Chapter5-TrigFunctions

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MCR3U

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5.1–ModelingPeriodicBehaviour

GraphingSineandCosineFunctionsWorksheet

5.3–TransformationsofSineandCosineWorksheet#1

5.3–TransformationsofSineandCosineWorksheet#2

5.5/5.6ApplicationsofSineandCosineWorksheet#1

5.5/5.6ApplicationsofSineandCosineWorksheet#1

5.1ModelingPeriodicBehaviourMCR3UJensen1)Classifyeachgraphasperiodicornotperiodic.

a) b) c) d)2)Determinetheamplitudeandperiodforanygraphinquestion1thatisperiodic.

3)Aperiodicfunction𝑓(𝑥)hasaperiodof8.Thevaluesof𝑓 1 , 𝑓 5 ,and𝑓(7)are-3,2,and8,respectively.Predictthevalueofeachofthefollowing.Ifapredictionisnotpossible,explainwhynot.a)𝑓(9) b)𝑓(29) c)𝑓(63) d)𝑓(40)4)Thepeoplemoveratanairportshuttlesbetweenthemainterminalandasatelliteterminal300metersaway.Aone-waytrip,movingataconstantspeed,takes1minute,andthecarremainateachterminalfor30secondsbeforeleaving.a)Sketchagraphtorepresentthedistanceofthecarfromthemainterminalwithrespecttotime.Includefourcompletecycles. b)Whatistheperiodofthemotion?c)Whatistheamplitudeofthemotion?

5)WhilevisitingtheeastcoastofCanada,Ranoufnoticesthatthewaterlevelatatowndockchangesduringthedayasthetidescomeinandgoout.Markingsononeofthepilessupportingthedockshowahightideof3.3metersat6:30a.m.,alowtideof0.7metersat12:40p.m.,andahightideagainat6:50p.m.

a) Estimatetheperiodofthefluctuationofthewaterlevelatthetowndock.

b) Estimatetheamplitudeofthepattern

c) PredictwhenthenextlowtidewilloccurAnswers1)a)periodicb)notperiodicc)periodicd)periodic2)a)Theperiodis5units.Theamplitudeis1.5units.c)Theperiodis6units.Theamplitudeis1unit.d)Theperiodofthegraphis3.5units.Theamplitudeis0.75units.3)a)𝑓 9 = 𝑓 1 + 8 = −3b)𝑓 29 = 𝑓 5 + 3×8 = 2c)𝑓 63 = 𝑓 7 + 7×8 = 8 d)Nopredictionpossible.𝑥 = 40 − 8𝑛;Therearenointegervaluesof𝑛thatmake𝑥 = 1,5,or7.4)a) b)3minutesc)150meters5)a)12hand20minb)1.3metersc)1:00a.m.

GraphingSineandCosineFunctionsWorksheetMCR3UJensen1)Graphthefunction𝑦 = 𝑠𝑖𝑛𝑥usingkeypointsbetween0°and360°.2)Graphthefunction𝑦 = 𝑐𝑜𝑠𝑥usingkeypointsbetween0°and360°. 3)Determinethephaseshiftandtheverticalshiftofy=sinx.a)𝑦 = sin 𝑥 − 50° + 3b)𝑦 = 2 sin 𝑥 + 45° − 1

𝒙 𝒚

𝒙 𝒚

4)Determinethephaseshiftandtheverticalshiftofy=cosx.a)𝑦 = −9 cos 𝑥 + 120° − 5b)𝑦 = 12 cos[5 𝑥 − 150° ] + 75)Determinetheamplitude,theperiod,phaseshift,verticalshift,maximumandminimumforeachofthefollowing.a)𝑦 = 5 sin[4 𝑥 + 60° ] − 2b)𝑦 = 2 cos[2 𝑥 + 150° ] − 5c)𝑦 = F

Gsin[F

G𝑥 − 60° ] + 1d)𝑦 = 0.8 cos[3.6 𝑥 − 40° ] − 0.4

Answers1) 2)3)a)phaseshift:right50°b)phaseshift:left45°verticalshift:up3unitsverticalshift:downoneunit4)a)phaseshift:left120°b)phaseshift:right150°verticalshift:down5unitsverticalshift:up7units5)a)amplitude:5period:90°phaseshift:left60°verticalshift:down2unitsmax:3min:-7b)amplitude:2period:180°phaseshift:left150°verticalshift:down5unitsmax:-3min:-7c)amplitude:F

Gperiod:720°phaseshift:right60°

verticalshift:up1unitmax:1.5min:0.5d)amplitude:0.8period:100°phaseshift:right40°verticalshift:down0.4unitsmax:0.4min:-1.2

5.3TransformationsofSineandCosineWorksheet#1MCR3UJensenInstructions:Foreachofthefollowingstartwithtwocyclesoftheparentgraph(-360to360).Transformkeypointstoproducethenewgraph.1)Usingtransformations,graphtwocyclesofthefollowingtrigonometricfunctions.Statetheperiod,phaseshift,amplitude,andverticaldisplacement.a) ( )= + +04cos 180 2y q

b) ( )é ù= + +ë û0sin 2 30 3y q

c) ( )é ù= -ë û

02sin 3 180y q

d) ( )q= - +3cos 2 4y

e) ( )= + -01 sin 45 32

y q

f) ( )é ù= - +ë û03cos 2 60 4y q

Answers

1a)

b)

c)

d)

e)

f)

5.3TransformationsofSineandCosineWorksheet#2MCR3UJensen1)Asinusoidalfunctionhasanamplitudeof5units,aperiodof120°,andamaximumat(0,3).

a) Representthefunctionwithanequationusingasinefunctionb) Representthefunctionwithanequationusingacosinefunction

2)AsinusoidalfunctionhasanamplitudeofF

Gunits,aperiodof720°,andamaximumat 0, I

G.

a) Representthefunctionwithanequationusingasinefunctionb) Representthefunctionwithanequationusingacosinefunction

3)Determinetheequationofacosinefunctionthatrepresentsthegraphshown.4)Therelationshipbetweenthestressontheshaftofanelectricmotorandtimecanbemodelledwithasinusoidalfunction.Determineanequationofafunctionthatdescribesstressintermsoftime.

5)Determinetheequationofthesinefunctionshown.6)Representthegraphofthefollowingfunctionsusingasineandcosinefunction.

Answers1)a)y=5sin[3(x+30°)]–2b)y=5cos3x–2

2)a) ( )1 1sin 180 12 2

y xé ù= + ° +ê úë ûb) 1 1cos 1

2 2y x= +

3)y=cos[4(x–82.5°)]+24)y=3sin(9000x)+8OR𝑦 = 3 cos 9000 𝑥 − 0.01 + 85)a)y=2sin[6(x–15°)]+1b)Ifthemaximumvaluesarehalfasfarapart,theperiodofthefunctionisreducedbyone-halfto30°.Thevalueofkdoublesfrom6to12.Theequationforthenewfunctionisy=2sin[12(x–15°)]+1.6)y=4cos2xandy=4sin[2(x+45°)].

5.5/5.6ApplicationsofSineandCosineFunctionsWorksheet#1MCR3UJensen1)Atamaximumheightof135m,theMillenniumWheel,inLondon,England,isthelargestcantileveredstructureintheworld.Itmovessoslowlythatthereisusuallynoneedtostopthewheeltoletpeopleonoroff.Lettheoriginbethecenterofthewheel.a)Startasketchoftheverticaldisplacementfromthecenterofthewheelofacaronthewheelasafunctionoftheanglethroughwhichthewheelrotates,usingthebottomofthewheelasthestartingpointofthetrip.Sketchtwocycles.

b)Determinetheamplitudeandperiodofthefunction.2)a)Repeatquestion1,exceptthistimegraphhorizontaldisplacementinsteadofverticaldisplacement.

b)Determinetheamplitudeandperiodofthefunction.

3)Thehourhandonaclockhasalengthof12cm.a)Sketchthegraphoftheverticalpositionofthetipofthehourhandfromthecenteroftheclockversustheanglethroughwhichthehandturnsforatimeperiodof72h.Assumethatthehourhandstartsat9.b)Sketchthegraphofthehorizontalpositionofthetipofthehourhandversustheanglethroughwhichthehandturnsforatimeperiodof72h.Assumethatthehourhandstartsat3.c)Howmanycyclesappearinthegraphinparta)andb)?

4)AFerriswheelhasadiameterof20mandis4mabovegroundlevelatitslowestpoint.Assumethatariderentersacarfromaplatformthatislocated30°aroundtherimbeforethecarreachesitslowestpoint.a)Modeltherider’sheightabovethegroundversusangleusingatransformedsinefunctionb)Modeltherider’sheightabovethegroundversusangleusingatransformedcosinefunction.c)Supposethattheplatformismovedto60°aroundtherimfromthelowestpositionofthecar.Howwilltheequationsinpartsa)andb)change?Writethenewequations.5)SupposethatthecenteroftheFerriswheelinthepreviousquestionismovedupward2m,buttheplatformisleftinplaceatapoint30°beforethecarreachesitslowestpoint.Howdotheequationsinpartsa)andb)change?Writethenewequations.

Answers1)a) b)amplitude67.5;period360°2)a) b)amplitude67.5;period360°3)a) b) c)Thegraphsinparta)andb)bothshow6fullcyclesbecausetherearesix12-hperiodsin72h.4)a)Theamplitudeis10mandthemidlineisat14m.Iftheriderbeginsherride30°beforetheminimum,thenshewillreachtherisingmidlinepointafterarotationof120°foraphaseshiftof120°totheright.Theperiodis360°.Anequationthatmodelstherider’sheightversustherotationangleisy=10sin(x–120°)+14.b)Foracosinefunction,theridermustrotate210°toreachthefirstmaximumpoint.Thisrequiresaphaseshiftof210°totheright.Theotherparametersremainthesame.Anequationthatmodelstherider’sheightversustherotationangleisy=10cos(x–210°)+14.c)Iftheinitialpositionisplaced30°sooner,thenthephaseshiftofbothcurvesmustincreaseby30°.Anewsineequationthatmodelstherider’sheightversustherotationangleisy=10sin(x–150°)+14.Anewcosineequationthatmodelstherider’sheightversustherotationangleisy=10cos(x–240°)+14.5)IfthecentreoftheFerriswheelisraisedby2m,thentheverticalshiftalsoincreaseby2from14to16.Therelativepositionoftheplatformdoesnotchange,sothephaseshiftisnotaffected.a)Anewsineequationthatmodelstherider’sheightversustherotationangleisy=10sin(x–120°)+16.b)Anewcosineequationthatmodelstherider’sheightversustherotationangleisy=10cos(x–210°)+16.

5.5/5.6ApplicationofSineandCosineFunctionsWorksheet#2MCR3UJensen1)Amotionsensorrecordedthemotionofachildonaswing.Thedatawasgraphed,asshown.

a)Findthemaxandminvalues.b)Findamplitudec)Determinetheverticalshiftofthefunction.d)Findtheperiodofthefunctione)Determinethephaseshift,ifthemotionweretobemodelledusingasinefunction.2)Theheightofthebladeofawindturbineasitturnsthroughanangleofqisgivenbythefunctionh(q)=8.5sin(q+

180°)+40,withheightmeasuredinmetres.a)Findthemaximumandminimumpositionsoftheblade.b)Explainwhatthevalueof40intheequationrepresents.c)Explainwhatthevalueoftheamplituderepresents.

d)Sketchthefunctionovertwocycles.3)Theheight,ℎ,inmeters,ofthetideinagivenlocationonagivendayat𝑡hoursaftermidnightcanbemodeledusingthesinusoidalfunctionℎ 𝑡 = 5 sin 30 𝑡 − 5 + 7.a)Findthemaxandminvaluesforthedepthofwater.b)Whattimeishightide?Whattimeislowtide?c)Whatisthedepthofthewaterat9:00am?d)Findallthetimesduringa24-hperiodwhenthedepthofthewateris3meters.

4)Thepopulation,𝑃,ofalakesidetownwithalargenumberofseasonalresidentscanbemodeledusingthefunction𝑃 𝑡 = 5000 sin 30 𝑡 − 7 + 8000,where𝑡isthenumberofmonthsafterNewYear’sDay.a)Findthemaxandminvaluesforthepopulationoverawholeyear.b)Whenisthepopulationamaximum?Whenisitaminimum?c)WhatisthepopulationonSeptember30th?5)Thepopulationofpreyinapredator-preyrelationisshown.Timeisinyearssince1985.a)Determinethemaxandminvaluesofthepopulation,tothenearest50.Usethesetofindtheamplitude.b)Determinetheverticalshift,𝑐.c)Determinethephaseshift,𝑑.d)Determinetheperiod.Usetheperiodtodeterminethevalueof𝑘.e)Modelthepopulationversustimewithasinusoidalfunction.

6)Thenumberofmillionsofvisitorsthatatouristattractiongetscanbemodeledusingtheequation𝑦 =2.3sin[30 𝑥 + 1 ] + 4.1,where𝑥 = 1representsJanuary,𝑥 = 2representsFebruary,andsoon.a)Determinetheperiodofthefunctionandexplainitsmeaning.b)Graphthefunctionfor12months.c)Whichmonthhasthemostvisitors?d)Whichmonthhastheleastvisitors?

Answers1)a)maximum2.25,minimum0.25b)amplitude1c)verticalshiftup1.25d)period5e)horizontalshift1.75totheright2)a)maximum48.5,minimum31.5 d)

b)Theheightofthecenteroftheturbinec)Theamplitudeof8.5representsthelengthoftheblade.

3)a)Fromtheequation,c=7anda=5,sothefunctionhasamidlinevalueof7andanamplitudeof5.Themaximumheightis12mandtheminimumheightis2m.b)Fromtheequation,k=30andd=5,sotheperiodis12handthephaseshiftis5hright.Thefirstmidlinevalueoccursat5:00a.m.Thefirstmaximumoccursone-quarterperiod,or3hafterthis,at8:00a.m.Thepreviousminimumis3hpriorto5:00a.m.,at2:00a.m.Becauseofthe12-hperiod,therewillalsobeamaximumat8:00p.m.andaminimumat2:00p.m.c)11.3md)Thesolutiongivesatimeofapproximately3:14a.m.Thistimeis1h14minafterthefirstminimumsothedepthshouldalsooccur1h14minbefore2:00a.m,at12:46a.m.Becauseofthe12-hperiod,thedepthwillalsooccurat12:46p.m.and3:14p.m.4)a)Fromtheequation,c=8000anda=5000,sothefunctionhasamidlinevalueof8000andanamplitudeof5000.Themaximumpopulationis13000andtheminimumpopulationis3000.b)Fromtheequation,k=30andd=7,sotheperiodis12monthsandthephaseshiftis7monthsright.Theinitialmidlinevalueoccursatt=7.Themaximumoccurs3monthslateratt=10(October)andtheminimum3monthsearlieratt=4(April).c)123305)a)Fromthegraphthemaximumpopulationisapproximately850animalsandtheminimumpopulationisapproximately250animals.Theamplitudeisapproximately300animals,soa=300.b)Theverticalshiftisthemaximumvalueminustheamplitude,soc=550.c)Themidlineintersectsthegraphatt=0sonohorizontalshiftisnecessary,sod=0.d)Thepatternrepeatsevery6years,sotheperiodis6years.k=60.e)Asinefunctionthatmodelsthepopulationofprey,N,withrespecttotime,t,isN=300sin60t+550.6)a)12months b)c)Februaryd)August