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Trigonometry
PolarGraphs
Typically,PolarGraphswillbeplottedonpolargraphssuchastheone
illustratedatright.Onthisgraph,apointሺ‫ݎ‬ǡ ߠሻcanbeconsideredtobethe
intersectionofthecircleofradius‫ݎ‬andtheterminalsideoftheangleߠ(see
theillustrationbelow).
PartsofthePolarGraph
గ
Theillustrationbelowshowsthekeypartsofapolargraph,alongwithapoint,ሺͶǡ ሻ.
ଷ
ThePoleisthepointሺͲǡ Ͳሻ(i.e.,theorigin).
ThePolarAxisisthe‫Ͳݔ‬axis.
TheLine:ߠ ൌ గisthe‫Ͳݕ‬axis.
ଶ
Manyequationsthatcontainthecosine
functionaresymmetricaboutthePolarAxis.
Manyequationsthatcontainthesine
functionaresymmetricaboutthelineߠ ൌ గ.
ଶ
PolarEquations–Symmetry
Followingarethethreemaintypesofsymmetryexhibitedinmanypolarequationgraphs:
Symmetryabout:
QuadrantsContainingSymmetry
SymmetryTest(1)
ThePole
Opposite(andorand)
ThePolarAxis
Leftorrighthemispheres(and
ReplaceߠwithȂ ߠintheequation
‘”and)
࣊
TheLineࣂ ൌ ૛
(1)
Upperorlowerhemispheres(and
orand)
Replace‫ݎ‬withȂ ‫ݎ‬intheequation
Replaceሺ‫ݎ‬ǡ ߠሻwithሺെ‫ݎ‬ǡ െߠሻinthe
equation
Ifperformingtheindicatedreplacementresultsinanequivalentequation,theequationpasses
thesymmetrytestandtheindicatedsymmetryexists.Iftheequationfailsthesymmetrytest,
symmetrymayormaynotexist.
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Trigonometry
GraphsofPolarEquations
GraphingMethods
Method1:Pointplotting
x CreateatwoͲcolumnchartthatcalculatesvaluesof‫ݎ‬forselectedvaluesofߠ.Thisisakintoa
twoͲcolumnchartthatcalculatesvaluesof‫ݕ‬forselectedvaluesof‫ݔ‬thatcanbeusedtoplota
rectangularcoordinatesequation(e.g.,‫ ݕ‬ൌ ‫ ݔ‬ଶ െ Ͷ‫ ݔ‬൅ ͵).
x TheߠͲvaluesyouselectforpurposesofpointplottingshouldvarydependingontheequation
youareworkingwith(inparticular,thecoefficientofߠintheequation).However,asafebet
istostartwithmultiplesofߨൗ͸(includingߠ ൌ Ͳ).Ploteachpointonthepolargraphand
seewhatshapeemerges.Ifyouneedmoreorfewerpointstoseewhatcurveisemerging,
adjustasyougo.
x Ifyouknowanythingaboutthecurve(typicalshape,symmetry,etc.),useittofacilitate
plottingpoints.
x Connectthepointswithasmoothcurve.Admiretheresult;manyofthesecurvesare
aestheticallypleasing.
Method2:Calculator
UsingaTIͲ84PlusCalculatororitsequivalent,dothefollowing:
x Makesureyourcalculatorissettoradiansandpolarfunctions.HittheMODE
key;selectRADIANSinrow4andPOLARinrow5.Afteryoudothis,hitting
CLEARwillgetyoubacktothemainscreen.
x HitY=andentertheequationintheform‫ ݎ‬ൌ ݂ሺߠሻ.UsetheX,T,ી,nkeyto
enterɅintotheequation.Ifyourequationisoftheform‫ ݎ‬ଶ ൌ ݂ሺߠሻ,youmay
needtoentertwofunctions,‫ ݎ‬ൌ ඥ݂ሺߠሻand‫ ݎ‬ൌ െඥ݂ሺߠሻ,andplotboth.
x HitGRAPHtoplotthefunctionorfunctionsyouenteredinthepreviousstep.
x Ifnecessary,hitWINDOWtoadjusttheparametersoftheplot.
o Ifyoucannotseethewholefunction,adjusttheXͲandYͲvariables(oruseZOOM).
o Ifthecurveisnotsmooth,reducethevalueoftheીstepvariable.Thiswillplotmore
pointsonthescreen.Notethatsmallervaluesofીsteprequiremoretimetoplotthe
curve,sochooseavaluethatplotsthecurvewellinareasonableamountoftime.
o Iftheentirecurveisnotplotted,adjustthevaluesoftheીminandીmaxvariablesuntil
youseewhatappearstobetheentireplot.
Note:Youcanviewthetableofpointsusedtographthepolarfunctionbyhitting2ND–TABLE.
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Trigonometry
GraphTypes(PolarEquations)
Circle
Equation:‫ ݎ‬ൌ ܽ •‹ ߠ
Equation:‫ ݎ‬ൌ ܽ …‘• ߠ Equation:‫ ݎ‬ൌ ܽ
Location:
Location:
abovePolarAxisifܽ ൐ Ͳ rightoflineߠ ൌ ߨȀʹifܽ ൐ Ͳ
belowPolarAxisifܽ ൏ Ͳ leftoflineߠ ൌ ߨȀʹifܽ ൏ Ͳ
Location:
CenteredonthePole
Radius:ܽȀʹ Symmetry:Lineߠ ൌ ߨȀʹ
Radius:ܽȀʹ
Radius:ܽ
Symmetry:PolarAxis Symmetry:Pole,PolarAxis,
Lineߠ ൌ ߨȀʹ
Rose
Characteristicsofroses:
x Equation:‫ ݎ‬ൌ ܽ •‹ ݊ߠ
o Symmetricaboutthelineߠ ൌ ߨȀʹ(‫Ͳݕ‬axis)
x Equation:‫ ݎ‬ൌ ܽ …‘• ݊ߠ
o SymmetricaboutthePolarAxis(‫Ͳݔ‬axis)
x Containedwithinacircleofradius‫ ݎ‬ൌ ܽ
x If݊isodd,therosehas݊petals.
x If݊iseventherosehasʹ݊petals.
x Notethatacircleisarosewithonepetal(i.e,݊ ൌ ͳ).
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Trigonometry
GraphsofPolarEquations
LimaçonofPascal
Equation:‫ ݎ‬ൌ ܽ ൅ ܾ •‹ ߠ
Equation:‫ ݎ‬ൌ ܽ ൅ ܾ …‘• ߠ Location:bulbabovePolarAxisifܾ ൐ Ͳ
bulbbelowPolarAxisifܾ ൏ Ͳ
Location:bulbrightofLineߠ ൌ ߨȀʹifܾ ൐ Ͳ
bulbleftofLineߠ ൌ ߨȀʹifܾ ൏ Ͳ
Symmetry:Lineߠ ൌ ߨȀʹ
Symmetry:PolarAxis
FourLimaçonShapes
ܽ ൏ ܾ
ܽ ൌ ܾ ܾ ൏ ܽ ൏ ʹܾ
ܽ ൒ ʹܾ
Innerloop
“Cardioid”
Dimple
Nodimple
FourLimaçonOrientations(usingtheCardioidasanexample)
sinefunction
sinefunction
cosinefunction cosinefunction
ܾ ൐ Ͳ
ܾ ൏ Ͳ ܾ ൐ Ͳ
ܾ ൏ Ͳ
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Trigonometry
GraphingPolarEquations–TheRose
Example:‫ ܚ‬ൌ ૝ ‫ ܖܑܛ‬૛ી
Thisfunctionisarose.Considertheforms‫ ݎ‬ൌ ܽ •‹ ܾɅand‫ ݎ‬ൌ ܽ …‘• ܾɅ.
Thenumberofpetalsontherosedependsonthevalueofܾ.
x Ifܾisaneveninteger,therosewillhaveʹܾpetals.
x Ifܾisanoddinteger,itwillhaveܾpetals.
Let’screateatableofvaluesandgraphtheequation:
‫ ܚ‬ൌ ૝ ‫ ܖܑܛ‬૛ࣂ
ࣂ
࢘
ࣂ
࢘
Ͳ
Ͳ
ߨȀͳʹ
ʹ
͹ɎȀͳʹ
െʹ
ߨȀ͸
͵ǤͶ͸Ͷ
ʹɎȀ͵
െ͵ǤͶ͸Ͷ
ߨȀͶ
4
͵ɎȀͶ
Ͳ4
ߨȀ͵
͵ǤͶ͸Ͷ
ͷߨȀ͸
െ͵ǤͶ͸Ͷ
ͷߨȀͳʹ
ʹ
ͳͳߨȀͳʹ
െʹ
ߨȀʹ
Ͳ
ߨ
Ͳ
Becausethisfunctioninvolvesan
argumentofʹɅ,wewanttostartby
lookingatvaluesofɅinሾͲǡ ʹߨሿ ൊ
ʹ ൌ ሾͲǡ ߨሿǤYoucouldplotmore
points,butthisintervalissufficient
toestablishthenatureofthecurve;
soyoucangraphtheresteasily.
Oncesymmetryis
established,thesevalues
areeasilydetermined.
Thevaluesinthetable
generatethepointsinthe
twopetalsrightofthe‫Ͳݕ‬axis.
Knowingthatthecurveisa
roseallowsustographthe
othertwopetalswithout
calculatingmorepoints.
Bluepointsonthegraph
correspondtobluevalues
inthetable.
Orangepointsonthe
graphcorrespondto
orangevaluesinthetable.
ThefourRoseforms:
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Trigonometry
GraphingPolarEquations–TheCardioid
Example:‫ ܚ‬ൌ ૛ ൅ ૛ ‫ ܖܑܛ‬ી
Thiscardioidisalsoalimaçonofform‫ ݎ‬ൌ ܽ ൅ ܾ •‹ ߠwithܽ ൌ ܾ.Theuseofthesinefunction
indicatesthatthelargeloopwillbesymmetricaboutthe‫Ͳݕ‬axis.The൅signindicatesthatthelarge
loopwillbeabovethe‫Ͳݔ‬axis.Let’screateatableofvaluesandgraphtheequation:
‫ ܚ‬ൌ ૛ ൅ ૛ ‫ࣂ ܖܑܛ‬
ࣂ
࢘
ࣂ
࢘
Ͳ
ʹ
ߨȀ͸
͵
͹ɎȀ͸
ͳ
ߨȀ͵
͵Ǥ͹͵ʹ
ͶɎȀ͵
ͲǤʹ͸ͺ
ߨȀʹ
4
͵ɎȀʹ
Ͳ
ʹߨȀ͵
͵Ǥ͹͵ʹ
ͷߨȀ͵
ͲǤʹ͸ͺ
ͷߨȀ͸
͵
ͳͳߨȀ͸
ͳ
ߨ
ʹ
ʹߨ
2
Generally,youwanttolookat
valuesofߠinሾͲǡ ʹߨሿ.However,
somefunctionsrequirelarger
intervals.Thesizeoftheinterval
dependslargelyonthenatureofthe
functionandthecoefficientofߠǤ
Oncesymmetryis
established,thesevalues
areeasilydetermined.
Theportionofthegraph
abovethexͲaxisresults
fromߠinQ1andQ2,
wherethesinefunctionis
positive.
Similarly,theportionof
thegraphbelowthexͲaxis
resultsfromߠinQ3and
Q4,wherethesine
functionisnegative.
Bluepointsonthegraph
correspondtobluevalues
inthetable.
Orangepointsonthe
graphcorrespondto
orangevaluesinthetable.
ThefourCardioidforms:
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