Elementary Education Chapter 9 Applications Trigonometric Functions 91 Applications Involving

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Ch.9 ApplicationsofTrigonometricFunctions

9.1 ApplicationsInvolvingRightTriangles

1 SolveRightTriangles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetherighttriangleusingtheinformationgiven.Roundanswerstotwodecimalplaces,ifnecessary.

1) a=8

B=30°;Findb

c,andA.

A) b=4.62

c=9.24

A=60°

B) b=5.62

c=9.24

A=60°

C) b =4.62

c=10.24

A=60°

D) b=5.62

c=10.24

A=60°

2) a=5

A=35°;Findb

c,andB.

A) b=7.14

c=8.72

B=55°

B) b=7.14

c=8.72

B=65°

C) b =7.14

c=9.72

B=55°

D) b=7.14

c=9.72

B=65°

3) a=8

b=5;Findc,A

andB.

A) c=9.43

A=57.99°

B=32.01°

B) c=6.24

A=58.99°

B=31.01°

C) c=6.24

A=57.99°

B=32.01°

D) c=9.43

A=58.99°

B=31.01°

4) a=2

c=6;Findb

A

andB.

A) b=5.66

A=19.47°

B=70.53°

B) b=5.66

A=70.53°

B=19.47°

C) b =6.32

A=19.47°

B=70.53°

D) b=6.32

A=20.47°

B=69.53°

2 SolveAppliedProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Asurveyorismeasuringthedistanceacrossasmalllake.Hehassetuphistransitononesideofthelake

130feetfromapilingthatisdirectlyacrossfromapierontheothersideofthelake.Fromhistransit,the

anglebetweenthepilingandthepieris50°.Whatisthedistancebetweenthepilingandthepiertothe

nearestfoot?

A) 155ft B) 100ft C) 84 ft D) 109 ft

2) Aradiotransmissiontoweris210feettall.Howlongshouldaguywirebeifitistobeattached8 feetfrom

thetopandistomakeanangleof20°withtheground?Giveyouranswertothenearesttenthofafoot.

A) 590.6ft B) 614.0 ft C) 215.0 ft D) 223.5 ft

3) Astraighttrailwithauniforminclinationof16° leadsfromalodgeatanelevationof500feettoa

mountainlakeatanelevationof5600feet.Whatisthelengthofthetrail(tothenearestfoot)?

A) 18,503ft B) 20,317 ft C) 5306 ft D) 5826 ft

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4) Abuilding160feettallcastsa90footlongshadow.Ifapersonlooksdownfromthetopofthebuilding,

whatisthemeasureoftheanglebetweentheendoftheshadowandtheverticalsideofthebuilding(to

thenearestdegree)?(Assumethepersonʹseyesarelevelwiththetopofthebuilding.)

A) 29° B) 61° C) 56° D) 34°

ohn(whoselineofsightis6ftabovehorizontal)istryingtoestimatetheheightofatalloaktree.Hefirst

measurestheangleofelevationfromwhereheisstandingas35°.Hewalks30feetclosertothetreeand

findsthattheangleofelevationhasincreasedby12°.Estimatetheheightofthetreeroundedtothenearest

wholenumber.

A) 67ft B) 90ft C) 86ft D) 61ft

6) Aphotographerpointsacameraatawindowinanearbybuildingforminganangleof42°withthe

cameraplatform.Ifthecamerais52mfromthebuilding,howhighabovetheplatformisthewindow,to

thenearesthundredthofameter?

A) 46.82m B) 0.9m C) 1.11m D) 57.75m

7) Atreecastsashadowof26meterswhentheangleofelevationofthesunis24°.Findtheheightofthetree

tothenearestmeter.

A) 12mB)11mC)10mD)13m

8) Atwenty–fivefootladderjustreachesthetopofahouseandformsanangleof41.5°withthewallofthe

house.Howtallisthehouse?Roundyouranswertothenearest0.1foot.

A) 18.7ft B) 18.6ft C) 18.8ft D) 19ft

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

9) In1838,theGermanmathematicianandastronomerFriedrichWilhelmBesselwasthefirstpersonto

calculatethedistancetoastarotherthantheSun.Heaccomplishedthisbyfirstdeterminingtheparallax

ofthestar,61Cygni,at0.314arcseconds(Parallaxisthechangeinpositionofthestarmeasuredagainst

backgroundstarsasEarthorbitstheSun.Seeillustration.)IfthedistancefromEarthtotheSunisabout

150,000,000kmand

θ=0.314seconds=0.314

minutes=0.314

60·60

degrees

determinethedistancedfromEarthto61CygniusingBesselʹsfigures.

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MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

10) Twohikersonoppositesidesofacanyoneachstandprecisely525metersabovethecanyonfloor.They

eachsightalandmarkonthecanyonflooronalinedirectlybetweenthem.Theanglesofdepressionfrom

eachhikertothelandmarkmeterare37°and21°.Howfarapartarethehikers?Roundyouranswertothe

nearestwholemeter.

A) 2064m B) 2065m C) 2063m D) 1064m

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

11) YosemiteFallsinCaliforniaconsistsofthreesections:UpperYosemiteFall(byitselfoneofthetenhighes

waterfallsintheworld),theMiddleCascade,andLowerYosemiteFall.Fromafootbridgeacrossthecreek

2500feetfromthefalls,theanglesofelevationtothetopandbottomofUpperYosemiteFallare45.74°and

24.42°,respectively.Howhighisthetotalseriesofthreefalls?HowhighisUpperYosemiteFall?Round

youranswerstothenearestfoot.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

12) AforestrangeratLookoutAsightsafiredirectlynorthofherposition.AnotherrangeratLookoutB,

exactly2kilometersdirectlywestofA,sightsthesamefireatabearingofN41.2°E.Howfaristhefire

fromLookoutA?Roundyouranswertothenearest0.01km.

A) 2.28km B) 2.32km C) 2.18km D) 2.25km

13) AsailboatleavesportonabearingofS72°W.Aftersailingfortwohoursat12knots,theboatturns90°

towardthesouth.Aftersailingforthreehoursat9knotsonthiscourse,whatisthebearingtotheship

fromport?Roundyouranswertothenearest0.1°.

A) S23.6°W B) S24.6°W C) N23.6°E D) N24.6°E

14) Fromtheedgeofa1000–footcliff,theanglesofdepressiontotwocarsinthevalleybeloware21° and28°.

Howfarapartarethecars?Roundyouranswerstothenearest0.1ft.

A) 724.4ft B) 724.5ft C) 714.4ft D) 713.4ft

9.2 TheLawofSines

1 SolveSAAorASATriangles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethetriangle.

95°

45°

A) B=40°

a=7.7

c=10.85 B) B =40°

a=10.85

c=7.7

C) B=45°

a=7.7

c=10.85 D) B =35°

a=10.85

c=7.7

35° 115°

A) C=30°

a=15.8

c=8.72 B) C =30°

a=8.72

c=15.8

C) C=35°

a=15.8

c=8.72 D) C =25°

a=8.72

c=15.8

3) A=10°

B=80°

a=4

A) C=90°

b=22.69

c=23.04 B) C=90°

b=23.04

c=22.69

C) C=90°

b=23.69

c=23.04 D) C=90°

b=23.04

c=21.69

4) B=20°

C=60°

a=4

A) A=100°

b=1.39

c=3.52 B) A =100°

b=3.52

c=1.39

C) A=100°

b=4.52

c=1.39 D) A =100°

b=2.39

c=3.52

2 SolveSSATriangles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Twosidesandananglearegiven.Determinewhetherthegiveninformationresultsinonetriangle,twotriangles,

ornotriangleatall.Solveanytriangle(s)thatresults.

1) a=7,b=9,B=49°

A) onetriangle

A=35.94°,C=95.06°,c=11.88

B) onetriangle

A=76.01°,C=54.99°,c=7.60

C) twotriangles

A1=76.01°,C1=54.99°,c1=7.60or

A2=103.99°,C2=27.01,c2=12.14

D) notriangle

2) b=3

c=7

B=70°

A) onetriangle

C=36°,A=74°,a=14

B) onetriangle

B=35°,A=75°,a=10

C) onetriangle

C=34°,A=76°,a=12

D) notriangle

3) a=23

b=12

B=10°

A) twotriangles

A1=19.44°,C1=150.56°,c1=33.97or

A2=160.56°,C2=9.44°,c2=11.33

B) onetriangle

A=19.44°,C=150.56°,c=33.97

C) onetriangle

A=160.56°,C=9.44°,c=11.33

D) notriangle

4) A=30°

a=2

b=4

A) B=90°

C=60°

c=3.5 B) B=60°

C=90°

c=3.5

C) B=60°

C=60°

c=3.5 D) notriangle

5) B=106°

b=4

a=24

A) onetriangle

A=54°,C=21°,c=32

B) onetriangle

A=53°,C=21°,c=28

C) onetriangle

A=52°,C=21°,c=30

D) notriangle

6) B=52°

b=9

a=30

A) onetriangle

A=48°,C=79°,c=36

B) onetriangle

A=50°,C=77°,c=39

C) onetriangle

A=51°,C=78°,c=40.5

D) notriangle

7) B=12°

b=11.8

a=28.38

A) twotriangles

A1=30°,C1=138°,c1=38;

A2=150°,C2=18°,c2=17.5

B) onetriangle

A=30°,C=138°,c=38

C) onetriangle

A=150°,C=18°,c=17.5

D) notriangle

8) B=80°

b=6

c=8

A) onetriangle

C=41°,A=59°,a=18

B) onetriangle

B=40°,A=60°,a=14

C) onetriangle

C=39°,A=61°,a=16

D) notriangle

9) C=35°

a=18.7,c=16.1

A) twotriangles

A1=42°,B1=103°,b1=27.4;

A2=138°,B2=7°,b2=3.4

B) twotriangles

A1=103°,B1=42°,b1=27.4;

A2=7°,B2=138°,b2=3.4

C) onetriangle

A=42°,B=103°,b=27.4

D) notriangle

10) B=41°

a=4,b=3

A) twotriangles

A1=61°,C1=78°,c1=4.5;

A2=119°,C2=20°,c2=1.6

B) twotriangles

A1=61°,C1=78°,c1=0.1;

A2=119°,C2=20°,c2=0.1

C) onetriangle

A=29°,C=110°,c=5.7

D) notriangle

Solvetheproblem.

11) Givenatrianglewitha=9,b=11

A=31°

whatis(are)thepossiblelength(s)ofc?Roundyouranswerto

twodecimalplaces.

A) 16.42or2.44 B) 6.61 C) 16.42or3.41 D) 14.21

Page5

3 SolveAppliedProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Anairplaneissightedatthesametimebytwogroundobserverswhoare2 milesapartandbothdirectly

westoftheairplane.Theyreporttheanglesofelevationas15°and24°.Howhighistheairplane?

A) 1.35mi B) 0.52 mi C) 2.12 mi D) 0.81 mi

2) Asurveyorstanding56metersfromthebaseofabuildingmeasurestheangletothetopofthebuilding

andfindsittobe38°.Thesurveyorthenmeasurestheangletothetopoftheradiotoweronthebuilding

andfindsthatitis47°.Howtallistheradiotower?

A) 16.3m B) 6.48 m C) 5.94 m D) 8.87 m

3) Ashipsailingparalleltoshoresightsalighthouseatanangleof14° fromitsdirectionoftravel.After

traveling4milesfarther,theangleis21°.Atthattime,howfaristheshipfromthelighthouse?

A) 7.94mi B) 2.7mi C) 11.76 mi D) 4mi

4) ArockettrackingstationhastwotelescopesAandBplaced1.7 milesapart.Thetelescopeslockontoa

rocketandtransmittheiranglesofelevationtoacomputerafterarocketlaunch.Whatisthedistanceto

therocketfromtelescopeBatthemomentwhenbothtrackingstationsaredirectlyeastoftherocket

telescopeAreportsanangleofelevationof25°andtelescopeBreportsanangleofelevationof60°?

A) 1.25mi B) 0.83 mi C) 2.57 mi D) 3.48 mi

5) Aguywiretothetopofatowermakesanangleof68° withthelevelground.Atapoint30feetfarther

fromthebaseofthetowerandinlinewiththebaseofthewire,theangleofelevationtothetopofthe

toweris25°.Whatisthelengthoftheguywire?

A) 18.59ft B) 13.67 ft C) 40.79 ft D) 65.82 ft

6) Ashipatsea,theAdmiral,spotstwootherships,theBarstowandtheCauldrewandmeasurestheangle

betweenthematbe45°.TheyradiotheBarstowandbycomparingknownlandmarks,thedistance

betweenthetheAdmiralandtheBarstowisfoundtobe323meters.TheBarstowreportsanangleof59°

betweentheAdmiralandtheCauldrew.Tothenearestmeter,whatisthedistancebetweentheBarstow

andtheCauldrew?

A) 235mB)81m C) 266mD)49m

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

7) Twosurveyors180metersapartonthesamesideofarivermeasuretheirrespectiveanglestoapoin

betweenthemontheothersideoftheriverandobtain54°and68°.Howfarfromthepoint(line–of–sight

distance)iseachsurveyor?Roundyouranswertothenearest0.1meter.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

8) Aflagpoleisperpendiculartothehorizontalbutisonaslopethatrises10° fromthehorizontal.Thepole

castsa43–footshadowdowntheslopeandangleofelevationofthesunmeasuredfromtheslopeis36°.

Howtallisthepole?Roundyouranswertothenearest0.1foot.

A) 36.4ft B) 35.4ft C) 33.5ft D) 36.2ft

9) Itis4.7kmfromLighthouseAtoPortB.ThebearingoftheportfromthelighthouseisN73°E.Ashiphas

sailedduewestfromtheportanditsbearingfromthelighthouseisN31°E.Howfarhastheshipsailed

fromtheport?Roundyouranswertothenearest0.1km.

A) 3.7km B) 3.5km C) 2.7km D) 3.1km

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

10) Apier1250meterslongextendsatananglefromtheshoreline.Asurveyorwalkstoapoint1500meters

downtheshorelinefromthepierandmeasurestheangleformedbytheendsofthepier.Ifisfoundtobe

53°.Whatacuteangle(correcttothenearest0.1°)doesthepierformwiththeshoreline?Istheremorethan

onepossibility?Ifso,howcanweknowwhichisthecorrectone?

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

11) Twotrackingstationsareontheequator152 milesapart.AweatherballoonislocatedonabearingofN

40°EfromthewesternstationandonabearingofN13°Efromtheeasternstation.Howfaristheballoon

fromthewesternstation?Roundtothenearestmile.

A) 326mi B) 335mi C) 267 mi D) 258 mi

12) TofindthedistanceABacrossariver,adistanceBCof955 mislaidoffononesideoftheriver.Itisfound

thatB=105.5°andC=17.2°.FindAB.Roundtothenearestmeter.

A) 336m B) 339m C) 283 m D) 280 m

9.3 TheLawofCosines

1 SolveSASTriangles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethetriangle.

40°

A) b=3.88

A=56°

C=84° B) b=3.88

A=84°

C=56°

C) b=4.88

A=56°

C=84° D) b=2.88

A=84°

C=56°

95°

A) c=6.05

A=29.6°

B=55.4° B) c=6.05

A=55.4°

B=29.6°

C) c=7.05

A=29.6°

B=55.4° D) c=5.05

A=55.4°

B=29.6°

3) b=4

c=5

A=75°

A) a=5.54

B=44.2°

C=60.8° B) a=5.54

B=60.8°

C=44.2°

C) a=6.54

B=44.2°

C=60.8° D) a=4.54

B=60.8°

C=44.2°

4) a=6,b=8,C=70°

A) c=8.2,A=43.5°

B=66.5° B) c=9,A=52.8°

B=57.2°

C) c=10,A=56.9°

B=53.1° D) c=6.3,A=28.6°

B=81.4°

Page7

5) a=80

b=8

C=115°

A) c=83.7

A=60°

B=5° B) c =86.6

A=62°

B=3°

C) c=89.5

A=58°

B=7° D) notriangle

6) a=4

c=3

B=90°

A) b=5

A=53.1°

C=36.9° B) b=5

A=36.9°

C=53.1°

C) b=6

A=53.1°

C=36.9° D) b=4

A=36.9°

C=53.1°

7) a=6

c=9

B=128°

A) b=13.5

A=21°

C=31° B) b=16.4

A=23°

C=29°

C) b=19.3

A=19°

C=33° D) notriangle

8) b=5

c=9

A=117°

A) a=12.1

B=22°

C=41° B) a=15

B=24°

C=39°

C) a=17.9

B=20°

C=43° D) notriangle

9) b=2

c=4

A=75°

A) a=3.98

B=29°

C=76° B) a=3.98

B=76°

C=29°

C) a=4.98

B=29°

C=76° D) a=2.98

B=76°

C=29°

2 SolveSSSTriangles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethetriangle.

A) A=57.1°

B=44.4°

C=78.5° B) A =44.4°

B=57.1°

C=78.5°

C) A=57.1°

B=78.5°

C=44.4° D) A =44.4°

B=78.5°

C=57.1°

A) A=125.1°

B=30.8°

C=24.1° B) A =30.8°

B=125.1°

C=24.1°

C) A=125.1°

B=24.1°

C=30.8° D) A =30.8°

B=24.1°

C=125.1°

3) a=5

b=5

c=2

A) A=78.5°

B=78.5°

C=23° B) A =79.5°

B=79.5°

C=21°

C) A=23°

B=78.5°

C=78.5° D) A =78.5°

B=23°

C=78.5°

Page8

4) a=8

b=5

c=4

A) A=125.1°

B=30.8°

C=24.1° B) A =30.8°

B=125.1°

C=24.1°

C) A=125.1°

B=24.1°

C=30.8° D) A =30.8°

B=24.1°

C=125.1°

5) a=8

b=14

c=16

A) A=30°

B=61°

C=89° B) A =32°

B=59°

C=89°

C) A=28°

B=61°

C=91° D) notriangle

6) a=19,b=16,c=11

A) A=87.4°

B=57.3°

C=35.3° B) A =87.4°

B=35.3°

C=57.3°

C) A=57.3°

B=87.4°

C=35.3° D) A =35.3°

B=57.3°

C=87.4°

3 SolveAppliedProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Inflyingthe76milesfromChampaigntoPeoria,astudentpilotsetsaheadingthatis12°offcourseand

maintainsanaveragespeedof140milesperhour.After15minutes,theinstructornoticesthecourseerror

andtellsthestudenttocorrecthisheading.Throughwhatanglewilltheplanemovetocorrecttheheading

andhowmanymilesawayisPeoriawhentheplaneturns?

A) 21.9°;42.39mi B) 158.1°;42.39 mi C) 21.9°;66.33 mi D) 158.1°;66.33 mi

2) TwopointsAandBareonoppositesidesofabuilding.AsurveyorselectsathirdpointCtoplacea

transit.PointCis48feetfrompointAand67feetfrompointB.TheangleACBis50°.Howfarapartare

pointsAandB?

A) 51.6ft B) 104.5 ft C) 68.7 ft D) 94.1 ft

3) Thedistancefromhomeplatetodeadcenterfieldinacertainbaseballstadiumis407feet.Abaseball

diamondisasquarewithadistancefromhomeplatetofirstbaseof90feet.Howfarisitfromfirstbaseto

deadcenterfield?

A) 349.2ft B) 384.5 ft C) 474.9 ft D) 332.1 ft

4) Afamousgolferteesoffonalong,straight459 yardpar4andsliceshisdrive10°totherightoftheline

fromteetothehole.Ifthedrivewent288yards,howmanyyardswillthegolferʹssecondshothavetobe

toreachthehole?

A) 182.4yd B) 744.3 yd C) 404.3 yd D) 651 yd

5) Afamousgolferteesoffonastraight390 yardpar4andsliceshisdrive totheright.Thedrivegoes270

yardsfromthetee.Usinga7–irononhissecondshot,hehitstheball170yardsanditlandsinchesfrom

thehole.Howmanydegrees(tothenearestdegree)totherightofthelinefromtheteetotheholedidhe

slicehisdrive?

A) 21° B) 123° C) 35° D) 52°

6) IslandAis150milesfromislandB.Ashipcaptaintravels250milesfromislandAandthenfindsthatheis

offcourseand160milesfromislandB.Whatangle,indegrees,mustheturnthroughtoheadstraightfor

islandB?Roundtheanswertotwodecimalplaces.(Hint:Becarefultoproperlyidentifywhichangleisthe

turningangle.)

A) 145.08° B) 55.08° C) 34.92° D) 110.17°

Page9

7) Aladderleansagainstabuildingthathasawallslantingawayfromtheladderatanangleof96° withthe

ground.Ifthebottomoftheladderis23feetfromthebaseofthewallanditreachesapoint52feetupthe

wall,howtallistheladdertothenearestfoot?

A) 59ft B) 60ft C) 61ft D) 58ft

8) AplanetakesofffromanairportonthebearingS29°W.Itcontinuesfor20minutesthenchangesto

bearingS52°Wandfliesfor2hours20minutesonthiscoursethenlandsatasecondairport.Iftheplaneʹs

speedis420mph,howfarfromthefirstairportisthesecondairport?Roundyouranswercorrecttothe

nearestmile.

A) 1110mi B) 1111mi C) 1010mi D) 1011mi

9) Aboxhasdimensions2ʺ×3ʺ×4ʺ.(Seeillustration.)

Determinetheangleθformedbythediagonalofthe2ʺ×3ʺsideandthediagonalofthe3ʺ×4ʺside.

Roundyouranswertothenearestdegree.

A) 60° B) 50° C) 65° D) 62°

10) Aplaneflyingastraightcourseobservesamountainatabearingof30.2° totherightofitscourse.Atthat

timetheplaneis9kilometersfromthemountain.Ashorttimelater,thebearingtothemountainbecomes

40.2°.Howfaristheplanefromthemountainwhenthesecondbearingistaken(tothenearesttenthofa

km)?

A) 7km B) 11.5 km C) 13.1 km D) 4.7km

11) TwosailboatsleaveaharborintheBahamasatthesametime.Thefirstsailsat25mphinadirection350°.

Thesecondsailsat33mphinadirection200°.Assumingthatbothboatsmaintainspeedandheading,

after5hours,howfarapartaretheboats?

A) 280.3mi B) 193.5 mi C) 210.5 mi D) 194.9 mi

12) TwopointsAandBareonoppositesidesofabuilding.AsurveyorselectsathirdpointCtoplacea

transit.PointCis46feetfrompointAand64feetfrompointB.TheangleACBis46°.Howfarapartare

pointsAandB?

A) 46.1ft B) 101.5 ft C) 64.6 ft D) 90.9 ft

9.4 AreaofaTriangle

1 FindtheAreaofSASTriangles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheareaofthetriangle.Ifnecessary,roundtheanswertotwodecimalplaces.

125°

A) 12.29 B) 15.12 C) 49.15 D) 8.6

25°

A) 8.45 B) 16.9 C) 33.81 D) 18.13

3) A=40°

b=10

c=9

A) 28.93 B) 34.47 C) 26.93 D) 36.47

4) A=83°

b=9,c=6

A) 26.80 B) 3.29 C) 27.01 D) 53.60

5) A=23°

b=9,c=2

A) 3.52 B) 3.51 C) 3.50 D) 3.53

6) a=12,b=15,C=52°

A) 70.92 B) 35.46 C) 141.84 D) 88.80

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Solvetheproblem.

7) Findtheareaoftheshadedportion(seeillustration)ofacircleofradius25cm,formedbyacentralangle

of115°.Roundyouranswertothenearestsquarecm.

[Hint:Subtracttheareaofthetrianglefromtheareaofthesectorofthecircletoobtaintheareaofthe

shadedportion.]

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

8) Anewhomeownerhasatriangular–shapedbackyard.Twoofthethreesidesmeasure65ftand80ftand

formanincludedangleof125°.Theownerwantstoapproximatetheareaoftheyard,sothathecan

determinetheamountoffertilizerandgrassseedtobepurchased.Findtheareaoftheyardroundedto

thenearestsquarefoot.

A) 2130sq.ft B) 4260sq.ft C) 2129sq.ft D) 5200sq.ft

Page11

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

9) PenrosetilesareformedfromarhombusWXYZwithsidesoflength1andinteriorangles72° and108°.

(Refertotheillustration.)ApointOischosenonthediagonal1unitfromY.LinesegmentsOXandOZare

drawntotheothervertices.Thetworesultingtilesarecalledakite(figureOXYZ)andadart(figure

OXWZ).Findtheareaofthekitetileandthedarttile,correcttothenearest0.01.

2 FindtheAreaofSSSTriangles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheareaofthetriangle.Ifnecessary,roundtheanswertotwodecimalplaces.

A) 14.98 B) 195.03 C) 4.86 D) 44.74

A) 32.25 B) 367.77 C) 8.94 D) 72.12

3) a=18

b=20

c=13

A) 114.67 B) 117.67 C) 120.67 D) 123.67

4) a=4,b=5,c=7

A) 9.80 B) 16.01 C) 10.01 D) 3.46

5) a=6,b=6,c=7

A) 17.06 B) 21.14 C) 15.54 D) 18.25

6) a=14,b=32,c=26

A) 177.99 B) 5280.01 C) 3219.69 D) 181.99

Page12

Solvetheproblem.

7) Aroomintheshapeofatrianglehassidesoflength6 yd,9 yd,and14 yd.Ifcarpetingcosts$17.50 a

squareyardandpaddingcosts$2.25asquareyard,howmuchtothenearestdollarwillitcosttocarpet

theroom,assumingthatthereisnowaste?

A) $364 B) $322 C) $354 D) $350

8) FindtheareaoftheBermudaTriangleifthesidesofthetrianglehavetheapproximatelengths841 miles,

924miles,and1317miles.

A) 386,115mi B) 1,544,461 mi C) 488,510 mi D) 510,605 mi

9) Apainterneedstocoveratriangularregion60 metersby68 metersby72 meters.Acanofpaintcovers70

squaremeters.Howmanycanswillbeneeded?

A) 28cans B) 311cans C) 14 cans D) 3cans

9.5 SimpleHarmonicMotion;DampedMotion;CombiningWaves

1 BuildaModelforanObjectinSimpleHarmonicMotion

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Anobjectattachedtoacoiledspringispulleddownadistanceafromitsrestpositionandthenreleased.Assuming

thatthemotionissimpleharmonicwithperiodT,writeanequationthatrelatesthedisplacementdoftheobject

fromitsrestpositionaftertseconds.Alsoassumethatthepositivedirectionofthemotionisup.

1) a=6; T=5seconds

A) d=–6cos2

πt B) d=–5cos1

πt C) d=–6sin2

πt D) d=–6cos1

πt

2) a=11; T=9seconds

A) d=–11cos2

πt B) d=–9cos2

πt C) d=–11sin2

πt D) d=–11cos1

πt

3) a=17; T=4πseconds

A) d=–17cos1

t B) d=–4cos2

t C) d=–17sin1

πt D) d=–17cos1

πt

4) a=5; T=10seconds

A) d=–5cosπ

t B) d=5cos(10t) C) d= –5cos(10t) D) d=5cosπ

t

Page13

2 AnalyzeSimpleHarmonicMotion

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Thedisplacementd(inmeters)ofanobjectattimet(inseconds)isgiven.Describethemotionoftheobject.What

isthemaximumdisplacementfromitsrestingposition,thetimerequiredforoneoscillation,andthefrequency?

1) d=5sin(3t)

A) simpleharmonic;5m; 2

πsec; 3

2π

oscillations/sec

B) simpleharmonic;5m; 3

2π

sec; 2

πoscillations/sec

C) simpleharmonic;–5m; 2

πsec; 3

2π

oscillations/sec

D) simpleharmonic;5m;3πsec; 3

oscillations/sec

2) d=–4sin(3t)

A) simpleharmonic;4m; 2

πsec; 3

2π

oscillations/sec

B) simpleharmonic;4m; 3

2π

sec; 2

πoscillations/sec

C) simpleharmonic;–4m; 2

πsec; 3

2π

oscillations/sec

D) simpleharmonic;–4m;3πsec; 3

oscillations/sec

3) d=3cos(5t)

A) simpleharmonic;3m; 2

πsec; 5

2π

oscillations/sec

B) simpleharmonic;3m; 5

2π

sec; 2

πoscillations/sec

C) simpleharmonic;–3m; 2

πsec; 5

2π

oscillations/sec

D) simpleharmonic;3m;5πsec; 5

oscillations/sec

4) d=–2sinπ

A) simpleharmonic;2m;4sec; 1

oscillation/sec

B) simpleharmonic;–2m;4sec; 1

oscillation/sec

C) simpleharmonic;2m; 1

sec;4oscillations/sec

D) simpleharmonic;2m;2sec; 1

oscillation/sec

5) d=2–7sin(2πt)

A) simpleharmonic;7m;1sec;1 oscillation/sec

B) simpleharmonic;–7m;1sec;1 oscillation/sec

C) simpleharmonic;7m; 1

sec;2oscillations/sec

D) simpleharmonic;2m;1sec;1 oscillation/sec

3 AnalyzeanObjectinDampedMotion

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphthedampedvibrationcurvefor0≤t≤2π.

1) d(t)=e–t

4πcost

2

-1

2

-1

2

-1

2

-1

2

-1

2

-1

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Solvetheproblem.

2) Anobjectissuspendedfromacoiledspring.Itispulleddownwardandreleased.ThepositionPaboveor

belowitsrestpositionaftertsecondsisgivenby

P=–e–x/2πcosx

Findallvaluesoft(0≤t<4π)forwhichtheobjectisattherestpositionandgraphtheequationfor

0≤t<4π.

1234

-1

1234

-1

3) Anobjectofmassmattachedtoacoiledspringwithdampingfactorbispulleddownadistanceafromits

restpositionandthenreleased.Assumethepositivedirectionofthemotionisupandtheperiodofthe

firstoscillationisT.Writeanequationthatrelatesthedistancedoftheobjectfromitsrestpositionaftert

seconds.

m=15g;a=11cm;b=0.7g/sec;T=3sec

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

4) Thedistanced(inmeters)ofthebobofapendulumfromitsrestpositionattimet(inseconds)isgivenby:

d=–9e–0.6t/60cos2π

2–0.36

3600

t

Whatisthemaximumdisplacementofthebobafterthefirstoscillation?

A) about0.13m B) about8.39m C) about5.18m D) about8.91m

Page16

4 GraphtheSumofTwoFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethemethodofaddingy–coordinatestographthefunction.

1) f(x)=x–sinx

–





–



-2



–





–



-2



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



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

-2



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



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

-2



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



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

-2



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



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

-2



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



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

-2



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



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

-2



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



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

-2



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



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-2



Page17

2) g(x)=cosx–cos(2x)

–





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

-2



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



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

-2



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



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-2



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



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-2



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



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-2

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

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-2

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



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-2

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

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-2

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



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-2



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



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-2



SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Solvetheproblem.

3) Asquarewaveisbuiltupfromsinusoidalcurvesofvaryingperiodsandamplitudes.

Graphthefollowingfunction,whichcanbeusedtoapproximatethesquarewave.

f(x)=4

πsin(πx)+1

sin(3πx) 0≤x≤4

1234

f(x)

-1

1234

f(x)

-1

Abetterapproximationtothesquarewaveisgivenby

f(x)=4

πsin(πx)+1

sin(3πx)+1

sin(5πx) 0≤x≤4

1234

f(x)

-1

1234

f(x)

-1

Graphthisfunctionandcomparetheresulttothepreviousgraph.Addinganothertermwillimprovethe

approximationevenmore.Writethisnewfunctionwithfourterms.

Ch.9 ApplicationsofTrigonometricFunctions

AnswerKey

9.1 ApplicationsInvolvingRightTriangles

1 SolveRightTriangles