin(180°2A).
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/ A 5 180° 2 39.4153…°
/ A 5 140.5846…°
Themeasureoftheangleseemsappropriate,
accordingtomydiagram.
/ A measures 140.6°.
Your Turn
2
Solving a problem using the sine law
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example
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Determine the length of side AB in ^ ABC above, to the nearest tenth of a centimetre.
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Colleen and Juan observed a tethered balloon advertising the opening of a new
fitness centre. They were 250 m apart, joined by a line that passed directly below
the balloon, and were on the same side of the balloon. Juan observed the balloon
at an angle of elevation of 7o while Colleen observed the balloon at an angle of
elevation of 82o. Determine the height of the balloon to the nearest metre.
Colleen’s Solution
B
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7°
J
250 m
166
Chapter 4 Oblique Triangle Trigonometry
C D
82°
Idrewadiagramto
represent the situation.
Theheightoftheballoonis
represented by BD. I need
todetermine thelengthof
BC in order to determine
thelengthofBD.Icanuse
the sine law in ∆BJC.
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/ BCJ 5 180° 2 82°
/ BCJ 5 98°
Ideterminedthesupplementof82°todeterminethe
measureofasecondanglein^ BJC.Thisisanobtusetriangle.
/ JBC 5 180° 2 98° 2 7°
/ JBC 5 75°
Ideterminedthemeasureofthethirdanglein∆BJC.This
gave meaknownside,JC,andaknownangleopposite
this side,/JBC,inthistriangle.
JC
BC
5
sin /BJC
sin /JBC
250
BC
5
sin 1 7° 2
sin 1 75° 2
Isubstitutedtheknowninformationintothe
equationandsolvedforBC.
250
b
sin 1 75° 2
BC 5 sin 1 7° 2 a
IwroteanequationthatinvolvedBD,BC,and
theknownanglein∆BCD.
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a
BC 5 31.542....
BD
sin 1 /BCD 2 5
BC
BD
sin (82°) 5
31.542...
Isubstitutedtheknowninformationintothe
equationandsolvedforBD.
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(31.542...) (sin (82°)) 5 BD
31.235... m 5 BD
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Iusedthesinelawtowriteanequationthat
involvedBCandtheknownside-anglepair.
Your Turn
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The advertising balloon is 31 m above the ground.
Determine the distance between Juan and the balloon.
3
Use reasoning to demonstrate the cosine law for obtuse triangles
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example
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Show that the cosine law holds for obtuse triangles, using ^ ABC.
B
c
A
b
a
C
Hyun Yoon’s Solution
c
B
a
h
ACB
A
NEL
b
C
D
x
180°  ACB
IextendedthebaseofthetriangletoD.Thiscreated
twooverlappingrighttriangles,^CBDand
^ ABD,withheightBD.Italsocreatedtwo
anglesatC,/ ACBand/DCB,suchthat
/DCB5180°2/ ACB.
4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles
167
In ^ ABD
h2 5 c2 2 (b 1 x)2
In ^CBD
h2 5 a2 2 x 2
IusedthePythagoreantheoremtowritetwo
expressionsforh2,usingthetworighttriangles.
Theexpressionsthatequalh2equaleachother
(transitiveproperty).
c 2 2 (b 1 x)2 5 a2 2 x 2
c 2 5 (b 1 x)2 1 a2 2 x 2
c 2 5 b2 1 2bx 1 x 2 1 a2 2 x 2
c 2 5 a2 1 b2 1 2bx
Isolvedforc2.
Theacuteanglein^CBDhasameasureof
180°2/ ACB.
x
a
a cos (180° 2 / ACB) 5 x
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cos (180° 2 / ACB) 5
Iusedthecosineratiotowriteanexpressionforx.
Isubstitutedmyexpressionforxintomyequation.
c 2 5 a2 1 b2 1 2b[a cos (180° 2 / ACB)]
Towriteanequationthatcontainedonlymeasures
foundintheoriginaltriangle,Iusedthefollowingfact:
cos(180°2/ ACB)52cos/ ACB
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a
c 2 5 a2 1 b2 2 2ab cos / ACB
I have demonstrated the cosine law.
Your Turn
Review the proof of the cosine law for acute triangles in Lesson 3.3, pages 130 to
131. Explain how Hyun Yoon modified this proof to deal with an obtuse triangle.
4
Using reasoning and the cosine law to determine the measure
of an obtuse angle
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example
roofing cap
17.0 ft
20.3 ft
33.5 ft
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The roof of a house consists of two slanted sections, as shown.
A roofing cap is being made to fit the crown of the roof, where the
two slanted sections meet. Determine the measure of the angle
needed for the roofing cap, to the nearest tenth of a degree.
Pr
Maddy’s Solution: Substituting into the cosine law and then rearranging
a  17.0 ft

b  20.3 ft
c  33.5 ft
c2 5 a2 1 b2 2 2ab cos u
168
Chapter 4 Oblique Triangle Trigonometry
Isketchedatriangletorepresenttheproblem
situation.
Thelargestangleisu,becauseitisoppositethe
longestside.
Threesidelengthsaregiven,soIknewthatIcould
usethecosinelaw.
NEL
(33.5)2 5 (17.0)2 1 (20.3)2 2 2(17.0)(20.3) cos u
(33.5)2 2 (17.0)2 2 (20.3)2 5 22(17.0)(20.3) cos u
Isubstitutedtheknownvaluesintotheformulafor
thecosinelawandisolatedu.
1122.25 2 289 2 412.09 5 −690.2 cos u
cos−1 a2
421.16
b5u
690.2
127.6039…° 5 u
An angle of 127.6° is needed for the roofing cap.
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421.16
5 cos u
2 690.2
Myanswerisreasonable,giventhediagram.
a  17.0 ft
b  20.3 ft

c  33.5 ft
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Georgia’s Solution: Rearranging the cosine law before substituting
c2 5 a2 1 b2 2 2ab cos u
Isketchedatriangletorepresenttheproblem
situation.
Iknewthelengthsofallthreesides,soIusedthe
cosinelaw.
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c2 1 2ab cos u 5 a2 1 b2 2 2ab cos u 1 2ab cos u
SinceIwantedtosolveforu,Irearrangedthe
formulatoisolatecosu.
Isubstitutedthevaluesofa,b,andcintothe
rearrangedformula.
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Pu
c2 2 c2 1 2ab cos u 5 a2 1 b2 2 c2
2ab cos u
a2 1 b2 2 c2
5
2ab
2ab
2
a 1 b2 2 c2
cos u 5
2ab
1 17.0 2 2 1 1 20.3 2 2 2 1 33.5 2 2
cos u 5
2 1 17.0 2 1 20.3 2
cos u 5 20.6101…
u 5 cos21(20.6101…)
u 5 127.6039…°
The angle for the roofing cap should measure 127.6°.
Your Turn
Determine the angle of elevation for each roof section, to the nearest tenth
of a degree.
NEL
4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles
169
In Summary
Key Idea
• Thesinelawandcosinelawcanbeusedtodetermineunknownside
lengthsandanglemeasuresinobtusetriangles.
Need to Know
Use the sine law when
you know …
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•Thesinelawandcosinelawareusedwithobtusetrianglesinthesame
waythattheyareusedwithacutetriangles.
-thelengthsoftwosidesand
themeasureofthecontained
angle
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-thelengthsoftwosidesand
themeasureoftheanglethat
isoppositeaknownside
Use the cosine law when
you know …
-thelengthsofallthreesides
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-themeasuresoftwoangles
andthelengthofanyside
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or
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•Becarefulwhenusingthesinelawtodeterminethemeasureofanangle.
Theinversesineofaratioalwaysgivesanacuteangle,butthesupplementary
anglehasthesameratio.Youmustdecidewhethertheacuteangle,u,orthe
obtuseangle,180°2u,isthecorrectangleforyourtriangle.
•Becausethecosineratiosforanangleanditssupplementarenotequal
(theyareopposites),themeasuresoftheanglesdeterminedusingthe
cosinelawarealwayscorrect.
CHECK Your Understanding
1. There are errors in each application of the sine law or cosine law.
Identify the errors.
a)
100°
32°
5m
Chapter 4 Oblique Triangle Trigonometry
x
12 cm
x
5
x
5
sin 100°
sin 32°
170
b)
12 cm
115°
122 5 x2 1 122 2 2(12)(x) cos 115°
NEL
2. Which law could be used to determine the unknown angle measure
or side length in each triangle? For your answer, choose one of the
following: sine law, cosine law, both, neither. Explain your choice.
a)
d)
95°
12 m
28°
33 in.
x
15 m
x
b)
5 cm
25 m
x
110°
12°
x
18°
150°
3 cm
22 m
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a
c)
e)
7 cm
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35°
x
PRACTISING
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3. Determine the unknown side length in each triangle, to the nearest tenth
of a centimetre.
b)
a)
x
Pu
24.0 cm
28.0°
32.0°
2.0 cm
30.0 cm
130.0°
x
1.4 cm
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x
c)
101.0°
Pr
4.0 cm
4. Determine the unknown angle measure in each triangle, to the nearest degree.
a)
b)
44 m
118°
68 m
c)
106 cm
2 cm
x
180 cm
150°
x
5 cm
x
4 cm
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4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles
171
5. Determine each unknown angle measure to the nearest degree and
each unknown side length to the nearest tenth of a centimetre.
a)
c)
L
105°
11.2 cm
M
N
8 cm
C
10 cm
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7.5 cm
5 cm
A
B
b) R
d)
25.6 cm
X
Y
120°
S
18.7 cm
35°
Z
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a
28°
21°
T
6. A triangle has side lengths of 4.0 cm, 6.4 cm, and 9.8 cm.
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a) Sketch the triangle, and estimate the measure of the largest angle.
b) Calculate the measure of the largest angle to the nearest tenth of
a degree.
c) How close was your estimate to the angle measure you calculated?
How could you improve similar estimates in the future?
Pu
7. Wei-Ting made a mistake when using the cosine law to determine the
unknown angle measure below. Identify the cause of the error message
on her calculator. Then determine u to the nearest tenth of a degree.
12
Q
20.5 m
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12.8 m
R
10.2 m
S
H
T
172

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10
23°
B
20
202 5 102 1 122 2 2(10)(12) cos u
400 5 100 1 144 2 240 cos u
400 5 244 2 240 cos u
400 5 4cos u
100 5 cos u
−1
cos (100) 5 u
<error!> 5 u
8. In ^QRS, q 5 10.2 m, r 5 20.5 m, and s 5 12.8 m. Solve ^QRS
by determining the measure of each angle to the nearest tenth of a
degree.
9. While golfing, Sahar hits a tee shot from T toward a hole at H. Sahar
hits the ball at an angle of 23° to the hole and it lands at B. The
scorecard says that H is 295 yd from T. Sahar walks 175 yd to her ball.
How far, to the nearest yard, is her ball from the hole?
Chapter 4 Oblique Triangle Trigonometry
NEL
10. The posts of a hockey goal are 6 ft apart. A player attempts to score
by shooting the puck along the ice from a point that is 21 ft from one
post and 26 ft from the other post. Within what angle, u, must the
shot be made? Express your answer to the nearest tenth of a degree.
11. In ^DEF, /E 5 136°, e 5 124.0 m, and d 5 68.4 m. Solve the
triangle. Round each angle measure or side length to the nearest tenth.
E
136°
F
124.0 m
5.0 m
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68.4 m
D
15.0 m
12. A 15.0 m telephone pole is beginning to lean as the soil erodes.
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A cable is attached 5.0 m from the top of the pole to prevent the pole
from leaning any farther. The cable is secured 10.2 m from the base of
the pole. Determine the length of the cable that is needed if the pole is
already leaning 7° from the vertical.
cable
10.2 m
13. A building is observed from two points, P and Q, that are 105.0 m
apart. The angles of elevation at P and Q measure 40° and 32°, as
shown. Determine the height, h, of the building to the nearest tenth of
a metre.
32°
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14. A surveyor in an airplane observes
Closing
9750 m
32°
40°
Q
P 105.0 m
45°
A
B
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that the angles of depression to
points A and B, on opposite shores
of a lake, measure 32° and 45°, as
shown. Determine the width of the
lake, AB, to the nearest metre.
h
15. In ^PQR, /Q is obtuse, /R 5 12°, q 5 15.0 m, and r 5 10.0 m.
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Explain to a classmate the steps required to determine the measure of /Q.
Extending
16. Two roads intersect at an angle of 15°. Darryl is standing on one of
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the roads, 270 m from the intersection.
a) Create a problem that must be solved using the sine law.
Include a sketch and a solution.
b) Create a problem that must be solved using the cosine law.
Include a sketch and a solution.
17. The interior angles of a triangle measure 120°, 40°, and 20°.
The longest side of the triangle is 10 cm longer than the shortest side.
Determine the perimeter of the triangle, to the nearest centimetre.
NEL
4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles
173
4
Mid-Chapter Review
FREQUENTLY ASKED Questions
Q:
How are the primary trigonometric ratios for an obtuse
angle related to the primary trigonometric ratios for its
supplementary acute angle?
A:
The primary trigonometric ratios for supplementary angles (one being
acute and the other being obtuse) are either equal or opposite. The
sine ratios for supplementary angles are equal. The cosine and tangent
ratios for supplementary angles are opposites.
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Study Aid
• SeeLesson4.1.
• TryMid-ChapterReview
Questions1to3.
150°
30°
A:
cos 150° 5 20.8660...
tan 150° 5 20.5773...
sin 30° 5 0.5
cos 30° 5 0.8660...
tan 30° 5 0.5773...
Why do you sometimes need a diagram when you are using
the sine law to determine the measure of an angle, whereas
you do not need a diagram when you are using the cosine
law?
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Q:
sin 150° 5 0.5
When you use the sine law or the inverse sine to determine the
measure of an angle, there are always two possibilities for the measure:
Pu
Study Aid
• SeeLesson4.2,Examples
1,2,and4.
• TryMid-ChapterReview
Questions4to9.
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For example, 150° and 30° are supplementary angles.
sin u 5 sin (180° 2 u)
e-
The angles that give equal sine ratios are supplementary, so you
may need to check a diagram or interpret the problem carefully to
determine which angle measure is appropriate.
When you use the cosine law or the inverse cosine to determine the
measure of an angle, there is only one possible measure:
Pr
cos u 5 2cos (180° 2 u)
174
Chapter 4 Oblique Triangle Trigonometry
NEL
PRACTISING
1. Determine each ratio to four decimal places.
Then identify another angle that has an equal or
opposite ratio. Verify your answers.
a) sin 75°
d) sin 172°
b) cos 100°
e) cos 38.5°
c) tan 32°
f ) tan 122.3°
2. Draw obtuse triangle ABC and acute triangle
DEF to satisfy each of the following conditions:
a) No internal angles have the same measure.
b) Two angles, one from each triangle, have
equal sine ratios.
3. Determine all the angles that satisfy each ratio
d 5 15 cm. Determine the measure of
/F to the nearest degree.
6. a) Determine the measure of the indicated
angle in each triangle, to the nearest tenth
of a degree.
i)
ii)
5.0
4. Calculate the indicated angle measure or side
Pu
length in each triangle, to the nearest tenth.
a)
11.0 m
10.7 m
e-

15.0 m
b)
Pr
y
2.5 km
3.0
2.5

46°
b) Which angle measure in part a) could you
not have determined without a diagram?
Explain.
7. Draw an obtuse triangle, and measure two sides
(with a ruler) and one angle (with a protractor).
a) If possible, use the sine law to determine the
measure of another angle. If you cannot use
the sine law, explain why.
b) Determine a different angle measure another
way. Use a protractor to verify the measure.
8. In ^ ABC, / A is obtuse, /C 5 15°, c 5 3.0 cm,
and a 5 4.7 cm. Determine the measure of / A
to the nearest degree.
9. A boat travels 60 km due east. It then adjusts
its course by 25° northward and travels another
90 km in this new direction. How far is the
boat from its initial position, to the nearest
kilometre?
155.0°
55.0°
121.0°
NEL

bl
Lesson 4.2
c)
7.1
2.7
ic
a
below (0 , u , 180°). Round the angles to the
nearest degree.
1
a) sin u 5 0.362
d) sin u 5
2
b) cos u 5 20.75
e) cos u 5 0.214
5
c) tan u 5
f ) tan u 5 1
2
5. In ^DEF, /E 5 132°, e 5 20 cm, and
tio
n
Lesson 4.1
2.0 cm
x
Mid-Chapter Review
175
4.3
The Ambiguous Case
of the Sine Law
YOU WILL NEED
GOAL
• calculator
• ruler
• protractor
tio
n
Analyze the ambiguous case of the sine law, and solve problems
that involve the ambiguous case.
INVESTIGATE the Math
• Twosidesinanobtuse
triangleare3mand4m
inlength.Theanglethat
isoppositethe3mside
measures40°.Determinethe
measureoftheanglethatis
oppositethe4mside.
Naomi works for a company that makes supporting braces for solar panels.
She is drawing a scale diagram to show solar panels that are going to be
installed on the flat roof of a downtown high-rise. Each panel is 5.5 m
long and must be tilted at 40° to the horizontal in order to maximize
the strength of the Sun’s rays. Naomi needs to choose the length of the
supporting brace for each panel. Supporting braces are available in 1 m
increments, starting at 2 m and going up to 6 m.
ic
a
EXPLORE…
Naomi started with a 2 m brace and discovered that she could not complete
the triangle.
supporting
brace
bl
solar
panel
2 cm
40°
Idon’tthinkatrianglewith
thesemeasurementsexists.
e-
Pu
5.5 cm
Idrewascalediagramof
thesituation,using1cmto
represent1m.Idrewanangle
of40°first.ThenImeasured
5.5cmalongoneofthearmsto
representthesolarpanel.
Pr
?
176
How many different scale diagrams are possible, with the
supporting braces that are available?
A.
Work with a partner. Use a ruler and a protractor to construct a 40°
angle connected to a 5.5 cm side, as shown in Naomi’s diagram.
B.
Calculate the height of any triangle formed using the 5.5 cm side and
40° angle.
C.
The 2 cm side is too short. Try side lengths from 3 cm to 6 cm. The side
that is opposite the 40° angle can be at any angle to the 5.5 cm side.
D.
What length of supporting brace is necessary in order to have two
possible triangles? Explain.
Chapter 4 Oblique Triangle Trigonometry
NEL
Reflecting
What range of supporting brace lengths result in two possible
triangles?
F.
What information was Naomi originally given? Will this type of
information always lead to the ambiguous case of the sine law ?
G.
When dealing with a SSA situation, how does the height of the
triangle help you determine the number of possible triangles?
APPLY the Math
example
1
Asituationinwhichtwo
trianglescanbedrawn,given
theavailableinformation;the
ambiguouscasemayoccur
whenthegivenmeasurements
arethelengthsoftwosides
andthemeasureofanangle
thatisnotcontainedbythe
twosides(SSA).
tio
n
E.
ambiguous case
of the sine law
Connecting the SSA situation to the number
of possible triangles
ic
a
Given each SSA situation for ^ ABC, determine how many triangles are
possible.
a) / A 5 30°, a 5 4 m, and b 5 12 m
c) / A 5 30°, a 5 8 m, and b 5 12 m
b) / A 5 30°, a 5 6 m, and b 5 12 m
d) / A 5 30°, a 5 15 m, and b 5 12 m
Saskia’s Solution
Idrewthebeginningofatrianglewitha30°angle
anda12mside.
bl
b  12
h
30°
A
e-
Pu
h
sin 30° 5
12
12 sin 30° 5 h
6m5h
a) / A 5 30°, a 5 4 m, and b 5 12 m
12 m
4m
30°
Pr
A
No triangles are possible.
b) / A 5 30°, a 5 6 m, and b 5 12 m
12 m
6m
30°
A
One triangle is possible.
NEL
Iusedthesineratiotocalculatetheheightofthe
triangle.
Ican usethisheightasabenchmarktodecideon
sidelengthsoppositethe30°anglethatwillresult
inzero,one,ortwotriangles.
Sincea,banda,h,Iknewthatnotrianglesare
possible.
Iusedacompasstobecertain.Isetthecompasstipsto
represent4m.Iplacedonetipofthecompassatthe
openendofthe12msideandswungthepencil
tiptowardtheotherside.Thepencilcouldn’treach
thebase,soa4msidecouldnotclosethetriangle.
Sincea,banda5h,thereisonlyonepossible
triangle,arighttriangle.
Acompassarcintersectsthebaseatonlyonepoint.
4.3 The Ambiguous Case of the Sine Law
177
c) / A 5 30°, a 5 8 m, and b 5 12 m
12 m
Sincea,banda.h,therearetwopossible
triangles.
8m
Acompassarcintersectsthebaseattwopoints.
30°
A
Two triangles are possible.
d) / A 5 30°, a 5 15 m, and b 5 12 m
15 m
Sincea.b,onlyonetriangleispossible.
tio
n
A
12 m
30°
Acompassarcintersectsthebaseatonlyonepoint.
One triangle is possible.
Your Turn
example
ic
a
Determine how many triangles are possible, given / A 5 120°, a 5 15 m,
and b 5 12 m.
2
Solving a problem using the sine law
Pu
bl
Martina and Carl are part of a team that is studying weather patterns.
The team is about to launch a weather balloon to collect data. Martina’s
rope is 7.8 m long and makes an angle of 36.0° with the ground. Carl’s
rope is 5.9 m long. Assuming that Martina and Carl form a triangle in
a vertical plane with the weather balloon, what is the distance between
Martina and Carl, to the nearest tenth of a metre?
Sandra’s Solution: Using the sine law and then the cosine law
e-
Let h represent the height of the weather balloon.
Let u represent the angle for Carl’s rope.
Situation 1:
balloon
7.8 m
Pr
5.9 m
h
36.0°
Martina

Carl
h
sin 36.0 5
7.8
7.8(sin 36.0) 5 7.8a
4.5847… 5 h
Situation 2:
balloon
5.9 m
7.8 m

Carl
178
36.0°
Idrewthetriangle.
h
b
7.8
InoticedthatthisisaSSAsituation.
Ihadtodeterminetheheightofthe
triangletodetermineifthisisan
ambiguouscase.
Carl’sropeislongerthanthe
heightandshorterthanMartina’s
rope,sotherearetwopossible
triangles.Idrewthesecondtriangle.
Martina
Chapter 4 Oblique Triangle Trigonometry
NEL
Situation 1:
B
C
7.8 m

x
36.0°
M
sin u
sin 36°
5
7.8
5.9
7.8 sin 36°
sin u 5
5.9
sin u 5 0.7770…
u 5 sin−1 (0.7770…)
u 5 50.9932…°
Isubstitutedthesidelengthsandangles(includingu)
intotheformulaforthesinelawandisolatedu.
Themeasuresoftheanglesinatriangle
sumto180°.
/B 5 180° 2 36.0° 2 50.9932…°
/B 5 93.0067…°
ic
a
x 2 5 5.92 1 7.82 2 2(5.9)(7.8) cos 93.0067…°
x 2 5 100.4777…
x 5 10.0238…
In Situation 1, Martina and Carl are
10.0 m apart.
B
IalsoconsideredthesituationinwhichCarliscloser
toMartina.
Pu
7.8 m
Iusedthecosinelawtodeterminethedistance,x,
betweenMartinaandCarl.Isubstitutedtheknown
measurementsintothecosinelaw.
bl
Situation 2:
5.9 m
tio
n
5.9 m
36.0°

y
C
M
Pr
e-
sin u
sin 36°
5
7.8
5.9
7.8 sin 36°
sin u 5
5.9
sin u 5 0.7770…
u 5 sin−1 (0.7770…)
u 5 50.9932…°
u 5 180° 2 50.9932…°
u 5 129.0067… °
Iusedthesinelawtodetermineu.
Ideterminedthemeasureofthesupplementary
angle,whichissuitableforthissituation.
/B 5 180° 2 36.0° 2 129.0067… °
/B 5 14.9932…°
Themeasuresoftheanglesinatrianglesum
to180°.
y 2 5 5.92 1 7.82 2 2(5.9)(7.8) cos 14.9932…°
y 2 5 6.7433…
y 5 2.5968…
Icanuse/Binthecosinelawtodeterminethe
distance,y,betweenMartinaandCarl.
NEL
Isubstitutedthemeasureof/Bandthegivenside
lengthsintothecosinelaw.
4.3 The Ambiguous Case of the Sine Law
179
In the second situation, Martina and Carl are
2.6 m apart.
Martina and Carl are either 10.0 m
apart or 2.6 m apart.
Your Turn
example
3
Reasoning about ambiguity
tio
n
What length would Carl’s rope need to be in order for there to be only one
possible triangle that could model this situation?
bl
ic
a
Leanne and Kerry are hiking in the mountains. They left Leanne’s car in
the parking lot and walked northwest for 12.4 km to a campsite. Then
they turned due south and walked another 7.0 km to a glacier lake. The
weather was taking a turn for the worse, so they decided to plot a course
directly back to the parking lot. Kerry remembered, from the map in the
parking lot, that the angle between the path to the campsite and the path to
the glacier lake measures about 30°. What compass direction should they
follow to return directly to the parking lot?
Austin’s Solution
N
Pu
campsite
W
E
S
7.0 km
12.4 km
e-
lake

30°
Pr
current
position
180
parking
lot
Chapter 4 Oblique Triangle Trigonometry
SinceIamgivenspecificdirections,Iknowexactly
howtodrawasketchofthesituation.Thereis
onlyonewaytodrawthesketch,sothisisnot
ambiguous.
LeanneandKerrylefttheparkinglotandwalked
northwestandthensouth.
Becausethecampsiteisduenorthofthelake,
Iknewthattheangleatthelakevertexofthe
triangle,u,wouldhelpmedeterminethecompass
directionthatLeanneandKerryneedtotravel.
MydiagramshowsthatLeanneandKerryneedto
travelapproximatelysoutheast.
NEL
sin 30°
sin u
5
12.4
7.0
sin u
sin 30°
12.4a
b 5 12.4a
b
12.4
7.0
sin u 50.8857…
u 5 sin−1(0.8857)
u 5 62.3395…°
Inoticedtwoside-anglepairs,soIsubstitutedthe
valuesintothesinelawandsolvedforu.
Theangleseemedtoosmall,accordingtomy
diagram.Tocorrectthe anglemeasure,Ineeded
thesupplementaryangle.
tio
n
Correction:
u 5 180° 2 62.3395…°
u 5 117.6604…°
118°
W
E
S
bl
S62°E
ic
a
N
Your Turn
Isubtracted themeasureoftheangleinmytriangle
from180°todeterminethedirectionoftravel.
Pu
180° 2 117.6604…° 5 62.3395…°
Leanne and Kerry would need to travel in the
direction S62°E to reach the parking lot.
Pr
e-
How far would Leanne and Kerry need to travel to reach the parking lot?
NEL
4.3 The Ambiguous Case of the Sine Law
181
In Summary
Key Idea
Need to Know
tio
n
• The ambiguous case of the sine law may occur when you are given two
side lengths and the measure of an angle that is opposite one of these
sides. Depending on the measure of the given angle and the lengths of
the given sides, you may need to construct and solve zero, one, or two
triangles.
• In ^ ABC below, where h is the height of the triangle, / A and the
lengths of sides a and b are given, and / A is acute, there are four
possibilities to consider:
If / A is acute and a , h, there is
no triangle.
b
h
b
bl
A
ic
a
a
Pu
If / A is acute and a . b or
a 5 b, there is one triangle.
b
If / A is acute and a 5 h,
there is one right triangle.
h
A
If / A is acute and h , a , b,
there are two possible triangles.
a
b
A
e-
ha
a'
h
a
A
Pr
• If / A, a, and b are given and / A is obtuse, there are two possibilities
to consider:
If / A is obtuse and
a , b or a 5 b, there
is no triangle.
If / A is obtuse and a . b,
there is one triangle.
a
a
b
b
A
A
182
Chapter 4 Oblique Triangle Trigonometry
NEL
CHECK Your Understanding
1. Given each set of measurements for ^ ABC, determine if there are zero,
one, or two possibilities. Draw the triangle(s) to support your answer.
a) / A 5 75°, a 5 4 m, and b 5 12 m
b) / A 5 50°, a 5 10 m, and b 5 6 m
c) / A 5 115°, a 5 3.0 m, and b 5 9.0 m
d) / A 5 62°, a 5 2.8 m, and b 5 3.0 m
a)
b)
c)
d)
e)
f)
tio
n
2. Decide whether each description of a triangle involves the SSA situation.
In ^ ABC, /B 5 100°, a 5 8 cm, and b 5 10 cm.
In ^ DEF, /D 5 81°, e 5 9 cm, and f 5 8 cm.
In ^GHI, /G 5 40°, i 5 5 cm, and g 5 4 cm.
In ^ JKL, /L 5 15°, j 5 71 cm, and k 5 36 cm.
In ^ MNO, /O 5 28°, m 5 8.4 cm, and o 5 4.0 cm.
In ^ PQR, /Q 5 95°, q 5 1.0 cm, and r 5 0.5 cm.
ic
a
3. Calculate the height of each triangle in question 2. Determine the number
of triangles that are possible (zero, one, or two). Justify your answers.
PRACTISING
4. Decide whether each description of a triangle involves the SSA
Pu
bl
situation. If it does, determine the number of triangles (zero, one,
or two) that are possible with the given measurements. Draw the
triangle(s), and justify your answer.
a) In ^ ABC, / A 5 51°, a 5 5 m, and b 5 14 m.
b) In ^ ABC, /C 5 30°, a 5 6 mm, and c 5 12 mm.
c) In ^ ABC, /B 5 40°, a 5 12 cm, and b 5 10 cm.
d) In ^ ABC, / A 5 155°, b 5 15 m, and c 5 12 m.
5. In ^ DEF, EF 5 15.0 cm and /E 5 37°.
Pr
e-
a) Calculate the height of the triangle from
base ED.
b) Determine the possible lengths of side FD, so that there are zero,
one, or two triangles that satisfy these conditions. Draw each
triangle to support your answer.
6. A landowner claims that his property is triangular, with one side that
is 430 m long and another side that is 110 m long. The angle that is
opposite one of these sides measures 35°.
a) Determine the length of the third side of the property, to the
nearest metre.
b) Improve the description of the property to avoid confusion.
NEL
4.3 The Ambiguous Case of the Sine Law
183
7. The Raven’s Song, a traditional Tsimshian cedar canoe, is paddled away
tio
n
from a dock, directly toward a navigational buoy that is 5 km away.
After reaching the buoy, the direction of the canoe is altered and it is
paddled another 3 km. From the dock, the angle between the buoy
and the canoe’s current position measures 12°.
a) How far is the Raven’s Song from the dock?
b) Is this the only possible solution? Explain.
ic
a
BillHelincarvedtheRaven’s Songfroma600-year-oldcedartakenfromthe
NimpkishValley.Thecanoewascreatedtocarryamessageofgoodwillfrom
theFirstNationsPeoplesoftheWestCoastofBritishColumbiatothe1994
CommonwealthGamesinVictoria.
8. An obtuse triangle has two known side lengths: 4.0 m and 4.2 m.
bl
The angle that is opposite the shorter side measures 64.0°.
a) Calculate the obtuse angle in the triangle, to the nearest tenth of a
degree.
b) Is there only one possible answer? Explain.
9. Part of a highway is to be cantilevered out from a mountainside,
10. A farmer finishes repairing a fence post and then walks 250 yd through
Pr
e-
attachment point
for support beam
Pu
as shown. The width of the highway is 22 m, and the angle of the
mountain slope at the road measures 51°. An 18 m beam needs to
be installed to support the highway. Calculate possible distances,
downhill from the highway, where the support post could be fastened.
What distance would you recommend? Explain.
184
his corn field. He turns and walks another 300 yd east, until he can see
the fence post southwest of him. He realizes that he left some of his
tools at the fence post and heads directly back to it. How far does he
need to walk, to the nearest metre?
11. In an extreme adventure triathalon, participants swim 1.7 km from a
dock to one end of an island, run 1.5 km due north along the length
of the island, and then kayak back to the dock. From the dock, the
angle between the lines of sight to the ends of the island measures 15°.
How long is the kayak leg of the race?
Chapter 4 Oblique Triangle Trigonometry
NEL
12. Carol is flying a kite on level ground. The string of the kite forms
an angle of 50° with the ground. Two other girls, standing different
distances from Carol, see the kite at angles of elevation of 66° and 35°.
One girl is 11 m from Carol. All three girls are standing in a line. For
each question below, state all possible answers to the nearest metre.
a) How high is the kite above the ground?
b) How long is the string?
c) How far is the second girl from Carol?
tio
n
13. The Huqiu Tower in China was built in 961 CE. When the tower was
first built, its height was 47 m. Since then, it has tilted 2.8°. It is now
called China’s Leaning Tower. There is a point on the ground where
you can be equidistant from both the top and the bottom of the tower.
How far is this point from the base of the tower? Round your answer
to the nearest metre.
14. Create a SSA problem with zero, one, or two possible triangles. Exchange
ic
a
problems with a classmate. Sketch the situation described in your
classmate’s problem, and determine the number of possible triangles.
15. Draw a SSA situation in which there is no possible triangle.
Closing
Pu
bl
a) Label the sides and angle, and use trigonometry to confirm that
there is no possible triangle.
b) Determine the angle that would be necessary for there to be one
possible triangle.
c) What angle would be necessary for there to be two possible
triangles?
16. In ^LMN, /L is acute. Using a sketch, explain the relationship
Pr
e-
among /L, sides l and m, and the height of ^LMN for each situation
below.
a) Only one triangle is possible.
b) Two triangles are possible.
c) No triangle is possible.
Extending
17. In ^DEF, d 5 13.0 cm, f 5 15.0 cm, and /D 5 26°. The two
possible locations for vertex F are F1 and F2.
a) Calculate the area of ^DEF1.
b) Calculate the area of ^DEF2.
c) Calculate the area of ^ F1EF2.
d) Discuss with a classmate an alternative solution for determining
the area of ^ F1EF2.
NEL
4.3 The Ambiguous Case of the Sine Law
185
History Connection
YOU WILL NEED
Dioptras and Theodolites
• protractor
• string
• straw
• metre stick or tape measure
bl
ic
a
tio
n
Surveying has played an important role in most cultures, including ancient
cultures. For example, surveying tools and techniques were used to design the
pyramids in ancient Egypt, map North America, and determine the boundaries
of many nations. Tape measures, plumb lines, and levels were some of the
original surveying tools. With the development of trigonometry, tools for
measuring angles became important. One of these tools was the dioptra.
It consisted of a sighting tube attached to a protractor, and it was used in
ancient Greece and Rome to measure angles in the vertical and horizontal
planes. Historians speculate that the Romans used dioptras as early as 2600
years ago, when building tunnels and aqueducts.
This instrument was modified
to include a sighting telescope,
which can measure angles to
the nearest 2” or 0.06% of a
degree.
Use a straw, a protractor, and a plumb line to construct your own
dioptra.
e-
A.
Pu
In the 16th century, more accurate tools were developed. A polimetrum,
later called a theodolite, was used to measure horizontal angles. In the 18th
century, the theodolite was combined with the altazimuth, an instrument for
measuring vertical angles. The new, combined instrument became known as
the modern theodolite or the transit theodolite.
Pr
straw tube
protractor
view through
here
45°
0°
B.
186
45°
string and weight
Theodolites are still used in
mapping and building. They
can cost upward of $10 000.
Use your dioptra and concepts of trigonometry to determine the height
of a building or a tree.
Chapter 4 Oblique Triangle Trigonometry
NEL
Applying Problem-Solving Strategies
Analyzing an Area Puzzle
YOU WILL NEED
• scissors
• calculator
B
S
A
ic
a
C
bl
F
tio
n
E
D
Pu
^ ABC is an equilateral triangle with side lengths of 5 cm. Each side
has been extended to the vertices of ^DEF. All the extended segments
(CF, AD, and BE) are also 5 cm.
The Puzzle
Estimate how many ^ ABCs could fit into the area of ^DEF.
B.
Using scissors and extra cutouts of ^ ABC, determine exactly how
many ^ ABCs fit into ^DEF.
e-
A.
The Strategy
Describe the strategy you used to solve this puzzle.
Pr
C.
Variation
D.
Try using trigonometry to solve this puzzle.
E.
Create a similar puzzle using a different regular polygon.
NEL
4.3 The Ambiguous Case of the Sine Law
187
4.4
YOU WILL NEED
Solving Problems Using
Obtuse Triangles
GOAL
• calculator
• ruler
Solve problems that can be modelled by one or more obtuse triangles.
ic
a
A surveyor in a helicopter would like to know the width of Garibaldi Lake
in British Columbia. When the helicopter is hovering at 1610 m above the
forest, the surveyor observes that the angles of depression to two points on
opposite shores of the lake measure 45° and 82°. The helicopter and the
two points are in the same vertical plane.
Pu
bl
• Thecross-sectionofa
canalhastwoslopesandis
triangularinshape.Theangles
ofinclinationfortheslopes
measure28°and49°.When
thecanalisfullofwater,the
lengthofoneoftheslopesis
12m.Whatisthewidthof
thesurfaceofthewaterwhen
thecanalisfull?
tio
n
LEARN ABOUT the Math
EXPLORE…
Thisviewofthesouthpartof
GaribaldiLakewascapturedfrom
thePanoramaRidgetrail.
What is the width of Garibaldi Lake?
e-
?
example
1
Visualizing a triangle to solve a problem
Pr
Determine the width of the lake, to the nearest metre.
Spencer’s Solution: Creating right triangles
45°
82°
lake
188
Chapter 4 Oblique Triangle Trigonometry
Idrewadiagramofthe
helicopterovertheforest,with
itssightlines.
Anglesofdepressionarealways
measuredagainstthehorizontal,
soIdrewahorizontallineand
placedtheangles.
NEL
Becausethelakeisalsohorizontal,thealternate
interioranglesareequal.
45°
82°
Idrewthealtitudeofthehelicopteronthetriangle.
Irealizedthatthesightlinesformtworight
triangles.
82°
45°
Let a represent the distance from one end of the
lake to the point directly below the helicopter.
ic
a
Let b represent the distance from the other side of
the lake to the point directly below the helicopter.
tio
n
Lake
1610
45°
bl
a
Iredreweachrighttriangle.
Pu
1610
82°
b
1610
a
1610
a tan 45° 5 aa
b
a
1610
b
1610
b tan 82° 5 ba
b
b
1610
b5
tan 82°
b 5 226.270…
tan 82° 5
e-
tan 45° 5
1610
tan 45°
a 5 1610
Pr
a5
Width of lake 5 a 2 b
Width of lake 51610 2 226.270…
Width of lake 51383.730… m
Forbothrighttriangles,themeasureofanangle
andthelengthofitsoppositesideareknown.
Theunknownbaseistheadjacentsideofthe
angle.Iusedthetangentratiotodeterminethe
lengthofthebaseineachtriangle.
Sincea representsthewidthofthelakeanda
smallpieceoflandbeneaththehelicopter,and
brepresentsthesmallpieceoflandbeneaththe
helicopter, thewidthofthelakeisa2b.
The width of the lake is about 1384 m.
NEL
4.4 Solving Problems Using Obtuse Triangles
189
Emily’s Solution: Using the sine law
A
45°
45° 98°
D
C
lake
A
1610
45°
D
In ^ ABD:
1610
AB
1610
AB sin 45° 5 a
bAB
AB
1610
AB 5
sin 45°
AB 5 2276.883…
Pu
bl
sin 45° 5
I calculated the angle at the
helicopter, between the sight
lines, by subtraction.
ic
a
B
I used parallel lines to determine
the measure of / B. Then I
calculated the remaining angle
in the base to be 98°, since the
measures of angles in a triangle
add to 180°.
tio
n
B
I drew a diagram to represent
the situation.
37°
I used the primary trigonometric
ratios to determine the length of
AB. AB is a side in both ^ ABD
and ^ ABC.
In ^ ABC:
AB
BC
5
sin 98°
sin 37°
Pr
e-
2276.883...
sin 37°a
b 5 BC
sin 98°
190
I used the sine law to determine
the width of the lake, BC.
1383.729… 5 BC
The width of the lake is about 1384 m.
Reflecting
A.
Could Emily have used the cosine law to calculate the width
of the lake?
B.
Does Emily need to worry about the ambiguous case when
using the sine law in this situation? Explain.
Chapter 4 Oblique Triangle Trigonometry
NEL
APPLY the Math
example
2
Solving a 3-D problem
tio
n
A wind turbine called the Eye of the Wind is located at the top
of Grouse Mountain in Vancouver. Rae is standing in the viewing
pod at an altitude of 1272 m above sea level. She observes two ships
in the harbour below. The first ship is at S3.3°E, with an angle of
depression that measures 6.9°. The second ship is at S15.5°E, with
an angle of depression that measures 7.3°. Determine the distance
between the two ships, to the nearest metre.
ic
a
The Eye of the Wind was built in 2009.
The power that it generates is about
20% of the total power required for
Grouse Mountain. There is an elevator
up to the viewing pod, where visitors
can see Vancouver and the surrounding
mountains.
Rae’s Solution
E
I sketched a 3-D diagram of this situation. I noticed
that there are two right triangles.
bl
1272 m
G
H
ship 1
Pu
F
ship 2
I decided to draw the right triangles separately.
Let a represent the horizontal distance from
Rae to ship 1.
e-
Let b represent the horizontal distance from
Rae to ship 2.
E
a
Pr
G
6.9°
83.1°
1272 m
H
ship 1
The angle between the altitude of the viewing
platform and the horizontal measures 90°. If
the angle of depression measures 6.9°, then the
measure of the complementary angle in the triangle
is 83.1° because these measures must add to 90°.
In ^ EGH:
a
1272
a
1272 tan 83.1° 5 a
b 1272
1272
10 511.2416… 5 a
tan 83.1° 5
NEL
These are right triangles, so I used the tangent ratio
to determine the horizontal distance from the base
of the mountain, a and b, to each ship.
4.4 Solving Problems Using Obtuse Triangles
191
82.7°
E
7.3°
F
ship 2
b
1272 m
G
In ^ EGF:
b
1272
1272 tan 82.7° 5 a
9929.5133… 5 b
tio
n
tan 82.7° 5
b
b 1272
1272
N
12.2°
3.3°
15.5°
Idrewthesituation,asseenfromabovethewind
turbine.
E
ic
a
G
W
Bothcompassdirectionsaremeasuredagainstsouth,
soIdrew anorth–southlineandtheapproximate
sightlinestoeachship.
9930 m
Todeterminethemeasureoftheanglebetween
thetwosightlines,Isubtracted:
bl
10 511 m
15.5°23.3°512.2°
x
Pu
S
H
ship 1
F
ship 2
e-
x 2 5 (9930)2 1 (10 511)2 2 2(9930)(10 511) cos 12.2°
x 2 5 5 052 701.96
x 5 2247.8216…
Thisdistanceappearsappropriate,accordingtomy
diagram.
Pr
The distance between the two ships is
about 2248 m.
InoticedthatIhadtwoknownsidesandacontained
angle,soIusedthecosinelawtodeterminethe
distancebetweenthetwoships.
Your Turn
If you were on the bridge of ship 2, in what direction would ship 1 be?
192
Chapter 4 Oblique Triangle Trigonometry
NEL
In Summary
Key Idea
• Thesinelaw,thecosinelaw,theprimarytrigonometricratios,andthe
sumofthemeasuresoftheanglesinatrianglemayallbeusefulwhen
solvingproblemsthatcanbemodelledusingobtusetriangles.
Need to Know
tio
n
• Whensolvingproblemsthatinvolvetrigonometry,thefollowing
decisiontreemaybeusefulforchoosinganappropriatestrategy.
Draw and label a diagram with
all the given information.
Acute
triangle
Use the
sine law* or cosine law.
bl
Use the
primary
trigonometric
ratios.
Obtuse
triangle
ic
a
Right
triangle
Pu
* When you know the lengths of two sides and the measure
of an angle that is not contained by the two sides, the case
may be ambiguous.
CHECK Your Understanding
1. a) Explain how you would determine the indicated side length or
e-
angle measure in each triangle.
i)
ii)
Pr
14°
x
iii)
8 cm

1.0 m

15 cm
20°
30°
18 m
1.3 m
0.9 m
b) Use the strategies you described to determine the indicated side
lengths and angle measure in part a). Round your answers to the
nearest tenth of a unit.
c) Compare your strategies with a classmate’s strategies. Which
strategy seems to be more efficient for each triangle?
NEL
4.4 Solving Problems Using Obtuse Triangles
193
PRACTISING
2. Two forest-fire towers, A and B, are 20.3 km apart. From tower A, the
compass heading for tower B is S80°E. The ranger in each tower sees
the same forest fire. The heading of the fire from tower A is N50°E.
The heading of the fire from tower B is N60°W. How far, to the
nearest tenth of a kilometre, is the fire from each tower?
fire
50°
tower A
tio
n
N
N
60°
20.3 km
tower B
ic
a
3. The Leaning Tower of Pisa is 55.9 m tall and leans 5.5° from the
vertical. What is the distance from the top of the tower to the tip of
its shadow, when its shadow is 90.0 m long? (Assume that the ground
around the tower is level.) Round your answer to the nearest metre.
4. Shannon wants to build a regular pentagonal sun deck. She is going to
e-
5.5°
Pu
55.9 m
bl
use five 2-by-6s, each 12 ft long, to frame the perimeter. She plans to
finish the deck with 4 in. cedar planks, laid side by side and parallel
to one of the sides. Determine the length of the longest cedar plank.
Pr
shadow
194
5. Bijan is hiking in Manning Park, British Columbia. He is hiking
alone, but he has a walkie-talkie so that he can keep in touch with
his friends at the camp. The walkie-talkies have a range of 6 km.
Bijan hikes 5 km along the Skagit Bluffs Trail in a S60°E direction.
He then hikes 2 km along the Hope Pass Trail in a N30°E direction.
a) Draw a diagram to show Bijan’s hiking route. Estimate his distance
from the camp. Is he still in the range to communicate with his
friends at the camp?
b) Calculate Bijan’s distance from the camp. Can he still
communicate with his friends at the camp? Explain.
Chapter 4 Oblique Triangle Trigonometry
NEL
6. On February 28, 2010, Earth was equidistant from the spacecraft
tio
n
Dawn and the Sun, forming an isosceles triangle. The distance from
Earth to Dawn and Earth to the Sun was 0.99 AU (astronomical
units). The distance from Dawn to the Sun was 1.84 AU.
a) Draw a diagram to show Dawn, Earth, and the Sun.
b) Determine the angle between the sight lines from Earth to Dawn
and the Sun.
ic
a
Dawn was launched by NASA on September 27, 2007, with the goal of investigating
two of the largest objects in the main asteroid belt: Vesta and Ceres. Dawn was to
arrive at Vesta in July 2011 and at Ceres in February 2015.
7. A surveyor is measuring the length of a lake. He takes angle
Pu
bl
measurements from two positions, A and B, that are 136 m apart
and on the same side of the lake. From B, the measure of the angle
between the sight lines to the ends of the lake is 130°, and the
measure of the angle between the sight lines to A and one end of
the lake is 120°. From A, the measure of the angle between the
sight lines to the ends of the lake is 65°, and the measure of the
angle between the sight lines to B and the same end of the lake is
20°. Calculate the length of the lake, to the nearest metre.
8. From an airplane, the angles of depression to two forest fires measure
e-
18° and 35°. One fire is on a heading of N15°W. The other fire is on
a heading of S70°E. The airplane is flying at an altitude of 3000 ft.
What is the distance between the two fires, to the nearest foot?
9. Bert wants to calculate the height of a tree on the opposite bank
Pr
of a river. To do this, he lays out a baseline that is 80 m long and
measures the angles shown in the diagram. Is the information that
Bert has gathered sufficient to determine the height of the tree?
Justify your answer.
B
h
30°
85° 80 m
28°
A
NEL
4.4 Solving Problems Using Obtuse Triangles
195
10. Two towns, Smith Falls and Chester, are 20 km apart. From Smith
Falls, the direction to Chester is N70°E. A grass fire has been reported
on a bearing of N30°E from Smith Falls and N12°E from Chester.
Which town’s fire department is closer to the fire? How much closer
is it, to the nearest kilometre?
11. Mount Logan, in Yukon Territory, is Canada’s highest peak. In North
tio
n
America, it is second in height only to Mount McKinley. An amateur
climber is trying to calculate the height of Mount Logan. From her
campsite, the angle of elevation to the summit measures 35°. She
walks 500 m closer, up a 10° inclined slope, and measures the new
angle of elevation as 38°. Her campsite is at an altitude of 1834 m.
Determine the height of Mount Logan, to the nearest 10 m.
12. Brit and Tara are standing 8.8 m apart on a dock when they observe a
ic
a
sailboat moving parallel to the dock. When the sailboat is equidistant
from both girls, the angle of elevation to the top of its 8.0 m mast is
51° for both girls. Describe how you would determine the measure of
the angle, to the nearest degree, between Tara and the boat, as viewed
from Brit’s position. Justify your answer.
13. A zip line is going to be suspended between two trees. From the forest
Pu
bl
floor, 12 m from the base of the smaller tree, the angles of elevation to the
tree platforms measure 33° and 35°. The distance between the two trees
is 35 m.
a) Draw a diagram to represent this situation. What assumptions did
you make?
b) Calculate the length of the zip line needed.
Pr
e-
14. Determine the angle of depression for the zip line in question 13.
196
Chapter 4 Oblique Triangle Trigonometry
NEL
Closing
15. Sketch an obtuse oblique triangle, and label any three measurements
(side lengths or angles). Exchange triangles with a classmate. Solve
your classmate’s triangle, if possible. If your classmate’s triangle is
impossible to solve, explain to your classmate why it is impossible
to solve.
tio
n
Extending
16. A sailor, out on a lake, sees two lighthouses that are 11 km apart.
ic
a
Lighthouse A is in the direction N47°W and lighthouse B is in the
direction N5°W. As seen from lighthouse A, lighthouse B is in the
direction N8°E.
a) How far, to the nearest kilometre, is the sailor from each
lighthouse?
b) Assuming that the shore runs on a straight line through both
lighthouses, what is the least distance from the sailor to the shore?
Round your answer to the nearest kilometre.
17. An airport radar operator locates two airplanes that are flying toward
Pu
bl
the airport. The first airplane, P, is 120 km from the airport, A, in a
N70°E direction and at an altitude of 2.7 km. The other airplane, Q,
is 180 km away, in a S40°W direction and at an altitude of 1.8 km.
Calculate the distance between the two airplanes to the nearest tenth
of a kilometre.
Math in Action
Measuring the Viewing Angle of a Screen
viewing
angle
Pr
e-
Televisionsets,computermonitors,andotherscreensare
oftenrankedaccordingtotheirviewingangle.Ifascreen
hasaviewingangleof80°,thismeansthatthecolour,
brightness,andimagequalitydonotappeartodegrade
untilyouareviewingthescreenatanangleof80°ormore,
measuredfromthecentralaxisofthescreen.
• Withapartnerorinasmallgroup,makeaplantomeasure
theviewingangleofascreen.
• Testyourplan.Whatadjustmentsdidyouneedtomake
asyoumeasured?
• Compareyourresultswiththeresultsofotherpairsor
groups.Areyousatisfiedthatyourplanworkedwell?
Explain.
NEL
4.4 Solving Problems Using Obtuse Triangles
197
4
Chapter Self-Test
1. Determine a, where 0 < a < 180°, for each trigonometric ratio below.
Round your answer to the nearest tenth of a degree.
1
c) sin a 5 0.015
e) sin a 5
2
3
b) tan a 5 2.314
d) cos a 5 2
f) sin a 5 0.600
4
2. For each description below, determine if there are zero, one, or two
possible triangles. Draw the triangle(s), if possible, including the
unknown measurements.
a) In ^DEF, d 5 5 cm, e 5 3 cm, and f 5 9 cm.
b) In ^ ABC, / A 5 25°, b 5 3 m, and c 5 10 m.
c) In ^ JKL, / J 5 55°, j 5 10.4 km, and k 5 11.6 km.
d) In ^ PQR, / P 5 17°, /Q 5 110°, and r 5 26 mm.
e) In ^ FUN, / F 5 75°, f 5 25 cm, and n 5 47 cm.
ic
a
tio
n
a) cos a 5 0.235
3. Two workers are helping a crane operator lower a crate to the ground.
bl
The workers are standing in line with the crate and each other. Each of
them has a rope attached to the crate. The first worker has a 35 ft rope
that makes an angle of 50° with the ground. The second worker has
a 30 ft rope. Determine the distance, to the nearest foot, between the
two workers.
4. A vertical tree stands on the side of a ski run. The angle of inclination
Pu
for the ski run measures 40°. When the angle of elevation of the
Sun measures 65°, the tree casts a shadow 75.0 m down the slope.
Determine the height of the tree, to the nearest tenth of a metre.
5. ^ ABC is an equilateral triangle with a perimeter of 36 cm. Three
e-
triangles are created when / A is divided into three equal angles. Two
of these triangles are obtuse. Determine the side lengths of the obtuse
triangles, to the nearest centimetre.
Pr
6. The following describes the location of a buried treasure. From the
pine tree, walk 30 paces N20°E, then turn and walk 15 paces until the
tree is due south. How many paces would you need to walk due north
of the tree to reach the buried treasure?
7. From the window of a building, 45 m up, the angles of depression
to two different intersections measure 76° and 65°. The measure
of the angle between the lines of sight to the two intersections is
135°. Calculate, to the nearest metre, the distance between the two
intersections.
WHAT DO You Think Now? Revisit What Do You Think? on
page 161. How have your answers and explanations changed?
198
Chapter 4 Oblique Triangle Trigonometry
NEL
4
Chapter Review
FREQUENTLY ASKED Questions
In a SSA situation, how do you know how many triangles
are possible?
A:
In a SSA situation, you know the lengths of two sides and the measure
of an angle that is opposite one of the sides. After drawing a diagram
for the situation, you should first determine the height of the triangle,
opposite the known angle. Then you can determine the number of
possible triangles by comparing the height of the triangle with the
length of the side that is opposite the given angle.
• Iftheheightisgreaterthanthelengthofthesidethatisopposite
the given angle, no triangle is possible.
• Iftheheightequalsthelengthofthesidethatisoppositethegiven
angle, one triangle is possible.
• Iftheheightislessthanthelengthofthesidethatisoppositethe
given angle, two triangles may be possible:
- If the length of the side that is opposite the given angle is less
than the length of the other given side, two triangles are possible.
- If the length of the side that is opposite the given angle is greater
thanorequaltothelengthoftheothergivenside,onetriangleis
possible.
Q:
When solving a problem that can be modelled by an obtuse
triangle, how do you decide whether to use the sine law or
the cosine law?
A:
Use the same decision process that you used for acute triangles:
• Drawaclearlylabelleddiagram.Includegiveninformationand
information you can deduce.
• Usethesinelawifyouhaveeitherofthefollowingsituations:
Study Aid
• SeeLesson4.3,Examples
1to3.
• TryChapterReview
Questions5and7.
e-
Pu
bl
ic
a
tio
n
Q:
Study Aid
• SeeLesson4.4,Examples
1and2.
• TryChapterReview
Question8.
C
b
Pr
b
A
a
A
• Usethecosinelawifyouhaveeitherofthesesituations:
b
a
c
NEL
b
A
c
Chapter Review
199
PRACTISING
Lesson 4.1
Lesson 4.3
1. Describe, using examples, the relationships
5. For each description, determine the number
between the primary trigonometric ratios for
supplementary angles.
of possible triangles. Draw the triangle(s) to
support your answer.
a) In ^ ABC, / A 5 53°, a 5 7 m, and
b 5 15 m.
b) In ^ ABC, / A 5 27°, a 5 5 m, and
b 5 6 m.
c) In ^ ABC, / A 5115°, a 5 23.0 m, and
b 5 6.0 m.
2. Determine each trigonometric ratio. Predict
tio
n
another angle that has an equal or opposite
ratio. Check your prediction.
a) sin 122°
c) cos 100°
b) sin 58°
d) tan 15°
Lesson 4.2
6. Determine the unknown side length or angle
3. Determine the unknown angle measure that is
measure that is indicated in each triangle, to the
nearest tenth of a unit.
ic
a
indicated in each triangle, to the nearest tenth
of a unit.
a)
a)
4.0 m
x
42°
5.0 m
b)
b)
e-
16.2 cm

16.5 cm
Pr
21.2 cm
4. In ^ ABC, / A 5 125°, /B 5 30°, and the
side between these angles is 8.0 cm long. Solve
the triangle. Round each measure to the nearest
tenth of a unit, as necessary.
200
32°
Pu
 118°
4.8 cm
bl
5.3 cm
9.8 m
Chapter 4 Oblique Triangle Trigonometry
8.0 m

7. A 4.3 m ramp for a mountain-bike trail is inclined
at a 15° angle with the ground. The length of the
support that creates the incline is 1.3 m.
a) Determine the distance along the ground
between the base of the support and the
beginning of the ramp, to the nearest tenth
of a metre.
b) How high above the ground is the take-off
point on the ramp, to the nearest tenth of a
metre?
Lesson 4.4
8. An airplane passes over an airport and continues
flying on a heading of N70°W for 3 km. The
airplane then turns left and flies another 2 km
until the airport is exactly due east of its position.
What is the distance between the airplane and
the airport, to the nearest tenth of a kilometre?
NEL
4
Chapter Task
Stage Lighting and Trigonometry
Suppose that you are designing the lighting for
a presentation. You are using four automated
lights hung from a V-shaped bar in the ceiling,
directly above the stage. The presentation
requires two main focus areas on the stage,
spaced 5 m apart, as shown in the diagram.
tio
n
Whether you are at a rock concert or a play, lighting has an important role
in making the show enjoyable. When designing for a stage, the lighting
designer must consider the focus areas, which are the areas where the
director wants the audience to look. Each focus area may require light from
multiple sources.
1m
2m
1m
light 2
light 3
light 4
ic
a
light 1
2m
150°
Determine the angle measures from the
vertical so that lights 1 and 2 will shine on
focus area A.
A
B.
stage left
Determine the angle measures from the bar so that lights 3 and 4 will
shine on focus area A.
Determine the angle measures from the bar so that lights 1 to 4 will
shine on focus area B.
Pr
C.
D.
NEL
B
5m
e-
A.
What angle measures are needed for
each light to illuminate each focus
area?
Pu
?
bl
Lights 1 and 2 have already been set and
measured to focus on area A. The measures of
the angles are 70° and 55°, with respect to the
bar. You now need to program all four lights
so that they can be directed to each focus area.
Each light must be programmed with an angle
measure from the vertical, not from the bar.
Determine the angle measures from the vertical to program all four
lights so they can be directed to shine on both focus areas.
stage right
Task
Checklist
✔ Didyoudrawclear,labelled
diagrams?
✔ Didyouexplainyour
solutionclearly?
Chapter Task
201
4
Project Connection
Carrying Out Your Research
As you continue with your project, you will need to conduct research and
collect data. The strategies that follow will help you collect data.
tio
n
Considering the Type of Data You Need
There are two different types of data that you need to consider: primary
and secondary. Primary data is data that you collect yourself using surveys,
interviews, and direct observations. Secondary data is data you obtain
through other sources, such as online publications, journals, magazines,
and newspapers.
ic
a
Both primary data and secondary data have their pros and cons. Primary
data provides specific information about your research question or
statement, but may take time to collect and process. Secondary data is
usually easier to obtain and can be analyzed in less time. However, because
the data was gathered for other purposes, you may need to sift through it to
find what you are looking for.
Pu
bl
The type of data you choose can depend on many factors, including the
research question, your skills, and available time and resources. Based on
these and other factors, you may choose to use primary data, secondary
data, or both.
Assessing the Reliability of Sources
Pr
e-
When collecting primary data, you must ensure the following:
• Forsurveys,thesamplesizemustbereasonablylargeandtherandom
sampling technique must be well designed.
• Forsurveysorinterviews,thequestionnairesmustbedesignedto
avoid bias.
• Forexperimentsorstudies,thedatamustbefreefrommeasurementbias.
• Thedatamustbecompiledaccurately.
202
Chapter 4 Oblique Triangle Trigonometry
NEL
Pu
bl
ic
a
tio
n
When obtaining secondary data, you must ensure that the source of your
data is reliable:
• Ifthedataisfromareport,determinewhattheauthor’scredentialsare,
how up-to-date the data is, and whether other researchers have cited the
same data.
• Beawarethatdatacollectionisoftenfundedbyanorganizationwith
an interest in the outcome or with an agenda that it is trying to further.
When presenting the data, the authors may give higher priority to the
interests of the organization than to the public interest. Knowing which
organization has funded the data collection may help you decide how
reliable the data is, or what type of bias may have influenced the
collection or presentation of the data.
• IfthedataisfromtheInternet,checkitagainstthefollowingcriteria:
- Authority: The credentials of the author are provided and can be
checked. Ideally, there should be a way to contact the author with
questions.
- Accuracy: The domain of the web address may help you determine
the accuracy of the data. For example, web documents from academic
sources (domain .edu), non-profit organizations and associations
(domains .org and .net) and government departments (domains such
as .gov and .ca) may have undergone vetting for accuracy before being
published on the Internet.
- Currency: When pages on a site are updated regularly and links are
valid, the information is probably being actively managed. This could
mean that the data is being checked and revised appropriately.
Accessing Resources
Pr
e-
To gather secondary data, explore a variety of resources:
• textbooks
• scientificandhistoricaljournalsandotherexpertpublications
• newsgroupsanddiscussiongroups
• librarydatabases,suchasElectricLibraryCanada,whichisadatabaseof
books, newspapers, magazines, and television and radio transcripts
NEL
Project Connection
203
project example
tio
n
People may be willing to help you with your research, perhaps by providing
information they have or by pointing you to sources of information. Your
school or community librarian can help you locate relevant sources, as can
the librarians of local community colleges or universities. Other people,
such as teachers, your parents or guardians, local professionals, and Elders
and Knowledge Keepers may have valuable input. (Be sure to respect local
community protocols when approaching Elders or Knowledge Keepers.)
The only way to find out if someone can and will help is to ask. Make a list
of people who might be able to help you obtain the information you need,
and then identify how you might contact each person on your list.
Carrying out your research
ic
a
Sarah chose, “Which Western province or territory grew the fastest over
the last century, and why?” as her research question. She has decided to use
1900 to 2000 as the time period. How can she find relevant data?
Sarah’s Search
Pr
e-
Pu
bl
Sincemyquestioninvolvesahistoricaleventoverawidearea,Idecided
torelyonsecondarydata.IstartedmysearchusingtheInternet.Idida
searchfor“provincialpopulationsCanada1900to2000”andfound many
websites.IhadtolookatquiteafewuntilIfoundthefollowinglink:
204
ThisledmetoadocumentfromtheUniversityofBritishColumbia that cited
adocumentfromStatisticsCanada,basedoncensusdata,thatshowedthe
provincialpopulationsfrom1851to1976.IwenttotheStatisticsCanada
websiteandsearchedforthecensusdata,butIcouldn’tfindit.SoItried
anothergeneralsearch,“historicalstatisticsCanadapopulation,”andfound
thelinkbelow:
Chapter 4 Oblique Triangle Trigonometry
NEL
tio
n
ThisledmetodataIwaslookingfor:
ic
a
InowhavesomedataIcanuse.Ifeelconfidentthatthedataisauthoritative
andaccurate,becauseIbelievethatthesourceisreliable.Iwillcontinuelooking
formorecurrentdata,from1976to2000.Iwillalsoneedtosearchfor
informationaboutreasonsforpopulationchangesduringthistimeperiod.
Iwillasktheschoollibrariantohelpmelookforothersources.
bl
Your Turn
Decide if you will use primary data, secondary data, or both.
Explain how you made your decision.
B.
Make a plan you can follow to collect your data.
C.
Carry out your plan to collect your data. Make sure that you record
your successful searches, so you can easily access these sources at a later
time. You should also record detailed information about your sources,
so you can cite them in your report. See your teacher for the preferred
format for endnotes, footnotes, or in-text citations.
Pr
e-
Pu
A.
NEL
Project Connection
205