Unit 8 - Trigonometry

Unit 8 Trigonometry

MathIIIMrs.Valentine

*  IdentifyingCyclesandPeriods*  Aperiodicfunctionisafunctionthatrepeatsapatternofy-

values(outputs)atregularintervals.* Onecompletepattern=cycle* Horizontallengthofonepattern=period* Example:Analyzetheperiodicfunctionbelow.Identifythecycleintwodifferentways.Whatistheperiodofthefunction?

8A.1 Angles and Periodic Data

Eachcycleishighlightedinred.Notethatyoucanstartinmultipleplacestoindicateacycle,buttheperiodisthesame.Theperiodofthisfunctionis4.

*  IdentifyingPeriodicFunctions*  Analyzeeachgraphtodetermineifthefunctionisperiodic.If

itis,findtheperiod.


No

No

Yes,4

Yes,8

*  FindingAmplitudeandMidlineofaPeriodicFunction*  Theamplitudeofaperiodicfunctionmeasurestheamount

ofvariationinthefunctionvalues.* Midline:horizontallinemidwaybetweenthemaximumandminimumofthefunction* Amplitude:halfthedistancebetweentheminimumandmaximum.

* Example:Whatistheamplitudeoftheperiodicfunction?Whatistheequationofthemidline?


Amplitude=½(maximumvalue–minimumvalue)

Amplitude=½(4–(–2))=½(6)=3

Midline=½(4+(–2))y=½(2)y=1

*  UsingaPeriodicFunctiontoSolveaProblem*  Somedatacanbemodeledusingperiodicfunctions,suchas

heartbeats,thecyclesofaFerriswheel,etc.*  Example:Soundisproducedbyperiodicchangesinair

pressurecalledsoundwave.Theyellowgraphinthedigitalwavedisplayattherightshowsthegraphofapuretonefromatuningfork.Whataretheperiodandtheamplitudeofthesoundwave?


Onecycle:from0.004to0.008

Period=0.008–0.004=0.004

Amplitude=½(2.5–1.5)=½(1)=½

Theperiodofthesoundwaveis0.004s.Theamplitudeis½.

*  MeasuringAnglesinStandardPosition*  Anangleinthecoordinateplaneisinstandardpositionwhen

thevertexisattheoriginandonerayisonthepositivex-axis.*  Initialsideisonthex-axis.* Terminalsideistheotherrayoftheangle.

*  Themeasureoftheangleinstandardpositionistheinputfortwoimportantfunctions:cosineandsine.*  Themeasureoftheangleispositivewhentherotationis

counterclockwiseandnegativeintheclockwisedirection.


*  MeasuringAnglesinStandardPosition*  Examples:Whatarethemeasuresofeachangle?


Counterclockwise90°

Clockwise-90°+(-45°)=-135°

Counterclockwise180°+45°=225°

Clockwise-270°+(-45°)=-315°

*  SketchingAnglesinStandardPosition*  Whatisasketchofeachangleinstandardposition?*  36°

*  315°

*  –150°


*  IdentifyingCoterminalAngles*  Coterminalanglesaretwoanglesinstandardpositionwith

thesameterminalside.

*  Whichofthefollowinganglesisnotcoterminalwiththeotherthree:300°,–60°,60°,–420°?


300°,–60°,and–420°allhaveaterminalsideinquadrantIV(thesameterminalside)while60°hasaterminalsideinquadrantI.So60°isnotcoterminalwiththeothers.

*  FindingCosinesandSinesofAngles*  Ina360°angle,apoint1unitfromtheoriginontheterminal

raymakesonefullrotationaroundtheoriginàUNITCIRCLE

*  Anyrighttriangleformedbytheradiusoftheunitcirclehasahypotenuseof1.*  Supposeanangleinstandardpositionhasmeasureθ.* Cosineofθ(cosθ)isthex-coordinateofthepointatwhichtheterminalsideoftheangleintersectstheunitcircle.* Sineofθ(sinθ)isthey-coordinate.

8A.2 The Unit Circle and Radians

*  Example:Whatarecosθandsinθforθ=90°,θ=–180°,andθ=270°?


Cos(90°)=0Sin(90°)=1

Cos(270°)=0Sin(270°)=–1

Cos(–180°)=–1Sin(–180°)=0

*  FindingExactValuesofCosineandSine*  Youcanfindexactvaluesofsineandcosineforanglesthatare

multiplesof30°and45°.*  Example:Whatarethecosineandsineofθ=60°?


Thecosineof60°isthelengthoftheshorterlegoftherighttriangleformedusingtheradiusat60°.Thesineof60°isthelengthofthelongerleg.Recallthatina30°-60°-90°triangle:

cos(60)=½(1)=½sin(60)=√(3)*½=√(3)/2

*  Example:Whatarethecosineandsineofθ=225°?


Recallthatina45°-45°-90°triangle:

ThelegsareequaltoeachotherSincethisisinquadrantIII,bothxandyshouldbenegative.

*  UsingDimensionalAnalysis*  Centralangle–anglewithvertexatthecenter

ofacircle.*  Interceptedarc–portionofcirclebetweenthe

endpointsofacentralangle.*  Radian–measureofacentralanglethatintercept

anarcwithlengthequaltotheradiusofthecircle.* Usetoconvertbetweendegreesandradians.

* Examples:


Converttodegrees

=–135°

Convert27°toradians

*  FindingCosineandSineofaRadianMeasure

*  Whataretheexactvaluesofand?


Thiscreatesa45°-45°-90°triangleusingtheradiusasthehypotenuse.Therefore,

*  FindingtheLengthofanArc*  Thelength(s)ofaninterceptedarciss=rθ

whereristheradiusandθistheangleinradians.*  Example:Usethecirclebelow.Whatislengths

tothenearesttenth?Whatisthelengthofb?


r=3inθ=5π/6

s=rθs=3(5π/6)s=5π/2=7.9in.

r=3inθ=2π/3

b=rθb=3(2π/3)b=2π=6.3in.

*  UsingRadianMeasuretoSolveaProblem*  AweathersatelliteinacircularorbitaroundEarthcompletesone

orbitevery2h.Howfardoesthesatellitetravelin1h?


Anglefor1hoftravel:

Findthelengthofthearc:

*  TheUnitCircleinRadians


Itishighlyimportanttoknowyourunitcircle.Thisoneshowsmeasureofanglesinbothdegreesandradians,aswellasthecosineandsineofeachangle.Youwillbeexpectedtocommitthisunitcircletomemory(seehandout).

*  EstimatingSineValuesGraphically*  Thesinefunction,y=sinθ,matchesthemeasureofangleθ

ofanangleinstandardpositionwiththey-coordinateofapointontheunitcircle.*  Itismucheasiertographinradiansthanindegreesforsine

functions.

8A.3 Sine & Cosine Functions

*  Whatisareasonableestimateforeachvaluefromthegraph?Checkyourestimatewithacalculator.

sin2

sinπ


Thesinefunctionreachesismaximumatπ/2.sin2isslightlypastthat,soitisabout0.9Check:sin2=0.9092974268

Thesinefunctioncrossesthex-axisatπ,sosinπ=0Check:sinπ=0

*  FindingthePeriodofaSineCurve*  Thegraphofasinefunctioniscalledasinecurve.*  Examples:Forthegraphofy=sin4x,howmanycyclesoccur

inthewindowdescribed?

Whatistheperiodofy=sin4x?


Window:

Thereare4cycles

Dividetheintervalbythenumberofcycles.2π/4=π/2

Theperiodofy=sin4xisπ/2

*  FindingtheAmplitudeofaSineCurve*  Youcanalsovarytheamplitudeofasinewave.*  Example:Thegraphsshowy=asinx.Eachx-axisshowsvalues

from0to2π.Whatistheamplitudeofeachgraph?


a=½(3–(-3))=½6=3

a=½(0.6–(-0.6))=½1.2=0.6

*  SketchingaGraph*  Propertiesofsinefunctions:

*  Whatisthegraphofonecycleofasinecurvewithamplitude2,period4π,midliney=0,anda>0?Usingtheformy=asinbθ,whatistheequationforthesinecurve?


a=24π=2π/bb=½y=2sin½θ

*  GraphingFromaFunctionRule*  Whatisthegraphofonecycleofy=½sin2θ?

*  Whatisthegraphofonecycleofy=3sinπ/2θ?


|a|=|½|=½b=2,soitcycles2timesfrom0to2πPeriod:2π/b=2π/2=π

|a|=|3|=3b=π/2,soitcyclesπ/2timesfrom0to2πPeriod:2π/(π/2)=4

*  UsingtheSineFunctiontoModelLightWaves*  Thegraphsprovidedmodelwavesofred,blue,andyellow

light.Whatequationbestmodelseachcolorlight?


Blue:a=1Period=480=2π/bb=2π/480=π/240y=sinπ/240θ

Red:a=1Period=640=2π/bb=2π/640=π/320y=sinπ/320θ

Yellow:a=1Period=570=2π/bb=2π/570=π/285y=sinπ/285θ

*  InterpretingaGraph*  Thecosinefunction,y=cosθ,matchesθwiththex-coordinateof

thepointontheunitcirclewheretheterminalsideofangleθintersectstheunitcircle.

*  Thesymmetryofthesetofpoints(X,y)=(cosθ,sinθ)ontheunitcircleguaranteesthatthegraphsofsineandcosinearecongruenttranslationsofeachother.


*  Example:Whatarethedomain,period,range,andamplitudeofthecosinefunction?

*  Whereinthecycledothemaximumandminimumoccur?Thezeros?


ThedomainofthefunctionisallrealnumbersThefunctiongoesfromitsmaximumvalueof1toitsminimumvalueof-1andbackagaininanintervalfrom0to2π.Theperiodis2π.Themidlineisy=0.Therangeofthefunctionis-1≤y≤1Amplitude=½(max–min)=½(1–(-1))=1

Themaximumvalueoccursat0and2π.Theminimumvalueoccursatπ.Thezerosoccuratπ/2and3π/2.

*  SketchingtheGraphofaCosineFunction*  PropertiesofCosineFunctions:

*  Tographacosinefunction,locatefivepointsequallyspacedthroughonecycle.Fora>0,thisfive-pointpatternismax-zero-min-zero-max.*  Example:Sketchonecycleofy=1.5cos2θ.


|a|=|1.5|=1.5b=2,soitcycles2timesfrom0to2πPeriod:2π/b=2π/2=π

*  ModelingwithaCosineFunction*  Thewaterlevelvariesfromlowtidetohightideasshown.

Whatisacosinefunctionthatmodelsthewaterlevelininchesaboveandbelowtheaveragewaterlevel?Expressthemodelasafunctionoftimeinhourssince10:30AM.


Amplitudeis½(60)=30.Sincethetideisat-30inchesattime0,thecurvefollowsmin-zero-max-zero-minpattern,soa=–30.Thecycleishalf-waycompleteafter6hand10min,sothefullperiodis12hoursand20minutes,or12⅓hours.12⅓=2π/bb=6π/37f(t)=–30cos((6π/37)t)

*  SolvingaCosineEquation*  Youcansolveanequationbygraphingtofindanexact

locationalongasineorcosinecurve.*  Example:Supposeyouwanttofindthetimetinhourswhen

thewaterlevelfromthelastproblemisexactly10in.abovetheaveragelevelrepresentedbyf(t)=0.Whatareallthesolutionstotheequation–30cos((6π/37)t)=10intheintervalfrom0to25?


Y1=–30cos((6π/37)t)Y2=10Findthepointsofintersectiont≈3.75,8.58,16.08,and20.92

Thewaterlevelis10inaboveaverageatabout3.75h,8.58h,16.08h,and20.92hafter10:30AM

*  FindingTangentsGeometrically*  Thetangentfunctioniscloselyassociatedwithsineand

cosinebutisdifferentfromtheminthreedramaticways:1.  Thetangentfunctionhasinfinitelymanypointsof

discontinuitywithverticalasymptotesateachone2.  Range=allrealnumbers;domain=allrealnumbers

exceptoddmultiplesofπ/2.3.  Itsperiodisπ,halfthatofsineandcosine

*  Foranyangleintheunitcirclewherepoint(x,y)isthepointofintersectionbetweentheterminalsideandtheunitcircle,tangentofθ,tanθ,istheratioy/x.

8A.4 The Tangent Function

*  Example:Whatisthevalueofeachexpression?


tanπtan(-5π/6)

Anangleofπradiansinstandardpositionhasaterminalsidethatintersectsthecircleat(–1,0).tanπ=0/–1=0

Anangleof-5π/6radiansinstandardpositionhasaterminalsidethatintersectsthecircleat(–√(3)/2,-1/2).tan-5π/6=(-1/2)/(-√(3)/2)=1/√(3)=√(3)/3

*  GraphingaTangentFunction*  Anotherwaytofindthetangentofanangleistodrawthe

tangentlinetotheunitcircle,x=1,andseewheretheterminalsideofthatanglewillintersectwithitwhenextended.They-coordinateofthatpointisthetangentofthatangle.

*  Thegraphattherightshowsonecycleofthetangentfunction,y=tanθ,for–π/2<θ<π/2.


*  Propertiesoftangent:

*  Youcanuseasymptotesandthreepointstosketchonecycleofatangentcurve.*  Usethepatternasymptote-(–a)-zero-(a)-asymptote*  Theperiodcanhelptodeterminethe

positionsoftheasymptotes.


*  Example:Sketchtwocyclesofthegraphoftanπθ.


Period:π/b=π/π=1Cycle:-π/2b=-π/2π=-½π/2b=π/2π=½Asymptotesareatθ=-½,½,and3/2Divideperiodintofourthsandlocate3pointsbetweentheasymptotesforeachcycle.Points: (-¼,-1),(0,0),(¼,1)

(¾,-1),(1,0),(5/4,1)

*  UsingtheTangentFunctiontoSolveProblems*  Anarchitectisdesigningthefrontfaçadeofabuildingto

includeatriangle,similartotheoneshown.Thefunctiony=100tanθmodelstheheightofthetriangle,whereθistheangleindicated.Graphthefunctionusingthedegreemode.Whatistheheightofthetriangleifθ=16°?Ifθ=22°?


Graphthefunction.Usethetablefeaturetofindtheheights.Ifθ=16°,heightisabout28.7ft.Ifθ=22°,heightisabout40.4ft.

*  IdentifyingPhaseShifts*  Foranyfunctionf,youcangraphf(x–h)bytranslatingthe

graphoffbyhunitshorizontally.*  Eachhorizontaltranslationofcertainperiodicfunctionsisa

phaseshift.

*  f(x)+kwilltranslatethegraphoffbykunitsvertically.*  Eachverticaltranslationofcertainperiodicfunctionsisa

midlineshift.

8A.5 Translating Sine and Cosine Fns

*  Examples:Whatisthevalueofhineachtranslation?Describeeachphaseshift(useaphrasesuchas3unitstotheleft).


g(t)=f(t–2)y=cos(x+4)y=sin(x+π)

h=2Thephaseshiftis2unitstotheright

h=–4Thephaseshiftis4unitstotheleft

h=–πThephaseshiftisπunitstotheleft

*  GraphingTranslations*  Youcananalyzeatranslationtodeterminehowitrelatesto

theparentfunction.*  Example:Usethegraphoftheparentfunctiony=sinx.

Whatisthegraphofeachtranslationintheinterval0≤x≤2π?


y=sinx+3y=sin(x–π/2)

k=3Themidlineshiftis3unitsup.Thenewmidlineisy=3.

h=π/2Thephaseshiftisπ/2unitstotheright

*  GraphingaCombinedTranslation*  Youcantranslatebothverticallyandhorizontallytoproduce

combinedtranslations.*  Example:Usethegraphoftheparentfunctiony=sinx.

Whatisthegraphofthetranslationy=sin(x+π)–2intheinterval0≤x≤2π?


h=–πThephaseshiftisπunitstotheleft.k=–2Themidlineshiftis2unitsdown.Thenewmidlineisy=–2.

*  GraphingaTranslationofy=sin2x*  FamiliesofSineandCosineFunctions:

*  Example:Whatisthegraphofy=sin2(x–π/3)–3/2intheintervalfrom0to2π?


Sketchtheoriginalgraphofy=sin2x

Translateeachofthepointsπ/3totherightand3/2down.

*  WritingTranslations*  Whatisanequationthatmodelseachtranslation?


y=sinx,πunitsdowny=-cosx,2unitstothelefty=sinx,π/2unitsrightand3unitsdown

k=–πNewEquation:y=sinx–π

h=–2NewEquation:y=-cos(x+2)

h=π/2k=–3NewEquation:y=sin(x–π/2)–3

*  WritingaTrigonometricFunctiontoModelaSituation*  Thetablegivetheaveragetemperatureinyourtownxdays

afterthestartofthecalendaryear(0≤x≤365).Makeascatterplotofthedata.Whatcosinefunctionmodelstheaveragedailytemperatureasafunctionofx?


Amplitude=½(max–min)=½(77–33)=22Period=365=2π/b,sob=2π/365Phaseshift:h=198–0=198Verticalshift:k=77–22=55

y=22cos2π/365(x–198)+55

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Unit 8 - Trigonometry