Ch.7 AdditionalTopicsinTrigonometry
7.1 TheLawofSines
1 UsetheLawofSinestoSolveObliqueTriangles
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvethetriangle.
1)
65°
10
40°
A) B=75°
,
a=6.65
,
c=9.38 B) B =75°
,
a=9.38
,
c=6.65
C) B=80°
,
a=6.65
,
c=9.38 D) B =70°
,
a=9.38
,
c=6.65
2)
10
25° 105°
A) C=50°
,
a=22.86
,
c=18.13 B) C =50°
,
a=18.13
,
c=22.86
C) C=55°
,
a=22.86
,
c=18.13 D) C =45°
,
a=18.13
,
c=22.86
Solvethetriangle.Roundlengthstothenearesttenthandanglemeasurestothenearestdegree.
3) B=27°
C=108°
b=11
A) A=45°
,
a=17.1
,
c=23 B) A=43°
,
a=23
,
c=17.1
C) A=45°
,
a=19.1
,
c=25 D) A=43°
,
a=25
,
c=19.1
4) A=37°
B=36°
a=30.7
A) C=107°
,
b=30
,
c=48.8 B) C=108°
,
b=30
,
c=48.8
C) C=107°
,
b=48.8
,
c=30 D) C=108°
,
b=48.8
,
c=30
5) A=26°
,
B=51°
,
c=29
A) C=103°
,
a=13
,
b=23.1 B) C=103°
,
a=23.1
,
b=13
C) C=103°
,
a=64.4
,
b=36.4 D) C=97°
,
a=12.8
,
b=22.7
6) A=11.2°
,
C=131.6°
,
a=89.8
A) B=37.2°
,
b=279.5
,
c=345.7 B) B=37.2°
,
b=345.7
,
c=279.5
C) B=37.2°
,
b=28.8
,
c=23.4 D) B=36.8°
,
b=276.9
,
c=345.7
Page1
2 UsetheLawofSinestoSolve,ifPossible,theTriangleorTrianglesintheAmbiguousCase
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Twosidesandanangle(SSA)ofatrianglearegiven.Determinewhetherthegivenmeasurementsproduceonetriangle,
twotriangles,ornotriangleatall.Solveeachtrianglethatresults.Roundlengthstothenearesttenthandangle
measurestothenearestdegree.
1) A=30°
,
a=8
,
b=16
A) B=90°
,
C=60°
,
c=13.9 B) B=60°
,
C=90°
,
c=13.9
C) B=60°
,
C=60°
,
c=13.9 D) notriangle
2) B=114°
,
b=5
,
a=24
A) notriangle B) A=58°
,
C=9°
,
c=33
C) A=57°
,
C=9°
,
c=29 D) A=56°
,
C=9°
,
c=31
3) B=41°
,
b=4
,
a=25
A) notriangle B) A=37°
,
C=101°
,
c=26
C) A=39°
,
C=99°
,
c=29 D) A=40°
,
C=100°
,
c=30.5
4) B=14°
,
b=5.3
,
a=10.95
A) A1=30°
,
C1=136°
,
c1=15.2;
A2=150°,C2=16°,c2=6
B) A=30°
,
C=136°
,
c=15.2
C) A=150°
,
C=16°
,
c=6D)notriangle
5) B=80°
,
b
=6
,
c=8
A) notriangle B) C =41°
,
A=59°
,
a=18
C) B=40°
,
A=60°
,
a=14 D) C =39°
,
A=61°
,
a=16
6) C=35°
,
a=18.7,c=16.1
A) A1=42°
,
B1=103°
,
b
1=27.4;
A2=138°,B2=7°,b2=3.4
B) A1=103°
,
B1=42°
,
b
1=27.4;
A2=7°,B2=138°,b2=3.4
C) A=42°
,
B=103°
,
b=27.4 D) notriangle
7) B=41°
,
a=4,b=3
A) A1=61°
,
C1=78°
,
c1=4.5;
A2=119°,C2=20°,c2=1.6
B) A1=61°
,
C1=78°
,
c1=0.1;
A2=119°,C2=20°,c2=0.1
C) A=29°
,
C=110°
,
c=5.7 D) notriangle
3 FindtheAreaofanObliqueTriangleUsingtheSineFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheareaofthetrianglehavingthegivenmeasurements.Roundtothenearestsquareunit.
1) A=34°
,
b=10inches,c=9inches
A) 25squareinches B) 37squareinches C) 23 squareinches D) 39squareinches
2) A=28°
,
b=13meters,c=12meters
A) 37squaremeters B) 19squaremeters C) 74 squaremeters D) 76squaremeters
Page2
3) C=115°
,
a=3yards,b=6yards
A) 8squareyards B) 16squareyards C) 33 squareyards D) 4squareyards
4) B=15°
,
a=1feet,c=7feet
A) 1squarefeet B) 2squarefeet C) 4 squarefeet D) 3squarefeet
4 SolveAppliedProblemsUsingtheLawofSines
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Asurveyorstanding63metersfromthebaseofabuildingmeasurestheangletothetopofthebuildingand
findsittobe38°.Thesurveyorthenmeasurestheangletothetopoftheradiotoweronthebuildingandfinds
thatitis48°.Howtallistheradiotower?
A) 20.75meters B) 8.03 meters C) 7.49 meters D) 11.11 meters
2) Twotrackingstationsareontheequator168 milesapart.AweatherballoonislocatedonabearingofN38°E
fromthewesternstationandonabearingofN21°Wfromtheeasternstation.Howfaristheballoonfromthe
westernstation?Roundtothenearestmile.
A) 183miles B) 192miles C) 154 miles D) 145 miles
3) TofindthedistanceABacrossariver,adistanceBCof1021 mislaidoffononesideoftheriver.Itisfound
thatB=113.7°andC=12.6°.FindAB.Roundtothenearestmeter.
A) 276meters B) 279meters C) 227 meters D) 224 meters
4) Aguywiretoatowermakesa75°anglewithlevelground.Atapoint31 ftfartherfromthetowerthanthe
wirebutonthesamesideasthebaseofthewire,theangleofelevationtothetopofthetoweris37°.Findthe
lengthofthewire(tothenearestfoot).
A) 30feet B) 35feet C) 60 feet D) 65feet
5 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Finda.Ifnecessary,roundyouranswertothenearesthundredth.
1)


45°
35°
 60
A) 17.99 B) 8.01 C) 6.72 D) 10.58
Page3
2)

51°24°
1.7
A) 1.52 B) 0.89 C) 2.91 D) 3.25
7.2 TheLawofCosines
1 UsetheLawofCosinestoSolveObliqueTriangles
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvethetriangle.Roundlengthstothenearesttenthandanglemeasurestothenearestdegree.
1)
65
7
A) A=57°
,
B=44°
,
C=79° B) A =44°
,
B=57°
,
C=79°
C) A=57°
,
B=79°
,
C=44° D) A =44°
,
B=79°
,
C=57°
2)
9
6
5
A) A=109°
,
B=39°
,
C=32° B) A =39°
,
B=109°
,
C=32°
C) A=109°
,
B=32°
,
C=39° D) A =39°
,
B=32°
,
C=109°
3) a=7
,
b=8
,
C=122°
A) c=13.1
,
A=27°
,
B=31° B) c=16
,
A=29°
,
B=29°
C) c=18.9
,
A=25°
,
B=33° D) notriangle
4) a=5
,
c=8
,
B=115°
A)
b
=11.1
,
A=24°
,
C=41° B)
b
=14
,
A=26°
,
C=39°
C)
b
=16.9
,
A=22°
,
C=43° D) notriangle
5)
b
=8
,
c=10
,
A=127°
A) a=16.1
,
B=23°
,
C=30° B) a=19
,
B=25°
,
C=28°
C) a=21.9
,
B=21°
,
C=32° D) notriangle
Page4
6)
b
=2
,
c=3
,
A=75°
A) a=3.1
,
B=38°
,
C=67° B) a=3.1
,
B=67°
,
C=38°
C) a=4.1
,
B=38°
,
C=67° D) a=2.1
,
B=67°
,
C=38°
7) a=4
,
c=2
,
B=90°
A)
b
=4.5
,
A=63°
,
C=27° B)
b
=4.5
,
A=27°
,
C=63°
C)
b
=5.5
,
A=63°
,
C=27° D)
b
=3.5
,
A=27°
,
C=63°
8) a=9
,
b=13
,
c=15
A) A=37°
,
B=60°
,
C=83° B) A=39°
,
B=58°
,
C=83°
C) A=35°
,
B=60°
,
C=85° D) notriangle
9) a=5
,
b=5
,
c=2
A) A=78°
,
B=78°
,
C=24° B) A =79°
,
B=79°
,
C=22°
C) A=24°
,
B=78°
,
C=78° D) A =78°
,
B=24°
,
C=78°
10) a=8
,
b=6
,
c=4
A) A=104°
,
B=47°
,
C=29° B) A =47°
,
B=104°
,
C=29°
C) A=104°
,
B=29°
,
C=47° D) A =47°
,
B=29°
,
C=104°
2 SolveAppliedProblemsUsingtheLawofCosines
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Aplaneflyingastraightcourseobservesamountainatabearingof31.2° totherightofitscourse.Atthattime
theplaneis6kilometersfromthemountain.Ashorttimelater,thebearingtothemountainbecomes41.2°.
Howfaristheplanefromthemountainwhenthesecondbearingistaken(tothenearesttenthofakm)?
A) 4.7kilometers B) 7.6kilometers C) 8.6 kilometers D) 3.1 kilometers
2) Twoairplanesleaveanairportatthesametime,onegoingnorthwest(bearing135 °)at421mphandtheother
goingeastat348mph.Howfarapartaretheplanesafter2hours(tothenearestmile)?
A) 1422miles B) 711miles C) 1183 miles D) 1268 miles
3) TwosailboatsleaveaharborintheBahamasatthesametime.Thefirstsailsat20mphinadirection340°.The
secondsailsat28mphinadirection200°.Assumingthatbothboatsmaintainspeedandheading,after2hours,
howfarapartaretheboats?
A) 90.4miles B) 67.9 miles C) 76.2 miles D) 70.6 miles
4) TwopointsAandBareonoppositesidesofabuilding.AsurveyorselectsathirdpointCtoplaceatransit.
PointCis50feetfrompointAand61feetfrompointB.TheangleACBis45°.HowfarapartarepointsAand
B?
A) 43.7feet B) 102.6 feet C) 63.8 feet D) 91.5 feet
5) ThedistancefromhomeplatetodeadcenterfieldinSunDevilStadiumis410 feet.Abaseballdiamondisa
squarewithadistancefromhomeplatetofirstbaseof90feet.Howfarisitfromfirstbasetodeadcenterfield?
A) 352.2feet B) 387.4 feet C) 477.9 feet D) 335.1 feet
6) Apainterneedstocoveratriangularregion62 metersby69 metersby73 meters.Acanofpaintcovers70
squaremeters.Howmanycanswillbeneeded?
A) 29cans B) 324cans C) 15 cans D) 3cans
Page5
3 UseHeronʹsFormulatoFindtheAreaofaTriangle
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
UseHeronʹsformulatofindtheareaofthetriangle.Roundtothenearestsquareunit.
1) a=20yards,b=13yards,c=13yards
A) 83squareyards B) 86squareyards C) 89 squareyards D) 92squareyards
2) a=9inches,
b
=13inches,c=5inches
A) 12squareinches B) 30squareinches C) 10 squareinches D) 32squareinches
3) a=8meters,
b
=14meters,c=7meters
A) 19squaremeters B) 10squaremeters C) 38 squaremeters D) 40squaremeters
7.3 PolarCoordinates
1 PlotPointsinthePolarCoordinateSystem
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
MatchthepointinpolarcoordinateswitheitherA,B,C,orDonthegraph.
1) (4
,
0)
-5 5
5
-5
CD
A
B
-5 5
5
-5
CD
A
B
A) C B) D C) A D) B
2) –4,–π
2
-5 5
5
-5
CD
A
B
-5 5
5
-5
CD
A
B
A) A B) D C) B D) C
Page6
Useapolarcoordinatesystemtoplotthepointwiththegivenpolarcoordinates.
3) 2,9π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page7
4) –2,–5π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page8
5) 2,–5π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page9
6) –2,9π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page10
7) (2
,
–135°)
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page11
8) (–4
,
–135°)
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page12
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
9) Plotthepoint4,π
6andfindotherpolarcoordinates(r,θ)ofthepointforwhich:
(a) r>0,–2π≤θ<0
(b) r<0,0≤θ<2π
(c) r>02π≤θ<4π
r
-5 5
5
-5
r
-5 5
5
-5
2 FindMultipleSetsofPolarCoordinatesforaGivenPoint
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findanotherrepresentation,(r,θ),forthepointunderthegivenconditions.
1) 7,π
3,r>0and2π<θ<4π
A) 7,7
3πB) 7,–5
3πC) 7,4
3πD) 7,–2
3π
2) 5,π
4,r<0and0<θ<2π
A) –5,5
4πB) –5,–3
4πC) –5,9
4πD) –5,–7
4π
3) 4,π
2,r>0and–2π<θ<0
A) 4,–3
2πB) 4,5
2πC) 4,–1
2πD) 4,3
2π
4) 9,π
3,r<0and2π<θ<4π
A) –9,10
3πB) –9,–8
3πC) –9,7
3πD) –9,4
3π
Selecttherepresentationthatdoesnotchangethelocationofthegivenpoint.
5) (2
,
20°)
A) (2
,
380)° B) (2
,
200)° C) (–2
,
380)° D) (–2
,
110)°
Page13
6) (–8
,
8π)
A) (8
,
7π)B)(
–8
,
9π)C)(
–8
,
7π)D)(8
,
6π)
3 ConvertaPointfromPolartoRectangularCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Polarcoordinatesofapointaregiven.Findtherectangularcoordinatesofthepoint.
1) (–1
,
–180°)
A) (1
,
0) B) (0
,
1) C) (–1
,
0) D) (0
,
–1)
2) (2
,
–180°)
A) (–2
,
0) B) (0
,
–2) C) (2
,
0) D) (0
,
2)
3) (–5
,
120°)
A) 5
2,–53
2B) –5
2,–53
2C) 5
2,53
2D) –5
2,53
2
4) (3
,
–135°)
A) –32
2,–32
2B) –32
2,32
2C) 32
2,32
2D) 32
2,–32
2
5) (5
,
–24°)
A) (4.6
,
–2) B) (–2
,
4.6) C) (–4.6
,
2) D) (2
,
–4.6)
6) –9,2π
3
A) 9
2,–93
2B) –9
2,–93
2C) 9
2,93
2D) –9
2,93
2
7) –9,3π
4
A) 92
2,–92
2B) –92
2,92
2C) –92
2,–92
2D) 92
2,92
2
8) 4.6,4π
9
A) (0.8
,
4.5) B) (4.5
,
0.8) C) (–0.8
,
–4.5) D) (–4.5
,
–0.8)
4 ConvertaPointfromRectangulartoPolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Therectangularcoordinatesofapointaregiven.Findpolarcoordinatesofthepoint.Expressθinradians.
1) (9
,
–9)
A) 9 2,7π
4B) 9 2,–7π
4C) 9,–7π
4D) 9,π
4
2) (6,–63)
A) 12,5π
3B) 6,5π
3C) 12,11π
6D) 6,11π
6
Page14
3) (4 3,4)
A) 8,π
6B) 4,π
6C) 8,π
3D) 4,π
3
4) (–4
,
0)
A) (4
,
π)B)4,π
2C) (4
,
0) D) 4,3π
2
5) (0,–7)
A) (–7,90°) B) (–7,270°) C) (–7,180°) D) ( 7,90°)
6) (–22
,–22)
A) (4
,
225°) B) (2 2,225°) C) (2 2,135°) D) (4
,
135°)
7) (3
,
–3)
A) (–32
,135°) B) (–32,225°) C) (–32,45°) D) (3 2,135°)
5 ConvertanEquationfromRectangulartoPolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Converttherectangularequationtoapolarequationthatexpressesrintermsofθ.
1) x=2
A) r=2
cosθ B) r=2
sinθ C) cosθ =2D)r=2
2) y=5
A) r=5
sinθ B) r=5
cosθ C) sinθ =5D)r=5
3) x2+y2=4
A) r=2B)r=4 C) r(cosθ +sinθ)=2 D) r(cosθ +sinθ)=4
4) 6x–7y+10=0
A) r=–10
(6cosθ–7sinθ)B) r=–10
(6sinθ–7cosθ)
C) 6cosθ–7sinθ=–10 D) 6 cosθ –7 sinθ =10
5) y2=5x
A) r=5cotxcscx B) r=25 cotxcscx
C) r2(cosθ+sinθ)=5D)r=5cot2x
6) (x–6)2+y2=36
A) r=12cosθ B) r=12 sinθ C) r2=12cosθ D) r= –12 sinθ +36
Page15
6 ConvertanEquationfromPolartoRectangularCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Convertthepolarequationtoarectangularequation.
1) r=6
A) x2+y2=36 B) x=6C)y
2=36 D) y=6
2) θ=5π
6
A) y=–3
3xB)y=3xC)y=–3
3x2D) x2+y2=1
3) rcosθ=3
A) x=3B)x
2+y2=3C)y
2=3D)y=3
4) r=9cscθ
A) y=9B)x=9C)y
2=9D)x
2+y2=9
5) r=–2cosθ
A) x+12+y2=1B)x=–2C)x–12+y2=4D)x
2+y2=2
6) r=8cosθ+2sinθ
A) x2+y2=8x+2y B) x2–y2=8x+2y C) x2+y2=2x+8y D) 8x+2y=0
7) r2sin2θ=8
A) xy=4B)xy=8C)y
2=8D)x
2+y2=8
7 SolveApps:PolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Thewindisblowingat10knots.Sailboatracerslookforasailingangletothe10–knotwindthatproduces
maximumsailingspeed.Inthisapplication,(r,θ)describesthesailingspeedr,inknots,atangleθtothe
10–knotwind.Fourpointsinthis10–knot–windsituationare(6.5,55°),(7.4,85°),(7.5,115°)and(7.3,130°).
Basedonthesespoints,whichsailingangletothe10–knotwindwouldyourecommendtoaserioussailboat
racer?Whatsailingspeedisachievedatthisangle?
A) 115°;7.5knots B) 130°;7.3 knots C) 55°;6.5 knots D) 85°;7.4 knots
8 Tech:PolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Polarcoordinatesofapointaregiven.Useagraphingutilitytofindtherectangularcoordinatesofthepointtotwo
decimalplaces.
1) 2.1,–4π
9
A) (0.36
,
–2.07) B) (–2.07
,
0.36) C) (0.36
,
2.07) D) (2.07
,
0.36)
2) –3.9,5π
9
A) (0.68
,
–3.84) B) (–3.84
,
0.68) C) (–0.68
,
3.84) D) (3.84
,
–0.68)
Page16
Rectangularcoordinatesofapointaregiven.Useagraphingutilityinradianmodetofindpolarcoordinatesofthe
point.
3) (12
,
16)Expressθtothreedecimalplaces.
A) (20
,
0.927) B) (20
,
0.644) C) (28
,
0.644) D) (20
,
0.848)
4) (–8
,
–6)Expressθtothreedecimalplaces.
A) (10
,
–2.498) B) (10
,
–2.214) C) (14
,
–2.214) D) (10
,
–5.356)
5) (–2
,
–4)Expressbothrandθtotwodecimalplaces.
A) (4.47
,
–2.03) B) (4.47
,
+2.03) C) (4.47
,
–1.11) D) (4.47
,
1.11)
9 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Convertthepolarequationtoarectangularequation.Thendeterminethegraphʹsslopeandy–intercept.
1) rsinθ
–π
4=2
A) y=x+22
;slope:1;y–intercept:22 B) y=x–22;slope:1;y–intercept:–22
C) y=–x–22;slope:–1;y–intercept:–22 D) y=–x+22;slope:–1;y–intercept:22
2) rcosθ
+π
6=2
A) y=x3
–4;slope:3;y–intercept:–4B)y=–x3–4;slope:–3;y–intercept:–4
C) y=x3
+4;slope:3;y–intercept:4D)y=–x3+4;slope:–3;y–intercept:4
7.4 GraphsofPolarEquations
1 UsePointPlottingtoGraphPolarEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Thegraphofapolarequationisgiven.Selectthepolarequationforthegraph.
1)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=3B)r=6 cosθ C) r=6 sinθ D) rsinθ =3
Page17
2)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=2sinθ B) r=2 cosθ C) r=1D)rsinθ =1
3)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=4cosθ B) r=4 sinθ C) r=2D)rsinθ =2
4)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=2+sinθ B) r=4 cosθ C) r=4 sinθ D) r=2+cosθ
Page18
5)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=3+cosθ B) r=6 cosθ C) r=6 sinθ D) r=3+sinθ
6)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=3cos(3θ)B)r=3 sin(3θ)C)r=3D)r=3+cos(3θ)
2 UseSymmetrytoGraphPolarEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Testtheequationforsymmetrywithrespecttothegivenaxis,line,orpole.
1) r=4cosθ; thepolaraxis
A) hassymmetrywithrespecttopolaraxis
B) mayormaynothavesymmetrywithrespecttopolaraxis
2) r=–4cosθ; thelineθ=π
2
A) mayormaynothavesymmetrywithrespecttothelineθ=π
2
B) hassymmetrywithrespecttothelineθ=π
2
3) r=–2sinθ; thepole
A) mayormaynothavesymmetryaboutthepole B) hassymmetryaboutthepole
4) r=2–2cosθ; polaraxis
A) hassymmetrywithrespecttothepolaraxis
B) mayormaynothavesymmetrywithrespecttothepolaraxis
Page19
5) r=4–4cosθ; thelineθ=π
2
A) mayormaynothavesymmetrywithrespecttothelineθ=π
2
B) hassymmetrywithrespecttothelineθ=π
2
6) r=4+2sinθ; thelineθ=π
2
A) hassymmetrywithrespecttothelineθ=π
2
B) mayormaynothavesymmetrywithrespecttothelineθ=π
2
7) r=4+2cosθ; thepole
A) mayormaynothavesymmetryaboutthepole B) hassymmetryaboutthepole
8) r=2+4sinθ; thepolaraxis
A) mayormaynothavesymmetrywithrespecttothepolaraxis
B) hassymmetrywithrespecttothepolaraxis
9) r2=sin2θ; thepole
A) hassymmetrywithrespecttothepole
B) mayormaynothavesymmetrywithrespecttothepole
10) r=2sin3θ; thelineθ=π
2
A) hassymmetrywithrespecttothelineθ
=π
2
B) mayormaynothavesymmetrywithrespecttothelineθ
=π
2
11) rcosθ=2; thepolaraxis
A) hassymmetrywithrespecttopolaraxis
B) mayormaynothavesymmetrywithrespecttopolaraxis
Page20
Graphthepolarequation.
12) r=2sinθ
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
C)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Page21
13) r=2cosθ
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
C)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Page22
14) r=1+sinθ
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
C)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Page23
15) r=1+cosθ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page24
16) r=1+sinθ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page25
17) r=3–cosθ
r
-10 -5 5 10
10
5
-5
-10
r
-10 -5 5 10
10
5
-5
-10
A)
r
-10 -5 5 10
10
5
-5
-10
r
-10 -5 5 10
10
5
-5
-10
B)
r
-10 -5 5 10
10
5
-5
-10
r
-10 -5 5 10
10
5
-5
-10
C)
r
-10 -5 5 10
10
5
-5
-10
r
-10 -5 5 10
10
5
-5
-10
D)
r
-10 -5 5 10
10
5
-5
-10
r
-10 -5 5 10
10
5
-5
-10
Page26
18) r=4sin2θ
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
C)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Page27
19) r2=4sin(2θ)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page28
20) rcosθ=–4
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page29
3 SolveApps:GraphsofPolarEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Thewindisblowingat10knots.Sailboatracerslookforasailingangletothe10–knotwindthatproduces
maximumsailingspeed.Thissituationisnowrepresentedbythepolargraphinthefigureshownbelow.Each
point(r,θ)onthegraphgivesthesailingspeed,r,inknots,atanangleθtothe10–knotwind.Whatisthe
speedtothenearestknot,ofthesailboatsailingat150°angletothewind?
A) 6knots B) 5knots C) 7 knots D) 8knots
2) Thewindisblowingat10knots.Sailboatracerslookforasailingangletothe10–knotwindthatproduces
maximumsailingspeed.Thissituationisnowrepresentedbythepolargraphinthefigureshownbelow.Each
point(r,θ)onthegraphgivesthesailingspeed,r,inknots,atanangleθtothe10–knotwind.Whatangleto
thewindproducesthemaximumsailingspeed?Whatisthespeedtothenearestknot,ofthesailboatsailingat
60°angletothewind?
A) 120°;5knots B) 60°;5 knots C) 120°;8knots D) 60°;7 knots
Page30
4 Tech:GraphsofPolarEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useagraphingutilitytographthepolarequation.
1) r=cos4θ
r
-1 -1
21
21
1
1
2
-1
2
-1
r
-1 -1
21
21
1
1
2
-1
2
-1
A)
r
-1 -1
21
21
1
1
2
-1
2
-1
r
-1 -1
21
21
1
1
2
-1
2
-1
B)
r
-1 -1
21
21
1
1
2
-1
2
-1
r
-1 -1
21
21
1
1
2
-1
2
-1
C)
r
-1 -1
21
21
1
1
2
-1
2
-1
r
-1 -1
21
21
1
1
2
-1
2
-1
D)
r
-1 -1
21
21
1
1
2
-1
2
-1
r
-1 -1
21
21
1
1
2
-1
2
-1
Page31
2) r=2sinθ
–π
4
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
A)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
B)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
C)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
D)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
Page32
3) r=1
2–4sinθ
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
A)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
B)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
C)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
D)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
Page33
4) r=sin45θ+cos3θ
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
A)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
B)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
C)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
D)
r
-2 -1 1 2
2
1
-1
-2
r
-2 -1 1 2
2
1
-1
-2
5) r=2
θ
rr
Page34
A)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
B)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
C)
r
-8 -4 4 8
8
4
-4
-8
r
-8 -4 4 8
8
4
-4
-8
D)
r
-8 -4 4 8
8
4
-4
-8
r
-8 -4 4 8
8
4
-4
-8
Page35
5 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphthepolarequation.
1) r=5cosθ+6sinθ
r
-8 -4 4 8
8
4
-4
-8
r
-8 -4 4 8
8
4
-4
-8
A)
r
-8 -4 4 8
8
4
-4
-8
r
-8 -4 4 8
8
4
-4
-8
B)
r
-8 -4 4 8
8
4
-4
-8
r
-8 -4 4 8
8
4
-4
-8
C)
r
-8 -4 4 8
8
4
-4
-8
r
-8 -4 4 8
8
4
-4
-8
D)
r
-8 -4 4 8
8
4
-4
-8
r
-8 -4 4 8
8
4
-4
-8
Page36
2) r=3–4sin2θ
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
A)
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
B)
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
C)
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
D)
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
r
-12 -8 -4 4 8 12
12
8
4
-4
-8
-12
Page37
3) r=5cos2θsinθ
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
A)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
B)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
C)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
D)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
Page38
4) r=4sin2θcosθ
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
A)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
B)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
C)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
D)
r
-4 -2 2 4
4
2
-2
-4
r
-4 -2 2 4
4
2
-2
-4
Page39
7.5 ComplexNumbersinPolarForm;DeMoivreʹsTheorem
1 PlotComplexNumbersintheComplexPlane
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Plotthecomplexnumber.
1) –3–3i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page40
2) 5i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page41
3) 5
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page42
4) –5+i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page43
5) –3–i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page44
6) –33–3i
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
A)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
B)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
C)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
D)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
Page45
7) 2 2–22i
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
A)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
B)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
C)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
D)
R
-10 -5 5 10
i
10
5
-5
-10
R
-10 -5 5 10
i
10
5
-5
-10
Page46
8) 9+7i
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
A)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
B)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
C)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
D)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
2 FindtheAbsoluteValueofaComplexNumber
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheabsolutevalueofthecomplexnumber.
1) z=15i
A) 15 B) –15 C) 30 D) 0
2) z=–25
A) 25 B) –25 C) 50 D) 0
3) z=–7–3i
A) 58 B) 2 10 C) i 10 D) 2i
Page47
4) z=–2–10i
A) 2 26 B) 4i 6 C) 2 2 D) 2i 3
3 WriteComplexNumbersinPolarForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writethecomplexnumberinpolarform.Expresstheargumentindegrees.
1) 2
A) 2(cos0°+isin0°) B) 2(cos180° +isin180°)
C) 2(cos90°+isin90°) D) 2(cos270° +isin270°)
2) –5i
A) 5(cos270°+isin270°) B) 5(cos90° +isin90°)
C) 5(cos180°+isin180°) D) 5(cos0° +isin0°)
3) 3–4i
A) 5(cos306.9°+isin306.9°) B) 5(cos126.9° +isin126.9°)
C) 5(cos53.1°+isin53.1°) D) 5(cos233.1° +isin233.1°)
Writethecomplexnumberinpolarform.Expresstheargumentinradians.
4) 3–3i
A) 3 2 cos7π
4+isin7π
4B) 3 cos7π
4+isin7π
4
C) 3 2 cos5π
4+isin5π
4D) 3 cos5π
4+isin5π
4
5) –23–2i
A) 4 cos7π
6+isin7π
6B) 2 3 cos13π
6+isin13π
6
C) 4 cos4π
3+isin4π
3D) 2 3 cos4π
3+isin4π
3
6) –6+63i
A) 12 cos2π
3+isin2π
3B) 6 3 cos5π
6+isin5π
6
C) 12 cos5π
6+isin5π
6D) 6 3 cos2π
3+isin2π
3
4 ConvertaComplexNumberfromPolartoRectangularForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writethecomplexnumberinrectangularform.
1) –9(cos120°+isin120°)
A) 9
2+–93
2iB)
–9
2+–93
2iC)
9
2+93
2iD)
–9
2+93
2i
2) 3(cos225°+isin225°)
A) –32
2+–32
2iB)
–32
2+32
2iC)
32
2+32
2iD)
32
2+–32
2i
Page48
3) 8(cos98°+isin98°)
A) –1.1+7.9i B) 7.9–1.1i C) –0.4 +2.8i D) 0.4 –2.8i
4) 9.37(cos239.2°+isin239.2°)
A) –4.8–8i B) –8–4.8i C) –1.6 –2.6i D) 1.6 +2.6i
5) 3(cos2π
3+isin2π
3)
A) –3
2+33
2iB)
–3
2+–33
2iC)
3
2+33
2iD)
3
2+–33
2i
6) –5(cos3π
4+isin3π
4)
A) 52
2+–52
2iB)
–52
2+52
2iC)
–52
2+–52
2iD)
52
2+52
2i
7) 8(cosπ+isinπ)
A) –8B)
–8i C) 8 D) 8i
5 FindProductsofComplexNumbersinPolarForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheproductofthecomplexnumbers.Leaveanswerinpolarform.
1) z1=5(cos20°+isin20°)
z2=4(cos10°+isin10°)
A) 20(cos30°+isin30°) B) 20(cos200° +isin200°)
C) 9(cos30°+isin30°) D) 9(–cos200° –isin200°)
2) z1=3(cos31°+isin31°)
z2=4(cos14°+isin14°)
A) 12(cos45°+isin45°) B) 12(cos17° +isin17°)
C) 7(cos45°+isin45°) D) 7(cos74° +isin74°)
3) z1=3 cosπ
3+isinπ
3
z2=2 cosπ
2+isinπ
2
A) 6 cos5π
6+isin5π
6B) 5 cosπ
6+isinπ
6
C) 5 cos5π
6+isin5π
6D) 6 cosπ
6+isinπ
6
4) z1=8 cosπ
6+isinπ
6
z2=3 cosπ
2+isinπ
2
A) 24 cos2π
3+isin2π
3B) 11 cosπ
12 +isinπ
12
C) 24 sin2π
3+icos2π
3D) 11 cos2π
3+isin2π
3
Page49
5) z1=3 cos7π
4+isin7π
4
z2=6 cos9π
4+isin9π
4
A) 3 2(cos0+isin0) B) 3 2(sin0+icos0)
C) 3 2 cos5π
4+isin5π
4D) 3 2 sin5π
4+icos5π
4
6) z1=6 cos3π
2+isin3π
2
z2=12 cos5π
6+isin5π
6
A) 72 cosπ
3+isinπ
3B) 18 cosπ
3+isinπ
3C) 72 cosπ
3–isinπ
3D) 18 cosπ
3–isinπ
3
7) z1=4i
z2=–6+6i
A) 24 2 cos5π
4+isin5π
4B) 24 2 sin5π
4+icos5π
4
C) 24 2 cos3π
8+isin3π
8D) 24 2 sin3π
8+icos3π
8
8) z1=2+2i
z2=3–i
A) 4 2 cosπ
12 +isinπ
12 B) 4 2 cos23π
12 +isin23π
12
C) 4 cosπ
12 +isinπ
12 D) 4 cos23π
12 +isin23π
12
6 FindQuotientsofComplexNumbersinPolarForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthequotientz1
z2
ofthecomplexnumbers.Leaveanswerinpolarform.
1) z1=30(cos32°+isin32°)
z2=5(cos14°+isin14°)
A) 6(cos18°+isin18°) B) 6(cos46° +isin46°)
C) 25 cos16
7°+isin16
7°D) 25(cos18° –isin18°)
2) z1=5(cos200°+isin200°)
z2=4(cos50°+isin50°)
A) 5
4(cos150°+isin150°) B) 5
4(cos250°+isin250°)
C) 5
4(cos150°–isin150°) D) 5
4(sin150°+icos150°)
Page50
3) z1=1
7cos2π
3+isin2π
3
z2=1
8cosπ
4+isinπ
4
A) 8
7cos5π
12 +isin5π
12 B) 1
56 cos11π
12 +isin11π
12
C) 8
7cos8
3+isin8
3D) 7
8cos–5π
12 +isin–5π
12
4) z1=8 cosπ
2+isinπ
2
z2=3 cosπ
6+isinπ
6
A) 8
3cosπ
3+isinπ
3B) 8
3cos2π
3+isin2π
3
C) 8
3cosπ
3–isinπ
3D) 8
3sinπ
3+icosπ
3
5) z1=3 cos7π
4+isin7π
4
z2=6 cos9π
4+isin9π
4
A) 2
2cos3π
2+isin3π
2B) 2
2cos3π
2–isin3π
2
C) 2
2sin3π
2+icos3π
2D) 2
2cosπ
2+isinπ
2
6) z1=6 cos3π
2+isin3π
2
z2=12 cos5π
6+isin5π
6
A) 1
2cos2π
3+isin2π
3B) 1
2cos2π
3–isin2π
3
C) 1
2cos4π
3–isin4π
3D) 1
2cos4π
3+isin4π
3
7) z1=4i
z2=–6+6i
A) 2
3cos7π
4+isin7π
4B) 2
3cosπ
4+isinπ
4
C) 2
3cos7π
4–isin7π
4D) 2
3cosπ
4–isinπ
4
Page51
7 FindPowersofComplexNumbersinPolarForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
UseDeMoivreʹsTheoremtofindtheindicatedpowerofthecomplexnumber.Writetheanswerinrectangularform.
1) (cos30°+isin30°)12
A) 1 B) –1C)i D)
–i
2) 3(cos15°+isin15°) 4
A) 81
2+81 3
2iB)
81
2+81
2iC)
81 3
2+81
2i D) 81i
3) 2 2(cos7π
4+isin7π
4)
5
A) –128+128i B) –64+64i C) –2+2iD)
–64 2+64 2i
4) 10(cos3π
4+isin3π
4)
3
A) –500 2+500 2iB)502+50 2iC)152+15 2iD)52+52i
5) (–2+2i 3)3
A) 64 B) 8 C) –2+2i 3 D) 8+6i 3
6) (1–i)10
A) –32i B) 32 C) 32–32i D) –32+32i
7) (1+i)20
A) –1024 B) 1024i C) –1024i D) 1024
8) (–3+i)6
A) –64 B) 64i C) –64 3+64i D) 64–64 3i
8 FindRootsofComplexNumbersinPolarForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findallthecomplexroots.Writetheanswerintheindicatedform.
1) Thecomplexsquarerootsof49(cos150° +isin150°) (polarform)
A) 7(cos75°+isin75°),7(cos255°+isin255°) B) 7(cos150° +isin150°),7(cos165° +isin165°)
C) 7(cos75°+isin75°),165(cos255° +isin255°) D) 7(cos150° +isin150°),–7(cos165° +isin165°)
2) Thecomplexcuberootsof125(cos237° +isin237°)(polarform)
A) 5(cos79°+isin79°),5(cos199°+isin199°),5(cos319° +isin319°)
B) 5(cos79°+isin79°),5(cos119°+isin119°),5(cos159° +isin159°)
C) –5(cos79°+isin79°),5(cos199°+isin199°),–5(cos319° +isin319°)
D) –5(cos79°+isin79°),5(cos119°+isin119°),–5(cos159° +isin159°)
Page52
3) Thecomplexsquarerootsof2(cos2π
3+isin2π
3)(rectangularform)
A) 2
2+6
2i,–2
2–6
2iB)
2
2–6
2i,–2
2+6
2i
C) 6+2i,–6–2iD)6–2i,–6–2i
4) Thecomplexcuberootsof8(rectangularform)
A) 2,–1+3i,–1–3iB)2,1+3i,1–3i
C) 2,1+3i,–1–3iD)2,–1–3i,1–3i
5) Thecomplexcuberootsof–8(rectangularform)
A) –2,1+3i,1–3iB)
–2,–1+3i,–1–3i
C) –2,1+3i,–1–3iD)
–2,–1–3i,1–3i
6) Thecomplexcuberootsof8i(rectangularform)
A) –2i,3+i,–3+iB)
–2i,–3–i,3–i
C) 2i,3–i,–3–i D) 2i,3+i,–3+i
7) Thecomplexcuberootsof–8i(rectangularform)
A) 2i,–3–i,3–i B) 2i,–3+i,–3+i
C) –2i,3–i,3–iD)
–2i,3+i,3+i
8) Thecomplexsquarerootsofi(rectangularform)
A) 2
2+2
2i,–2
2–2
2iB)
2
2–2
2i,–2
2+2
2i
C) –1,1D)
–i,i
9) Thecomplexfourthrootsof–16(rectangularform)
A) 2+2i,2–2i,–2+2i,–2–2i
B) 1+i,1–i,–1+i,–1–i
C) 2+i,2–i,–2+i,–2–i
D) 8 2+82i,82–82i,–82+82i,–82–82i
10) Thecomplexsquarerootsof4+43
i(rectangularform)
A) 6+2i,–6–2iB)6–2i,–6+2i
C) 2
2+6
2i,–2
2–6
2iD)
–6–2i,6–2i
Page53
9 SolveApps:ComplexNumbers
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usetheresulteiθ=cosθ+isinθtoplotthecomplexnumber.
1) 4e(πi)/2
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page54
2) 6e(πi)/2
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
10 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheequationinthecomplexnumbersystem.
1) x3–1=0
A) 1,–1
2+3
2i,–1
2–3
2iB)1,1
2+3
2i,–1
2+3
2i
C) 1,1
2–3
2i,–1
2–3
2iD)
–1,1
Page55
2) x5–1=0
A) 1,cos(72°)+isin(72°),cos(144°)+isin(144°),cos(216°)+isin(216°),cos(288°)+isin(288°)
B) –1,cos(72°)+isin(72°),cos(144°)+isin(144°),cos(216°)+isin(216°),cos(288°)+isin(288°)
C) 1,cos(72°)+isin(72°),cos(144°)+isin(144°),cos(216°)+isin(216°),–1
D) 1,cos(36°)+isin(36°),cos(108°)+isin(108°),cos(180°)+isin(180°),cos(252°)+isin(252°)
3) x7–1=0
A) 1,cos(51.4°)+isin(51.4°),cos(102.9°)+isin(102.9°),cos(154.3°)+isin(154.3°),cos(205.7°)+isin(205.7°),
cos(257.1°)+isin(257.1°),cos(308.6°)+isin(308.6°)
B) 1,cos(25.7°)+isin(25.7°),cos(51.4°)+isin(51.4°),cos(77.1°)+isin(77.1°),cos(102.9°)+isin(102.9°),
cos(128.6°)+isin(128.6°),cos(154.3°)+isin(154.3°)
C) 1,cos(51.4°)+isin(51.4°),cos(102.9°)+isin(102.9°),cos(154.3°)+isin(154.3°),cos(205.7°)+isin(205.7°),
cos(257.1°)+isin(257.1°),cos(308.6°)+isin(308.6°),–1
D) –1,cos(25.7°)+isin(25.7°),cos(51.4°)+isin(51.4°),cos(77.1°)+isin(77.1°),cos(102.9°)+isin(102.9°),
cos(128.6°)+isin(128.6°),cos(154.3°)+isin(154.3°)
4) x3=–64i
A) 4(cos90°+isin90°),4(cos210°+isin210°),4(cos330° +isin330°)
B) 4(cos210°+isin210°),4(cos270° +isin270°),4(cos330° +isin330°)
C) 4(cos30°+isin30°),4(cos60°+isin60°),4(cos90° +isin90°)
D) 4(cos90°+isin90°),4(cos180°+isin180°),4(cos270° +isin270°)
5) x3–64i=0
A) 4(cos30°+isin30°),4(cos150°+isin150°),4(cos270° +isin270°)
B) 4(cos0°+isin0°),4(cos120°+isin120°),4(cos240+isin240°)
C) 4(cos60°+isin60°),4(cos180°+isin180°),4(cos300° +isin300°)
D) 1,–1,–i
6) x3–(–63
+6i)=0
A) 312(cos50°+isin50°),312(cos170°+isin170°),312(cos290°+isin290°)
B) 36(cos70°+isin70°),36(cos190°+isin190°),36(cos310°+isin310°)
C) 312(cos50°+isin50°),312(cos170°+isin170°),312(cos270°+isin270°)
D) 6(cos70°+isin70°),6(cos190°+isin190°),6(cos310°+isin310°)
7.6 Vectors
1 UseMagnitudeandDirectiontoShowVectorsareEqual
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) LetvectoruhaveinitialpointP1=(0,2)andterminalpointP2=(–2
,
5). Letvectorvhaveinitialpoint
Q1=(3,0)andterminalpointQ2=(1,3).uandvhavethesamedirection.Finduandv.Isu=v?
A) u=13,v=13;yes B) u=13,v=13;no
C) u=5,v=5;yes D) u=5
,
v=5;no
Page56
2 VisualizeScalarMultiplication,VectorAddition,andVectorSubtractionasGeometricVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usethevectorsv,u,w,andztodrawtheindicatedvector.
1) 3w
A) B)
C) D)
Page57
2) –2v
A) B)
C) D)
3) –1
2u
Page58
A) B)
C) D)
Page59
4) u+z
A) B)
C) D)
Page60
5) v–w
A) B)
C) D)
Page61
6) z–v
A) B)
C) D)
Page62
3 RepresentVectorsintheRectangularCoordinateSystem
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Sketchthevectorasapositionvectorandfinditsmagnitude.
1) v=9i+12j
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
A) v=15
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
B) v=225
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
C) v=21
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
D) v=15
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
Page63
2) v=–4i+3j
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A) v=5
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B) v=7
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C) v=7
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D) v= –1
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page64
3) v=2i–2j
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A) v=22
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B) v=4
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C) v=2
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D) v=0
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page65
4) v=–i–j
x
-2 -1 1 2
y
2
1
-1
-2
x
-2 -1 1 2
y
2
1
-1
-2
A) v=2
x
-2 -1 1 2
y
2
1
-1
-2
x
-2 -1 1 2
y
2
1
-1
-2
B) v=2
x
-2 -1 1 2
y
2
1
-1
-2
x
-2 -1 1 2
y
2
1
-1
-2
C) v=0
x
-2 -1 1 2
y
2
1
-1
-2
x
-2 -1 1 2
y
2
1
-1
-2
D) v=1
x
-2 -1 1 2
y
2
1
-1
-2
x
-2 -1 1 2
y
2
1
-1
-2
LetvbethevectorfrominitialpointP1toterminalpointP2.Writevintermsofiandj.
5) P1=(6
,
–5);P2=(3
,
–1)
A) v=–3i+4jB) v=4i–3jC) v=8i–7jD) v= –7i+8j
6) P1=(0,0);P2=(2
,
–4)
A) v=2i–4jB) v=–4i–4jC) v= –2i+4jD) v=4i–2j
Page66
7) P1=(6
,
3);P2=(–2
,
–4)
A) v=–8i–7jB) v=–7i–8jC) v=8i+7jD) v=7i+8j
8) P1=(3
,
2);P2=(–1
,
2)
A) v=–4iB) v=–4jC) v=4iD) v=4j
4 PerformOperationswithVectorsinTermsofiandj
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthespecifiedvectororscalar.
1) u=6i–5j
,
v=–9i+7j;Findu+v.
A) –3i+2jB) –4i+2jC) 15i–4jD) –15i+2j
2) u=–9i–2j
,
v=5i+7j;Findu–v.
A) –14i–9jB) –4i+5jC) –15i+5jD) –16i+5j
3) v=8i+2j;Find3v.
A) 24i+6jB) 24i+2jC) 11i+5jD) 11i+2j
4) v=–7i+2j;Find9v.
A) 9 53 B) 27 5 C) 27i 5 D) –953
5) u=–7i+1jandv=8i+1j;Findu+v.
A) 5 B) 15 C) 115 D) 82
6) u=2i+7jandv=12i+42j;Findv–u.
A) 5 53 B) 6 53 C) 5 54 D) 53
5 FindtheUnitVectorintheDirectionofv
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheunitvectorthathasthesamedirectionasthevectorv.
1) v=2i
A) u=iB) u=2iC) u=4iD) u=1
2i
2) v=–9j
A) u=–jB) u=–9jC) u=81jD) u=–1
9j
3) v=3i–4j
A) u=3
5i–4
5jB) u=15i–20jC) u=4
5i–3
5jD) u=5
3i–5
4j
4) v=–12i+5j
A) u=–12
13 i+5
13 jB) u=–156i+65jC) u=–5
13 i+12
13 jD) u=–13
12 i+13
5j
5) v=3i+j
A) u=3
10 i+1
10 jB) u=310
i+10jC) u=3
11 i+1
11 jD) u=10
3i+10j
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6 WriteaVectorinTermsofItsMagnitudeandDirection
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writethevectorvintermsofiandjwhosemagnitudevanddirectionangleθ aregiven.
1) v=10,θ=120°
A) v=–5i+53
jB) v=53
i–5jC) v=–52
i+52
jD) v=5i–53
j
2) v=7,θ=225°
A) v=–72
2i–72
2jB) v=–73
2i–7
2j
C) v=–7
2i–73
2jD) v=72
2i+72
2j
3) v=8,θ=30°
A) v=43
i+4jB) v=4i+43
jC) v=42
i+42
jD) v=–43
i+4j
4) v=12
,
θ=270°
A) v=–12jB) v=–12iC) v=12 2
2i–2
2jD) v= –12i–12j
5) v=15
,
θ=0°
A) v=15iB) v=15jC) v= –15jD) v=15i+15j
7 SolveAppliedProblemsInvolvingVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Achildthrowsaballwithaspeedof7 feetpersecondatanangleof16° withthehorizontal.Expressthe
vectordescribedintermsofiandj.Ifexactvaluesarenotpossible,roundcomponentsto3decimals.
A) 6.729i+1.929jB) 1.929i+6.729jC) –6.729i+1.929jD) 1.929i–6.729j
2) Themagnitudeanddirectionoftwoforcesactingonanobjectare35pounds,N45°E
,
and55pounds,S30°E,
respectively.Findthemagnitude,tothenearesthundredthofapound,andthedirectionangle,tothenearest
tenthofadegree,oftheresultantforce.
A) F=57.04;θ=–23.6° B) F=65.19;θ = –7.5°
C) F=43.30;θ=2.7° D) F=49.17;θ= –11.3°
3) Twoforces,F1andF2
,
ofmagnitude60and70pounds,respectively,actonanobject.ThedirectionofF1is
N40°EandthedirectionofF2isN40°W.Findthemagnitudeandthedirectionangleoftheresultantforce.
Expressthedirectionangletothenearesttenthofadegree.
A) F=99.37;θ=93.7° B) F=92.20;θ =89.4°
C) F=92.20;θ=80° D) F=94.63;θ =87.2°
4) Oneropepullsabargedirectlyeastwithaforceof60 newtons,andanotherropepullsthebargedirectlynorth
withaforceof90newtons.Findthemagnitudeoftheresultantforceactingonthebarge.
A) 108newtons B) 150newtons C) 5400 newtons D) 30newtons
5) AnaircraftgoingfromAtlantatoSavannahonaheadingof125° (fromnorth)istravellingataspeedof590
milesperhour.Thewindisoutofthenorthataspeedof20milesperhour.Findtheactualspeedanddirection
oftheaircraft.
A) 602milesperhour;127°fromnorth B) 579 milesperhour;127°fromnorth
C) 585milesperhour;127°fromnorth D) 826 milesperhour;126°fromnorth
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6) Apowerboatinstillwatermaintainsaspeedof35 milesperhour.Theboatheadsdirectlyacrossariver
perpendiculartothecurrentwhichhasaspeedof8milesperhour.Findtheactualspeedanddirectionofthe
boat.
A) 36milesperhour;13°offcourse B) 35 milesperhour;13°offcourse
C) 18milesperhour;26°offcourse D) 15 milesperhour;32°offcourse
8 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Performtheindicatedoperation.
1) u=8i+j
,
v=–2i–6j,w=i–9j;Findv–(u–w).
A) –9i–16jB) –11i+2jC) –9i+16jD) 7i–14j
Findthemagnitudevanddirectionangleθ,tothenearesttenthofadegree,forthegivenvectorv.
2) v=–4i–3j
A) 5;216.9° B) 5;36.9° C) 5;233.1° D) 7;216.9°
3) v=–12i+5j
A) 13;157.4° B) 13;22.6° C) 15;157.4° D) 13;112.6°
7.7 TheDotProduct
1 FindtheDotProductofTwoVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usethegivenvectorstofindthespecifiedscalar.
1) u=5i+11jandv=–6i–11j;Findu·v.
A) –151 B) 91 C) –30 D) –121
2) u=8i+7jandv=–10i–6j;Findu·v.
A) –122 B) –38 C) –80 D) –42
3) v=5i–8j;Findv·v.
A) 89 B) 9 C) –80 D) 1600
4) u=–6i–9j
,
v=–2i+3j
,
w=3i–4j;Findu·(v+w).
A) 3 B) –3C)
–15 D) 18
5) u=–5i+3j
,
v=7i–6j
,
w=–3i+12j;Findu·w+v·w.
A) –42 B) –37 C) –46 D) –35
6) u=9i+10j
,
v=–3i+8j;Find(–5u)·v.
A) –265 B) –210 C) –70 D) –120
2 FindtheAngleBetweenTwoVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheanglebetweenthegivenvectors.Roundtothenearesttenthofadegree.
1) u=–3i+6j,v=5i+2j
A) 94.8° B) 104.8° C) 47.4° D) 37.4°
2) u=i–j
,
v=2i+3j
A) 101.3° B) 11.3° C) 106.1° D) –11.3°
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3) u=–i+5j
,
v=3i–4j
A) 154.4° B) 0.9° C) 49.3° D) 64.4°
4) u=2j
,
v=6i–2j
A) 108.4° B) –18.4° C) 129.2° D) 71.6°
3 UsetheDotProducttoDetermineifTwoVectorsareOrthogonal
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usethedotproducttodeterminewhetherthevectorsareparallel,orthogonal,orneither.
1) v=2i+j
,
w=i–2j
A) orthogonal B) parallel C) neither
2) v=4i+3j
,
w=3i–4j
A) orthogonal B) parallel C) neither
3) v=4i–j
,
w=8i–2j
A) parallel B) orthogonal C) neither
4) v=4i+3j
,
w=8i+6j
A) parallel B) orthogonal C) neither
5) v=3i+3j
,
w=3i–2j
A) parallel B) orthogonal C) neither
6) v=i+3j,w=i–4j
A) orthogonal B) parallel C) neither
7) v=4i
,
w=–3i
A) parallel B) orthogonal C) neither
8) v=2j,w=4i
A) orthogonal B) parallel C) neither
4 FindtheProjectionofaVectorontoAnotherVector
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findprojwv.
1) v=i–3j;w=5i+12j
A) –155
169 i–372
169 jB) –155
13 i–372
13 jC) –31
2i–186
5jD) –31
10 i+93
10 j
2) v=2i+3j;w=8i–6j
A) –1
25 (4i–3j) B) –8
5(i–6j) C) –4
13 (i–3j) D) –4
325 (i–6j)
3) v=2i–3j;w=–3i+j
A) –9
10 (–3i+j)B)
1
10 (–3i+j)C)
–1(–3i+j)D)
–21
10 (–3i+j)
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5 ExpressaVectorastheSumofTwoOrthogonalVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Decomposevintotwovectorsv1andv2,wherev1isparalleltowandv2isorthogonaltow.
1) v=i+9j
,
w=i+j
A) v1=5(i+j),v2=–4i+4jB) v1=11
2(i+j),v2=–9
2i+7
2j
C) v1=5(i+j),v2=4i–4jD) v1=10(i+j),v2=–8i+8j
2) v=i–4j
,
w=3i+j
A) v1=–1
10 (3i+j),v2=13
10 i–39
10 jB) v1=–1
10 (3i+j),v2=11
10 i–39
10 j
C) v1=–1
9(3i+j),v2=4
3i–35
9jD) v1=–1
10 (3i+j),v2=–11
10 i–47
10 j
3) v=3i+5j
,
w=–3i+j
A) v1=–2
5(–3i+j),v2=9
5i+27
5jB) v1=–2
5(–3i+j),v2=17
5i+24
5j
C) v1=–4
9(–3i+j),v2=5
3i+49
9jD) v1=–2
5(–3i+j),v2=–6
5i+32
5j
4) v=–2i–3j
,
w=–2i+j
A) v1=1
5(–2i+j),v2=–8
5i–16
5jB) v1=1
5(–2i+j),v2=–17
5i–22
5j
C) v1=1
4(–2i+j),v2=3
2i–19
4jD) v1=1
5(–2i+j),v2=4
5i–22
5j
6 ComputeWork
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Apersonispullingafreightcartwithaforceof45 pounds.Howmuchworkisdoneinmovingthecart50 feet
ifthecartʹshandlemakesanangleof22°withtheground?
A) 2086.2ft–lb B) 842.9 ft–lb C) 84.3 ft–lb D) 2151.7 ft–lb
2) Findtheworkdonebyaforceof7poundsactinginthedirectionof35° tothehorizontalinmovinganobject4
feetfrom(0,0)to(4,0).
A) 22.9ft–lb B) 16.1 ft–lb C) 45.9 ft–lb D) 24.2 ft–lb
3) AforceisgivenbythevectorF=2i+4j.Theforcemovesanobjectalongastraightlinefromthepoint(8
,
6)
tothepoint(19,16).Findtheworkdoneifthedistanceismeasuredinfeetandtheforceismeasuredin
pounds.
A) 62ft–lb B) –18ft–lb C) –62 ft–lb D) 64ft–lb
4) Aforceof5poundsactsinthedirectionof10° tothehorizontal.Theforcemovesanobjectalongastraightline
fromthepoint(10,3)tothepoint(17,18),withdistancemeasuredinfeet.Findtheworkdonebytheforce.
Roundtheanswertoonedecimalplace,ifnecessary.
A) 81.5ft–lb B) 82.8 ft–lb C) 16.3 ft–lb D) 108.3 ft–lb
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Ch.7 AdditionalTopicsinTrigonometry
AnswerKey
7.1 TheLawofSines
1 UsetheLawofSinestoSolveObliqueTriangles
2 UsetheLawofSinestoSolve,ifPossible,theTriangleorTrianglesintheAmbiguousCase
3 FindtheAreaofanObliqueTriangleUsingtheSineFunction
4 SolveAppliedProblemsUsingtheLawofSines
5 AdditionalConcepts
7.2 TheLawofCosines
1 UsetheLawofCosinestoSolveObliqueTriangles
2 SolveAppliedProblemsUsingtheLawofCosines
3 UseHeronʹsFormulatoFindtheAreaofaTriangle
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7.3 PolarCoordinates
1 PlotPointsinthePolarCoordinateSystem
2 FindMultipleSetsofPolarCoordinatesforaGivenPoint
3 ConvertaPointfromPolartoRectangularCoordinates
4 ConvertaPointfromRectangulartoPolarCoordinates
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5 ConvertanEquationfromRectangulartoPolarCoordinates
6 ConvertanEquationfromPolartoRectangularCoordinates
7 SolveApps:PolarCoordinates
8 Tech:PolarCoordinates
9 AdditionalConcepts
7.4 GraphsofPolarEquations
1 UsePointPlottingtoGraphPolarEquations
2 UseSymmetrytoGraphPolarEquations
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3 SolveApps:GraphsofPolarEquations
4 Tech:GraphsofPolarEquations
5 AdditionalConcepts
7.5 ComplexNumbersinPolarForm;DeMoivreʹsTheorem
1 PlotComplexNumbersintheComplexPlane
2 FindtheAbsoluteValueofaComplexNumber
3 WriteComplexNumbersinPolarForm
6) A
4 ConvertaComplexNumberfromPolartoRectangularForm
7) A
5 FindProductsofComplexNumbersinPolarForm
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6 FindQuotientsofComplexNumbersinPolarForm
7) A
7 FindPowersofComplexNumbersinPolarForm
8 FindRootsofComplexNumbersinPolarForm
9 SolveApps:ComplexNumbers
10 AdditionalConcepts
7.6 Vectors
1 UseMagnitudeandDirectiontoShowVectorsareEqual
2 VisualizeScalarMultiplication,VectorAddition,andVectorSubtractionasGeometricVectors
3 RepresentVectorsintheRectangularCoordinateSystem
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8) A
4 PerformOperationswithVectorsinTermsofiandj
6 WriteaVectorinTermsofItsMagnitudeandDirection
7 SolveAppliedProblemsInvolvingVectors
8 AdditionalConcepts
7.7 TheDotProduct
1 FindtheDotProductofTwoVectors
2 FindtheAngleBetweenTwoVectors
3 UsetheDotProducttoDetermineifTwoVectorsareOrthogonal
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4 FindtheProjectionofaVectorontoAnotherVector
5 ExpressaVectorastheSumofTwoOrthogonalVectors
6 ComputeWork
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