Chapter5-TrigFunctions

LessonPackage

MCR3U

Chapter4OutlineUnitGoal:Beabletoidentifyandrepresentsinusoidalfunctions,andsolveproblemsinvolvingsinusoidalfunctions,includingproblemsarisingfromreal-worldapplications.

Section Subject LearningGoals CurriculumExpectations

L1 ModelingPeriodicBehaviour

-describekeypropertiesofperiodicfunctionsandpredictfuturevaluesbyextrapolating D2.1,D2.2

L2 GraphingSineandCosineFunctions

-graphsin 𝑥andcos 𝑥foranglesgivenindegrees D2.3,D2.4

L3 TransformationsofSineandCosinePart1

-giventheequationoftheasinusoidalfunction,usetransformationstographit

D2.5,D2.6,D2.7,D2.8

L4 TransformationsofSineandCosinePart2

-giventhegraphofasinusoidalfunction,determineanequationthatdefinesit

D2.5,D2.6,D2.7,D2.8

L5 TrigApplicationsPart1

-solveproblemsthatarisefromrealworldapplicationsinvolvingperiodicphenomena

D3.2,D3.3,D3.4

L6 TrigApplicationsPart2

-solveproblemsthatarisefromrealworldapplicationsinvolvingperiodicphenomena

D3.2,D3.3,D3.4

Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P

PreTestReview F/A P Test–TrigGeometry O D2.1,D2.2,D2.3,D2.4,D2.5,

D2.6,D2.7,D2.8,D3.4 P K(21%),T(34%),A(10%),C(34%)

L1-ModelingPeriodicBehaviourMCR3UJensenSection1:Definitions__________________________:afunctionthathasapatternof𝑦-valuesthatrepeatsatregularintervals._______________:onecompleterepetitionofapattern._______________:thehorizontallengthofonecycleonagraph.____________________:halfthedistancebetweenthemaximumandminimumvaluesofaperiodicfunction.Section2:RecognizingPropertiesofPeriodicFunctions

Example1:Determinewhetherthefunctionsareperiodicornot.Ifitis,statetheperiodofthefunction.i)ii)

How to find the PERIOD of a function: choose a convenient x-coordinate to start at and then move to the right and estimate the x-coordinate of the where the next cycle begins. Find the difference of these x-coordinates to calculate the period of the function.

Example2:Isthefunctionperiodic?Ifso,whatistheamplitude?

Example3:Inthefollowingperiodicfunction,determinetheperiodandamplitude.Section3:PredictingValuesofaPeriodicFunctionExample4:Forthefollowingfunction…

How to find the AMPLITUDE of a function: the amplitude is half the difference between the max and min values. Use the formula: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = 234562378

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a)determine𝑓(2)and𝑓(5)b)determine𝑓 8 , 𝑓 −10 , 𝑎𝑛𝑑𝑓(14)c)determine4valuesof𝑥sothat𝑓 𝑥 = 2Example5:Acuttingmachinechopsstripsofplasticintotheirappropriatelengths.Thefollowinggraphshowsthemotionofthecuttingbladeonthemachineintermsoftime.

a)Statethemaxheightoftheblade,theminimumheight,andtheamplitudeofthefunction.b)Whatistheperiodofthisfunction?c)Statethenexttwotimesthatthebladewillstrikethecuttingsurface?

L2-GraphingSineandCosineFunctionsMCR3UJensenSection1:GraphingSineandCosine DESMOSdemonstration

Tographsineandcosine,wewillbeusingaCartesianplanethathasanglesfor𝑥values.

Example1:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = sin(𝑥).Usespecialtriangles,theunitcircle,oracalculatortofindvaluesforthefunctionat30°intervals.Usethetabletographthefunction.

𝑥 𝑓(𝑥)0 30 60 90 120 150 180 210 240 270 300 330 360

Example2:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = cos(𝑥).Usespecialtriangles,theunitcircle,oracalculatortofindvaluesforthefunctionat30°intervals.Usethetabletographthefunction.

𝑥 𝑓(𝑥)0 30 60 90 120 150 180 210 240 270 300 330 360

Section2:PropertiesofSineandCosineFunctionsDomain:Range:Period:Amplitude:Section3:TransformationsoftheSineandCosineFunctions𝑦 = 𝑎 sin 𝑘 𝑥 − 𝑑 + 𝑐 DesmosDemonstration

𝑎 𝑘 𝑑 𝑐Verticalstretchorcompressionbyafactorof𝑎.Verticalreflectionif𝑎 < 0𝑎 = 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒

HorizontalstretchorcompressionbyafactorofM

N.

Horizontalreflectionif𝑘 < 0.360|𝑘| = 𝑝𝑒𝑟𝑖𝑜𝑑

Phaseshift𝑑 > 0; 𝑠ℎ𝑖𝑓𝑡𝑟𝑖𝑔ℎ𝑡𝑑 < 0; 𝑠ℎ𝑖𝑓𝑡𝑙𝑒𝑓𝑡

Verticalshift𝑐 > 0; 𝑠ℎ𝑖𝑓𝑡𝑢𝑝𝑐 < 0; 𝑠ℎ𝑖𝑓𝑡𝑑𝑜𝑤𝑛

Example3:Forthefunction𝑦 = 3 sin 2 𝜃 + 60° − 1,statethe…Amplitude: Period:

Phaseshift: Verticalshift:

Max: Min:

L3–TransformationsofSineandCosinePart1 EquationàGraphMCR3UJensenSection1:ReviewofSineandCosineFunctions

𝑦 = 𝑎 sin 𝑘 𝑥 − 𝑑 + 𝑐OR𝑦 = 𝑎 cos 𝑘 𝑥 − 𝑑 + 𝑐

𝑎 𝑘 𝑑 𝑐Verticalstretchorcompressionbyafactorof𝑎.Verticalreflectionif𝑎 < 0𝑎 = 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒

HorizontalstretchorcompressionbyafactorofMN.Horizontalreflectionif𝑘 <0.360|𝑘|

= 𝑝𝑒𝑟𝑖𝑜𝑑

Phaseshift𝑑 > 0; 𝑠ℎ𝑖𝑓𝑡𝑟𝑖𝑔ℎ𝑡𝑑 < 0; 𝑠ℎ𝑖𝑓𝑡𝑙𝑒𝑓𝑡

Verticalshift𝑐 > 0; 𝑠ℎ𝑖𝑓𝑡𝑢𝑝𝑐 < 0; 𝑠ℎ𝑖𝑓𝑡𝑑𝑜𝑤𝑛

Graphsofparentfunctions𝑦 = sin 𝑥and𝑦 = cos 𝑥usingkeypoints:

Section2:GraphingTransformedSinusoidalFunctionsExample1:Graph𝑦 = 2 sin 𝑥 + 1usingtransformations.Thenstatetheamplitude,period,andnumberofcyclesbetween0°and360°.

Amplitude:Period:Numberofcyclesbetween0°and360°:

Example2:Graph𝑦 = −1.5 cos 3 𝑥 − 30° + 0.5usingtransformations.Thenstatetheamplitude,period,andnumberofcyclesbetween0°and360°.

Amplitude:Period:Numberofcyclesbetween0°and360°:

Example3:Graph𝑦 = sin −4 𝑥 − 60° + 2usingtransformations.Thenstatetheamplitude,period,andnumberofcyclesbetween0°and360°.

Amplitude:Period:Numberofcyclesbetween0°and360°:

L4–TransformationsofSineandCosinePart2 GraphàEquationMCR3UJensenSection1:HowtoDeterminetheEquationofaSineorCosineFunctionGivenitsGraph1)Findthemaxandminofthefunction2)Findtheamplitudeofthefunction(𝑎-value):𝑎 = [\]6[^_

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3)Findtheverticalshift(𝑐-value):𝑐 = 𝑚𝑎𝑥 − 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒(thisfindsthe‘middle’ofthefunction)4)Findtheperiod(indegrees)ofthefunctionusingastartingpointandendingpointofafullcycle5)Calculatethe𝑘-value.𝑘 = `ab

cde^fgà𝑝𝑒𝑟𝑖𝑜𝑑 = `ab

|N|

6)Determinethephaseshift(𝑑-value)

-forsin 𝑥:tracealongthecenterlineandfindthedistancebetweenthe𝑦-axisandthebottomleftoftheclosestrisingmidline.-forcos 𝑥:thedistancebetweenthe𝑦-axisandtheclosestmaximumpoint

Section2:DeterminingtheEquationofaSinusoidalFunctionGivenitsGraphExample1:Foreachofthefollowinggraphs,determinetheequationofasineandcosinefunctionthatrepresentseachgraph:a)b)c)

Example2:Asinusoidalfunctionhasanamplitudeof3units,aperiodof180degreesandamaxpointat(0,5).Representthefunctionwithanequationintwodifferentways.Example3:Asinusoidalfunctionhasanamplitudeof5units,aperiodof120degreesandamaximumat(0,3).Representthefunctionwithanequationintwodifferentways.

L5–TrigApplicationsPart1 MCR3UJensenBeforewedoapplicationquestions,itwillbegoodtoknowtheconnectionbetweenwhatwelearnedlastchapterandthefunctionsfromthischapter:Desmos-SineGraph Desmos-CosineGraphSection1:RememberingtheUnitCircleThecirclebeingusedhasradius𝑟.Theradiusandthecoordinatesofapointonthecircle(𝑥, 𝑦)arerelatedtotheprimarytrigratios.Studythecircleandwriteexpressionsforsin 𝜃,cos𝜃,andtan𝜃intermsof𝑥,𝑦,and𝑟.AUNITCIRLCEhasaradiusof1.Usetheunitcircletowriteexpressionsforsin 𝜃,cos 𝜃,andtan 𝜃intermsof𝑥,𝑦,and𝑟.

Summaryoffindingsfortrigratiosusingtheunitcircle:Thesinefunction:graphstherelationshipbetweentheangleandthe____________________displacementfromthex-axis.Thecosinefunction:graphstherelationshipbetweentheangleandthe____________________displacementfromthey-axis.Section2:ModelingwithGraphsExample1:YouareinacarofaFerriswheel.Thewheelhasaradiusof8mandturnscounterclockwise.Lettheoriginbeatthecenterofthewheel.Beginyoursketchwhentheradiusfromthecenterofthewheeltoyourcarisalongthepositivex-axis.a)Sketchthegraphofverticaldisplacementversustheangleofrotationfor1completerotation.b)Sketchthegraphofhorizontaldisplacementversustheangleofrotationfor1completerotationstartingalongthepositivex-axis.

c)Sketchthegraphofhorizontaldisplacementversustheangleofrotationfor1completerotationifyourcarstartsatthebottomoftheFerrisWheel.Example2:Acarouselrotatesataconstantspeed.Ithasadiameterof15m.Ahorsethatisdirectlyinlinewiththecenter,horizontally,rotatesaround3fulltimes.Createagraphthatmodelsthehorizontaldistancefromthecenterasthehorserotatesaround.

Section3:ModelingwithEquationsExample3:Agroupofstudentsistrackingafriend,John,whoisridingaFerriswheel.TheyknowthatJohnreachesthemaximumheightof11mat10seconds,andthenreachestheminimumheightof1mat55seconds.a)DevelopanequationofasineandcosinefunctionthatmodelsJohn’sheightabovetheground.b)WhatisJohn’sheightabovethegroundafter78seconds?

Example4:DonQuixote,afictionalcharacterinaSpanishnovel,attackedwindmillsbecausehethoughttheyweregiants.Atonepoint,hegotsnaggedbyoneofthebladesandwashoistedintotheair.Thecenterofthewindmillis10metersoffthegroundandeachbladeis7meterslong.Thebladepickedhimupwhenitwasatitslowestpoint.a)GraphDon’sheightabovethegroundduringonefullrotationaroundthewindmillb)Determineanequationforasineandcosinefunctionthatrepresentshisheightabovethegroundinrelationtotheangleofrotation.

L6–TrigApplicationsPart2 MCR3UJensenExample1:Theheight,h,inmeters,abovethegroundofarideronaFerriswheelaftertsecondscanbemodelledbythesinefunction:

ℎ 𝑡 = 10 sin 3 𝑡 − 30 + 12a)Graphthefunctionusingtransformationsb)Determinethemaxheight,minheight,andtimeforonerevolution.

c)Representthefunctionusingtheequationofacosinefunctiond)Whatistheheightoftheriderafter35seconds?Usebothequationstoverifyyouranswer.Example2:Skyscrapersswayinhigh-windconditions.Inonecase,at𝑡 = 2seconds,thetopfloorofabuildingswayed30cmtotheleft(-30cm)andat𝑡 = 12seconds,thetopfloorswayed30cmtotheright(+30cm)ofitsstartingposition.a)Whatistheequationofacosinefunctionthatdescribesthemotionofthebuildingintermsoftime?b)Whatistheequationofasinefunctionthatdescribesthemotionofthebuildingintermsoftime?

Example3:Theheightofthetideonagivendayat′𝑡′hoursaftermidnightismodelledby:

ℎ 𝑡 = 5 sin 30 𝑡 − 5 + 7a)Findthemaxandminvaluesfortheheightofthedepthofthewaterb)Whattimeishighttide?Whattimeislowtide?c)Whatisthedepthofthewaterat9am?

Example4a:Awindturbinehasaheightof55mfromthegroundtothecenteroftheturbine.Graphonecycleoftheverticaldisplacementofa10mbladeturningcounterclockwise.Assumethebladestartspointingstraightdown.Example4b:Modeltherider’sheightabovethegroundversusangleusingatransformedsineandcosinefunction.