Chapter4/5Part1-Trigonometryin

Radians

LessonPackage

MHF4U

Chapter4/5Part1OutlineUnitGoal:Bytheendofthisunit,youwillbeabletodemonstrateanunderstandingofmeaningandapplicationofradianmeasure.Youwillbeabletomakeconnectionsbetweentrigratiosandthegraphicalandalgebraicrepresentationsofthecorrespondingtrigfunctions.

Section Subject LearningGoals CurriculumExpectations

L1 RadianMeasure-recognizetheradianasanalternativeunittothedegreeforanglemeasurement-convertbetweendegreeandradianmeasures

B1.1,1.2

L2 TrigRatiosandSpecialAngles

-Determine,withouttechnology,theexactvaluesoftrigratiosforspecialangles B1.3,1.4

L3 GraphingTrigFunctions -sketchthegraphsofall6trigratiosandbeabletodescribekeyproperties B2.1,2.2,2.3

L4 TransformationsofTrigFunctions

-Givenequationstatepropertiesofsinusoidalfunction,andgraphitusingtransformations-Givengraph,writetheequationofthetransformedfunction

B2.4,2.5,2.6

L5 ApplicationsofTrigFunctions

-Solveproblemsarisingfromrealworldapplicationsinvolvingtrigfunctions B2.7

Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P Quiz–FindingTrigRatios F P PreTestReview F/A P Test–TriginRadians O B1.1,1.2,1.3,1.4

B2.1,2.2,2.3,2.4,2.5,2.6,2.7 P K(21%),T(34%),A(10%),C(34%)

L1–4.1RadianMeasureMHF4UJensenPart1:WhatisaRadian?Anglesarecommonlymeasuredin____________.However,inmathematicsandphysics,therearemanyapplicationsinwhichexpressinganangleasapurenumber,withoutunits,ismoreconvenientthanusingdegrees.Whenmeasuringin____________,thesizeofanangleisexpressedintermsofthelengthofanarc,𝑎,thatsubtendstheangle,𝜃,where𝜃 =1Radianisdefinedasthesizeofananglethatissubtendedbyanarcwithalength________________________________ofthecircle.Therefore,whenthearclengthandradiusareequal,𝜃 = $

%=

Howmanyradiansareinafullcircle?Orinotherwords,howmanytimescananarclengthequaltotheradiusfitaroundthecircumferenceofacircle?Remember:𝐶 = 2𝜋𝑟Soifweusethefullcircumferenceofthecircleforthearclength,𝜃 = $

%=

Part2:SwitchingBetweenDegreesandRadiansThekeyrelationshipyouneedtoknowinordertoswitchbetweendegreesandradiansis:

360° = 2𝜋radians

1degree=Toswitchfromdegreestoradians,multiplyby

1radian=Toswitchfromradianstodegrees,multiplyby

Example1:Startbyconvertingthefollowingcommondegreemeasurestoradians

a)180°

b)90°

c)60°

d)45°

e)30°

f)1°

Example2:Converteachofthefollowingdegreemeasurestoradianmeasures

Example3:Converteachofthefollowingradianmeasurestodegreemeasures

Part3:ApplicationExample4:Suzettechoosesacameltorideonacarousel.Thecamelislocated9mfromthecenterofthecarousel.Itthecarouselturnsthroughanangleof34

5,determinethelengthofthearctravelledbythecamel,

tothenearesttenthofameter.

a)225°

b)80°

c)450°

a)647radians

b)849radians

c)1radian

Part4:SpecialTrianglesUsingRadianMeasuresDrawthe45-45-90triangle Drawthe30-60-90triangle Also,youwillneedtoremembertheUNITCIRLCEwhichisacirclethathasaradiusof1.Usetheunitcircletowriteexpressionsforsin 𝜃,cos 𝜃,andtan 𝜃intermsof𝑥,𝑦,and𝑟.Ontheunitcircle,thesineandcosinefunctionstakeasimpleform:sin 𝜃 =cos 𝜃 =Thevalueofsin 𝜃isthe________________ofeachpointontheunitcircleThevalueofcos 𝜃isthe________________ofeachpointontheunitcircleTherefore,wecanusethepointswheretheterminalarmintersectstheunitcircletogetthesineandcosineratiosjustbylookingatthe𝑦and𝑥co-ordinatesofthepoints.

sin 𝜃 =cos 𝜃 =tan 𝜃 =

Example5:Filloutchartofratiosusingspecialtrianglesandtheunitcircle

0, 2𝜋𝜋6

𝜋4

𝜋3

𝜋2 𝜋 3𝜋

2

sin 𝑥

cos 𝑥

tan 𝑥

L2–4.2TrigRatiosandSpecialAnglesMHF4UJensenPart1:ReviewofLastYearTrigWhatisSOHCAHTOA?Ifweknowarightangletrianglehasanangleof𝜃,allotherrightangletriangleswithanangleof𝜃are_________andthereforehaveequivalentratiosofcorrespondingsides.Thethreeprimaryratiosareshowninthediagramtotheright.Whatisareferenceangle?Anyangleover90hasareferenceangle.Thereferenceangleisbetween0°and90°andhelpsusdeterminetheexacttrigratioswhenwearegivenanobtuseangle(angleover90degrees).Thereferenceangleistheanglebetweentheterminalarmandthe________________________(0/360or180).WhatistheCASTrule?Whenfindingthetrigratiosofpositiveangles,wearerotatingcounterclockwisefrom0degreestoward360.TheCASTrulehelpsusdeterminewhichtrigratios___________________ineachquadrantNote:Therearemultipleanglesthathavethesametrigratio.Youcanusereferenceanglesandthecastruletofindthem.

sinθ

cosθ

tanθ =

=

= opposite

opposite

hypotenuse

hypotenuse

adjacent

adjacent

Part2:FindingExactTrigRatiosforSpecialAnglesStartbydrawingbothspecialtrianglesusingradianmeasuresExample1:Findtheexactvalueforeachgiventrigratio.

a)tan (()* b)sin -)

*

c)cos -)0

d)sec 2)

0

Part3:ApplicationQuestionExample2:Findthevalueofall6trigratiosfor-)

2

Justinisflyingakiteattheendofa50-mstring.Thesunisdirectlyoverhead,andthestringmakesanangleof)*withtheground.Thewindspeedincreases,andthekiteflieshigheruntilthestringmakesanangleof)

2with

theground.Determineanexactexpressionforthehorizontaldistancebetweenthetwopositionsofthekitealongtheground.Thehorizontaldistancebetweenthetwokites=

L3–5.1/5.2GraphingTrigFunctionsMHF4UJensenPart1:RemembertheUnitCircleTheunitcircleisacircleisacirclethatiscenteredattheoriginandhasaradiusof________.Ontheunitcircle,thesineandcosinefunctionstakeasimpleform:sin 𝜃 =cos 𝜃 =Thevalueofsin 𝜃isthe________________ofeachpointontheunitcircleThevalueofcos 𝜃isthe________________ofeachpointontheunitcircle

Part2:GraphingSineandCosine

Tographsineandcosine,wewillbeusingaCartesianplanethathasanglesfor𝑥values.

Example1:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = sin(𝑥).Usespecialtriangles,theunitcircle,oracalculatortofindvaluesforthefunctionat30° = 0

1radianintervals.

Example2:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = cos(𝑥).Usespecialtriangles,theunitcircle,oracalculatortofindvaluesforthefunctionat30° = 0

1radianintervals.

𝑥 𝑓(𝑥)0 01

201= 0

3

301= 0

2

401= 20

3

501

101= 𝜋 701

801= 40

3

901= 30

2

:;01= 50

3

::01

:201= 2𝜋

𝑥 𝑓(𝑥)0 01

201= 0

3

301= 0

2

401= 20

3

501

101= 𝜋 701

801= 40

3

901= 30

2

:;01= 50

3

::01

:201= 2𝜋

PropertiesofbothSineandCosineFunctionsDomain:Range:Period:Amplitude:____________:thehorizontallengthofonecycleonagraph.____________:halfthedistancebetweenthemaximumandminimumvaluesofaperiodicfunction.Part3:GraphingtheTangentFunction

Recall:tan 𝜃 = ?@ABCD?B

Note:Sincecos 𝜃isinthedenominator,anytimecos 𝜃 = 0,tan 𝜃willbeundefinedwhichwillleadtoaverticalasymptote.

Sincesin 𝜃isinthenumerator,anytimesin 𝜃 = 0,tan 𝜃willequal0whichwillbean𝑥-intercept.

Example3:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = tan(𝑥).Usethequotientidentitytofind𝑦-values.

PropertiesoftheTangentFunctionDomain: Range:Period: Amplitude:

𝑥 𝑓(𝑥)0 01

201= 0

3

301= 0

2

401= 20

3

501

101= 𝜋 701

801= 40

3

901= 30

2

:;01= 50

3

::01

:201= 2𝜋

Part4:GraphingReciprocalTrigFunctionsThegraphofareciprocaltrigfunctionisrelatedtothegraphofitscorrespondingprimarytrigfunctioninthefollowingways:

• Reciprocalhasaverticalasymptoteateachzeroofitsprimarytrigfunction• Hasthesamepositive/negativeintervalsbutintervalsofincreasing/decreasingarereversed• 𝑦-valuesof1and-1donotchangeandthereforethisiswherethereciprocalandprimaryintersect• Localminpointsoftheprimarybecomelocalmaxofthereciprocalandviceversa.

Example4:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = csc(𝑥).Usethereciprocalidentitytofind𝑦-values.

PropertiesoftheCosecantFunctionDomain: Range:Period: Amplitude:

ReciprocalIdentities

𝒄𝒔𝒄𝜽 = 𝒔𝒆𝒄𝜽 = 𝒄𝒐𝒕𝜽 =

𝑥 𝑓(𝑥)0 01

201= 0

3

301= 0

2

401= 20

3

501

101= 𝜋 701

801= 40

3

901= 30

2

:;01= 50

3

::01

:201= 2𝜋

Example5:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = sec(𝑥).Usethereciprocalidentitytofind𝑦-values.

PropertiesoftheSecantFunctionDomain: Range:Period: Amplitude:

𝑥 𝑓(𝑥)0 01

201= 0

3

301= 0

2

401= 20

3

501

101= 𝜋 701

801= 40

3

901= 30

2

:;01= 50

3

::01

:201= 2𝜋

Example6:Completethefollowingtableofvaluesforthefunction𝑓 𝑥 = tan(𝑥).Usethereciprocalidentitytofind𝑦-values.

PropertiesoftheSecantFunctionDomain: Range:Period: Amplitude:

𝑥 𝑓(𝑥)0 01

201= 0

3

301= 0

2

401= 20

3

501= 20

3

101= 𝜋 701

801= 40

3

901= 30

2

:;01= 50

3

::01

:201= 2𝜋

L4–5.3TransformationsofTrigFunctionsMHF4UJensenPart1:TransformationProperties𝑦 = 𝑎 sin 𝑘 𝑥 − 𝑑 + 𝑐 DesmosDemonstration

𝑎 𝑘 𝑑 𝑐

Verticalstretchorcompressionbyafactorof𝑎.Verticalreflectionif𝑎 < 0𝑎 = 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒

Horizontalstretchorcompressionbyafactorof67.Horizontalreflectionif𝑘 <0.2𝜋|𝑘|

= 𝑝𝑒𝑟𝑖𝑜𝑑

Phaseshift𝑑 > 0; 𝑠ℎ𝑖𝑓𝑡𝑟𝑖𝑔ℎ𝑡𝑑 < 0; 𝑠ℎ𝑖𝑓𝑡𝑙𝑒𝑓𝑡

Verticalshift𝑐 > 0; 𝑠ℎ𝑖𝑓𝑡𝑢𝑝𝑐 < 0; 𝑠ℎ𝑖𝑓𝑡𝑑𝑜𝑤𝑛

Example1:Forthefunction𝑦 = 3 sin 6

G𝜃 + 𝜋

I− 1,statethe…

Amplitude:

Period:

Phaseshift:

Verticalshift:

Max:

Min:

Part2:GivenEquationàGraphFunctionExample2:Graph𝑦 = 2 sin 2 𝑥 − 𝜋

I+ 1usingtransformations.Thenstatetheamplitudeandperiodof

thefunction. Amplitude: Period:

𝑦 = sin 𝑥

𝒙 𝒚

𝑦 = 2 sin 2 𝑥 −𝜋3 + 1

Part3:GiventheGraphàWritetheEquation𝑦 = 𝑎 sin 𝑘 𝑥 − 𝑑 + 𝑐

𝑎 𝑘 𝑑 𝑐

Findtheamplitudeofthefunction:

Findtheperiod(inradians)ofthefunctionusingastartingpointandendingpointofafullcycle.

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

Findtheverticalshift

OR

(thisfindsthe‘middle’ofthefunction)

Example3:Determinetheequationofasineandcosinefunctionthatdescribesthefollowinggraph

Example4:DeterminetheequationofasineandcosinefunctionthatdescribesthefollowinggraphExample5:

a)Createasinefunctionwithanamplitudeof7,aperiodof𝜋,aphaseshiftof𝜋Rright,andavertical

displacementof-3.b)Whatwouldbetheequationofacosinefunctionthatrepresentsthesamegraphasthesinefunctionabove?

Part4:EvenandOddFunctions

EvenFunctions OddFunctionsEVENFUNCTIONif:

Linesymmetryoverthe____________

ODDFUNCTIONif:

Pointsymmetryaboutthe____________

Rule:𝑓 −𝑥 = 𝑓(𝑥)

Rule:−𝑓 𝑥 = 𝑓(−𝑥)

Example:

𝑦 = cos 𝑥

Example:

𝑦 = sin 𝑥𝑦 = tan 𝑥isalsoanoddfunction

L5–5.3TrigApplicationsMHF4UJensenExample1:AFerriswheelhasadiameterof15mandis6mabovegroundlevelatitslowestpoint.Ittakestherider30sfromtheirminimumheightabovethegroundtoreachthemaximumheightoftheferriswheel.Assumetheriderstartstherideattheminpoint.a)Modeltheverticaldisplacementoftheridervs.timeusingasinefunction.b)Sketchagraphofthisfunction

Example2:Mr.Ponsenhastheheatingsystem,inthisroom,turnonwhentheroomreachesaminof66oF,itheatstheroomtoamaximumtemperatureof76oFandthenturnsoffuntilitreturnstotheminimumtemperatureof66oF.Thiscyclerepeatsevery4hours.Mr.Ponsenensuresthatthisroomisatitsmaxtemperatureat9am.a)WriteacosinefunctionthatgivesthetemperatureatKing's,T,inoF,asafunctionofhhoursaftermidnight.b)Sketchagraphofthefunction.

Example3:ThetidesatCapeCapstan,NewBrunswick,changethedepthofthewaterintheharbor.OnonedayinOctober,thetideshaveahighpointofapproximately10metersat2pmandalowpointof1.2metersat8:15pm.Aparticularsailboathasadraftof2meters.Thismeansitcanonlymoveinwaterthatisatleast2metersdeep.Thecaptainofthesailboatplanstoexittheharborat6:30pm.Assuming𝑡 = 0isnoon,findtheheightofthetideat6:30pm.Isitsafeforthecaptaintoexittheharboratthistime?

Note:Horizontaldistancebetweenmaxandminpointsrepresenthalfacycle.Sincemaxandmintideare6.25hoursapart,onecyclemustbe______hours.