1 

IntegratedMath3Module6Honors

PolarandParametricFunctionsReady,Set,Go!Homework

Solutions

Adaptedfrom

TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,

TravisLemon,JanetSutorius

©2014MathematicsVisionProject|MVPInpartnershipwiththeUtahStateOfficeofEducation

LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense.

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SDUHSDMath3Honors

Name ModelingwithFunctions 6.1HReady,Set,Go!ReadyTopic:EquationsoflinesWritetheequationoftheline(inslope‐interceptform)thatisdefinedbythegiveninformation.1. A 5, 9 B 7, 17 2. 3, 8 Q 4, 13 3. 3, 10 H 1, 11

4. L 5, 6 M 8, 8 5. 6.

x y1 16 111 3

x y1 25 59 8

Topic:KeyFeaturesofCirclesReminder:Theequationofcircleis wherethecenterhascoordinates , andtheradiushasalengthofr.Theeccentricityofaconicsectiondescribeshowmuchtheconicsectiondeviatesfrombeingcircular(i.e.how“un‐circular”theconicsectionis).Thelargerthevalueofeccentricity,thestraighterthecurvewillbe.7. Predictthevalueoftheeccentricityofacircle. 0Foreachcirclebelow,identifythecenterandradius.8. 7 3 36 9. 9 2 4 Center: , Center: , Radius:6 Radius:210. 4 1 28 Center: , Radius: √

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SDUHSDMath3Honors

SetTopic:ParametricequationsThetwogivengraphsshowthemotionofanobjectwhosepositionattimetsecondsisgivenby and .Describethemotionoftheobject.Thengraphthetwofunctionsasasinglegraphinthe ‐plane.Connectthepointstoindicatethemotionateachsecond.11.

Describethemotion:Linesfrom , to , to , to ,

12.

Describethemotion:

Linesfrom . , to , to . , to , to . ,

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SDUHSDMath3Honors

13.

Describethemotion:

Linesfrom , to , to , to , to , 14.Fillinthetableofvaluesforthepairofparametricequations 3 sin and 2 cos .Sketchthe

graphofthetwoequationsinthexy‐plane.Indicatethedirectionofthecurve.Usethegraphtowriteanequationforyasafunctionofx.

Time(t)

3 sin 2 cos ,

0 3 3 ,

1 3.84 2.54 . , .

2 3.91 1.58 . , .

3 3.14 1.01 . , .

4 2.24 1.35 . , .

5 2.04 2.28 . , .

6 2.72 2.96 . , .

7 3.66 2.75 . .

Equation:

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SDUHSDMath3Honors

GoTopic:GraphsofthetrigonometricfunctionsGraphthefunctions.15. 4 sin 2 16. 3 cos 3

17. 3 tan 18. 2 csc

Topic:CompositionofinversetrigonometricfunctionsEvaluateeachexpression.

19.sin cos 20.cos tan

0

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SDUHSDMath3Honors

Name ModelingwithFunctions 6.2HReady,Set,Go!ReadyTopic:Measuresofcentraltendency1. Findthemean,median,andmodeofthefollowingtestscores:

98,74,70,68,85,82,85,94,90,91,99,85,88,79,96,98,85,82,80,86

Mean=85.75,Median=85,Mode=852. Thegraphtotherightshows12pointswhosey‐valuesaddupto48.

The 4lineisgraphed.Usethepositionofthepointsonthegraphtoexplainwhy4istheaverageofthe12numbers.

Answersmayvary.Possibleanswer:theaverageofthey‐coordinatesofthepointsis4.

Topic:Keyfeaturesofellipses

Reminder:Theequationsofellipsesare (horizontalellipse)and (verticalellipse).Thecenterhascoordinates , ,aisthedistancefromthecentertothevertices,andbisthedistancefromthecentertotheco‐vertices.Thedistancefromthecentertothefociisrepresentedbythevariablecandcanbefoundusingtheformula .Theeccentricityofellipsescanbefoundusingtheratio .Foreachellipsebelow,identifythecenter,coordinatesofthevertices,co‐vertices,andfoci,andthevalueoftheeccentricity.

3. 1 4. 1 Center: , Center: , Vertices: , & , Vertices: , & , Co‐vertices: , & , Co‐vertices: , & , Foci: , √ Foci: √ ,

Eccentricity:√ . Eccentricity:√ .

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SDUHSDMath3Honors

5. 1 6. 1 Center: , Center: , Vertices: , & , Vertices: , & , Co‐vertices: , & , Co‐vertices: , & , Foci: √ , Foci: , & ,

Eccentricity:√ . Eccentricity: . 7. Howdoestheeccentricityofanellipsecomparetotheeccentricityofacircle? Theeccentricityofanellipseisalwaysgreaterthan0butlessthan1.Theeccentricityofacircleis

always0.SetTopic:ParametricequationsForeachsetofparametricequations,a. Createatableofx‐andy‐valuesb. Plotthepoints , inyourtableandsketchthegraph.(Indicatethedirectionofthecurve.)c. Findtherectangularequation.8.

3 32 1

0 1

1 0 3

2 3 5

3 6 7

4 9 9

5 12 11

6 15 13

RectangularEquation:

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SDUHSDMath3Honors

9.2

0 4

1 1

0 2 0

1 3 1

2 4 4

3 5 9

4 6 16

RectangularEquation:

10.

4 sin 2 2 cos 2

0 2

4 0

0

0

0 0 2

4 0

0

   0

  0 2

RectangularEquation:

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SDUHSDMath3Honors

Matchtheparametricequationswiththeirgraphs.(Youmayuseacalculator.)11. 5 cos and sin 3 12. 6 sin 4 and 4 sin 6

D A13. 4 cos and 4 sin 14. cos ⋅ sin and sin ⋅ cos

B Ca.

b.

c.

d.

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SDUHSDMath3Honors

GoTopic:Compositionoffunctions15.Given 3 2and 4 ,findthefollowing.

a. ∘ b. ∘ c. ∘ 4 d. ∘ 2

16. Istherearestrictiononthedomaininanyoftheexercises(a–d)inquestion15?Explain.

Nobecausetherearenodenominatorsorsquareroots.17.Given and 3 3,findthefollowing.

a. ∘ b. ∘ c. ∘ 10 d. ∘ 4 17

18. Istherearestrictiononthedomaininanyoftheexercises(a–d)inquestion17?Explain.

a: b:

Topic:CompositionofinversetrigonometricfunctionsEvaluateeachexpression.

19.cos cos 20.sin cos Hint:drawapicture

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SDUHSDMath3Honors

Name TrigonometricFunctions 6.3HReady,Set,Go!ReadyTopic:KeyfeaturesofparabolasReminder:Theequationsofparabolasare and wherepisthedistancefromthevertextothefocusandthedistancefromthevertextothedirectrix.Theeccentricityofaparabolaistheratiobetweenthedistancesfromthefocusandanypointontheparabolaandthesamepointandthedirectrix.Foreachparabolabelow,identifythevertex,thecoordinatesofthefocus,theequationofthedirectrix,andthevalueoftheeccentricity.1. 8 5 3 2. 20 1 3 Vertex: , Vertex: , Focus: , Focus: , Directrix: Directrix: Eccentricity:1 Eccentricity:13. 1 5 4. 16 1 2 Vertex: , Vertex: ,

Focus: , Focus: ,

Directrix: Directrix: Eccentricity:1 Eccentricity:15. Howdoestheeccentricityofparabolascomparetotheeccentricityofcirclesandellipses?

Theeccentricityofparabolasisalways1whereastheeccentricityofcirclesisalways0andtheeccentricityofellipsesisgreaterthan0butlessthan1.

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SDUHSDMath3Honors

SetTopic:Applyingparametricfunctions6. EachyearSamlooksforwardtotheringtossattheSanDiegoCountyFair.Samisnothappytohitjust

anypeg,butinsteadprideshimselfonhittingthemiddlepeginthefrontrow.SupposeSamtossestheringwithaninitialhorizontalvelocityof8feet/secondandaninitialverticalvelocityof3feet/secondandthathe’llreleasetheringwhentheringis4feetabovetheground.Samlinesupdirectlyinfrontofthemiddleringatadistanceof4feetfromtheringplatform.TheRingTossgameisshownbelow.Itsitsonashortstoolmeasuring1footinheight.Thepegsaretwoincheshighandthebottomofthefirstrowofpegsis4inchesfromthebottomofthegame.

a. Writeanequationtomodelthehorizontalpositionofthering.

b. Theverticalpositionoftheringwillbegivenby ,wheregistheaccelerationduetogravity

32 / , istheinitialvelocityand istheinitialheightoftheringwhenSamreleasesit.Writetheequationfortheverticalpositionofthering.

c. Completethetablebelowforthehorizontalandverticalpositionsofthering.

Time( ) HorizontalPosition

VerticalPosition

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

d. Sketchagraphofthering’spositionforthevalues

inyourtable.Labelyouraxescarefully.

e. Usingyourparametricequations,determinewhetherornotSamwillhithistargetpeg.Explainyourreasoning.

InordertofindoutwhetherornotSam

hitshistarget,weneedtocalculatethepositionofthetopofthetargetpeg.Thegameis4feetfromSamandthegameisontheonefootstool.Thepegmeasures2inchesandislocated4inchesfromthebottomofthegame.Thisinformationindicatesthatthetopofthepegshouldbeatthecoordinates , . .Ifwetraceonourgraphtothepointwhere ,weseethatthey‐valueis1.5when

. ,soSamhashithismarkagain!

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SDUHSDMath3Honors

7. Abaseballishitandfollowsthemotionwherexandyaremeasuredinfeetandtismeasuredinseconds.

27 9 108

a. Howhighistheballafter4seconds? 288feetb. Whendoestheballhittheground? 12secondsc. Howfaristhebaseballwhenithitstheground? 324feetd. Whenisthebaseballatitshighest? 6seconds

Topic:ConvertingbetweenrectangularandparametricformsConverteachgivenrectangularequationtoparametricformandeachsetofparametricequationstorectangularform.8. 2 3 9. 3, 5

, Answersmayvary

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SDUHSDMath3Honors

GoTopic:Solvingtrigonometricequations.Solveeachequationinthedomain , .10.2 sin 3 1 0 11.cot cos 2 cot , , , , , , 12.2 sin sin 1 0 13.sec 2 tan 4 , , , & . , . Topic:Operationsonrationalexpressions.Simplifyeachrationalexpression.

14. 15.

16. ⋅ 17.

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SDUHSDMath3Honors

Name TrigonometricFunctions 6.4HReady,Set,Go!ReadyTopic:DistanceformulaFindthedistancefromtheorigintothegivenpoint.1. 8, 6 10

2. 5, 6

√ .

3. 7, 7

√ .

Topic:KeyfeaturesofhyperbolaReminder:

Theequationsofhyperbolasare (horizontalhyperbola)and (verticalhyperbola).Thecenterhascoordinatesis , ,aisthedistancefromthecentertothevertices,andtheslopesoftheasymptotesare (horizontalhyperbolas)or (verticalhyperbolas).Thedistancefromthecentertothefociisrepresentedbythevariablecandcanbefoundusingtheformula .Theeccentricityofhyperbolascanbefoundusingtheratio .Foreachhyperbolabelow,identifythecenter,coordinatesofthevertices,slopeofasymptotes,andcoordinatesoffoci,andthevalueoftheeccentricity.

4. 1 5. 1 Center: , Center: , Vertices: , and , Vertices: , and , SlopeofAsymptotes: SlopeofAsymptotes: Foci: √ , Foci: , √

Eccentricity:√ . Eccentricity:√ .

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SDUHSDMath3Honors

6. 1 7. 1 Center: , Center: , Vertices: , and , Vertices: , and , SlopeofAsymptotes: SlopeofAsymptotes: Foci: , √ Foci: √ ,

Eccentricity:√ . Eccentricity:√ . 8. Howdoestheeccentricityofahyperbolacomparetotheeccentricityofotherconicsections?

Theeccentricityofhyperbolasisgreaterthanone,whereasallotherconicsectionshaveeccentricitylessthanone.

SetTopic:ConvertingbetweenpolarandrectangularcoordinatesThegivenpolarpointcanbedescribedusingmanydifferentpairsofpolarcoordinates.Givethreedifferentpolarcoordinatesthat,whenplotted,areequivalenttothegivenpoint.9.

PolarCoordinates:Answerswillvary,

possibleanswers: , , ,

10.

PolarCoordinates:Answerswillvary,

possibleanswers: , , ,

11.

PolarCoordinates:Answerswillvary,

possibleanswers: , , ,

12.

PolarCoordinates:Answerswillvary,

possibleanswers: , , ,

13.Findtherectangularcoordinatesthat,whenplotted,areinthesamelocationasthepointinpolarform

giveninquestion12.(Hint:Drawatriangle) √ ,

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SDUHSDMath3Honors

GoTopic:ArclengthRecalltheformulaforarclength: ,where isalwaysinradians.Writeyouranswerswithπinit.Thenuseyourcalculatortofindtheapproximatelengthofthearcto2decimalplaces.14.Findthelengthofanarcgiventhat 10inand

7.85in15.Findthearclengthgiven 4cmand

10.47cm16.Findthearclengthgiven 72.0ftand

28.27ft17.Findtheradiusgiven 3.08mm.and

0.892mm

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SDUHSDMath3Honors

Name TrigonometricFunctions 6.5HReady,Set,Go!ReadyTopic:FindingthecentralanglewhengivenarclengthandradiusFindtheradianmeasureofthecentralangleofacircleofradiusrthatinterceptsanarcoflengths.

Roundanswersto4decimalplaces.

Radius ArcLength RadianMeasure

1. 35mm 11mm 0.3143

2. 14feet 9feet 0.6429

3. 16.5m 28m 1.6970

4. 45miles 90miles 2

Topic:Determiningthetypeofconicsectionbaseduponthevalueoftheeccentricity.Determinethetypeofconicsectionrepresentedbythegivenvaluesofeccentricity.5. 0.75 6. 0 7. 2.43 8. 1 ellipse circle hyperbola parabolaSetTopic:PolarcoordinatesPlotthegivenpoint , ,inthepolarcoordinatesystem.Giveanadditionalcoordinateinpolarformthatplotsthesamepoint.

9. , 5,

Answersvary: ,

10. , 3,

Answersvary: ,

11. , 8,

Answersvary: ,

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SDUHSDMath3Honors

12. , 9,

Answersvary: ,

13. , 7,

Answersvary: ,

14. , 6,

Answersvary: ,

Sketchtheindicatedgraphs.15. 7 sin 16. 7 sin

17. 4 sin 3 18. 4 sin 3

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SDUHSDMath3Honors

GoTopic:Theareaofasector

Findtheareaofasectorofacirclehavingradiusrandcentralangle . .Makeasketch

andshadeinthesector.Roundanswersto2decimals.19. 15 ; radians

. 20. 29.2 ; radians

. 21. 32 ; radians

.

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SDUHSDMath3Honors

Name PolarandParametricFunctions 6.6HReady,Set,Go!ReadyTopic:GraphingconicsectionsGrapheachconicsection.Labelallkeyfeaturesofeachincludingcoordinatesoffoci.

1. 1

2. 16

3. 4 1

4. 1

Topic:Rewritingequationsinpolarform

Recall: , , , .

Rewriteeachequationusingpolarcoordinates , .Expressanswersinas .5. 4 6. 16 7. 4 ________________ ________________ ________________

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SDUHSDMath3Honors

SetTopic:GraphingpolarfunctionsCompletethetableandsketchthegraphofeachpolarfunction.8. 3 sin 3

0 0

3

0

0

3

0

0

3

0

2 0

Sketch:

9. 1 2 cos

0 3

√

2

1

0

√

√

0

1

2

√

2 3

Sketch:

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SDUHSDMath3Honors

10. 16 cos 2

0

√

nonrealanswer

nonrealanswer

nonrealanswer

√

√

nonrealanswer

nonrealanswer

nonrealanswer

√

2

Sketch:

11. 3

0 0

2

Sketch:

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SDUHSDMath3Honors

GoTopic:PlottingpolarcoordinatesMatcheachofthepointsinpolarcoordinateswitheitherA,B,C,orDonthegraph.

12. 2, A 13. 2, B

14. 2, C 15. 2, C

16. 2, B 17. 2, D

18. 2, A 19. 2, D

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SDUHSDMath3Honors

Name TrigonometricFunctions 6.7HReady,Set,Go!ReadyTopic:OperationsoncomplexnumbersPerformeachoperationandwritetheanswerintheform .1. 4 7 8 3 2. 2 3 5 3 7 3. 2 9 3 6 4. 5. 3 5 Topic:DescribingcenterandspreadUsingthefollowingdatasetandgraph,identifythemean,median,mode,anddescribethespreadofthedata.Thencalculatethestandarddeviation.6. Thedatasetbelowgivestheagesofthefirst46vicepresidentsoftheUnitedStateswhentheyfirsttookoffice. 53,53,45,65,68,42,42,50,56,50,52,49,66,36,51,56,45,61,57,51,65,64,57,52,42,52,53,58,48,

59,69,64,52,60,71,40,52,53,51,60,66,49,56,41,44,59

Mean:54.02,Median:53,Mode:52 Dataappearstobenormallydistributed, StandardDeviation: .

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SDUHSDMath3Honors

SetTopic:IdentifyingandgraphingconicsectionsinpolarformForeachconicsectionbelow,writetheequationintheform

or

,ifnecessary.

Identifytheeccentricityoftheconicsectionandthetypeofconicsection.Thengraphtheconicsection.7. 8.

Re‐writtenEquation: Re‐writtenEquation:

Eccentricity:1 Eccentricity: TypeofConicSection:Parabola TypeofConicSection:Ellipse

9. Re‐writtenEquation:

Eccentricity: TypeofConicSection:Hyperbola

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SDUHSDMath3Honors

GoTopic:GraphingparametricfunctionsGrapheachparametricfunction.10. 2 3

3 7 2 2 1 0 01 32 2 63 7 9

11. √ 1 2

2 2.2361 41 1.4142 30 1 21 1.4142 12 2.2361 03 3.1623 4 4.1231 5 5.099 6 6.0928

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SDUHSDMath3Honors

Topic:ConvertingequationsandcoordinatesinrectangularformtopolarformWriteeachofthefollowingequationsinpolarform.12.5 7 12 13. 3 9 Writeeachorderedpairinpolarform , .14. 6√3, 18 15. 3, 4

, , .

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SDUHSDMath3Honors

Name TrigonometricFunctions 6.8HReady,Set,Go!ReadyTopic:ThecomplexplaneRecallfromIntegratedMath1H,(Module9),thatjustasrealnumberscanberepresentedbypointsontherealnumberline,youcanrepresentacomplexnumber astheorderedpair , inacoordinateplanecalledthecomplexplane.Thehorizontalaxisiscalledtherealaxisandtheverticalaxisiscalledtheimaginaryaxis.Acomplexnumber canalsoberepresentedbyapositionvectorwithitstaillocatedatthepoint 0, 0 anditsheadlocatedatthepoint , ,asshowninthediagram.Itwillbeusefultobeabletomovebackandforthbetweenbothgeometricrepresentationsofacomplexnumberinthecomplexplane‐sometimesrepresentingthecomplexnumberasasinglepointandsometimesasavector.Inthediagram,3complexnumbershavebeengraphedasvectors.Rewriteeachcomplexnumberasapointintheform , .1. 3 4 graphsas(___ __,__4___)2. 5 2 graphsas(___5__,__2___)3. 1 6 graphsas(__1___,___ __)Onthediagrambelow,graphthefollowingcomplexnumbersasvectors. 4. 5 3  5. 2 4 6. 6 7. 2

30 

SDUHSDMath3Honors

Topic:TheModulusofacomplexnumberWecancomparetherelativemagnitudesofcomplexnumbersbydetermininghowfartheylieawayfromtheorigininthecomplexplane.Werefertothemagnitudeofacomplexnumberasitsmodulusandsymbolizethislengthwiththenotation| |where| | √ .8. Findthemodulusofeachofthecomplexnumberslistedbelow. a. 3 4 b. 5 2 c. 1 6 d. 5 3

5 √ √ √ e. 2 4 f. 6 g. 2

√ √ √ SetTopic:TherelationshipbetweenpolarcoordinatesandrectangularcoordinatesCoordinateConversion:Thepolarcoordinates , arerelatedtotherectangularcoordinates , asfollows:

Convertthepointsfrompolartorectangularcoordinates.

9. 4,

,

10. √3,

, √

11. 2, ,

12. 5√2,

,

Convertthepointsfromrectangulartopolarcoordinates.13. 3, 3

√ ,

14. 6, 0 ,

15. 1, √3

,

16. 4√3, 4

,

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SDUHSDMath3Honors

Topic:PolarformofacomplexnumberConsiderthecomplexnumber .Byletting beanangleinstandardformassooninthediagramattheright,youcanwrite cos and

sin ,where √ .Byreplacingaandb,youhavecos sin .Wecanfactoroutther,obtainingthe

polarformofacomplexnumber.If thenthepolarformis .Writethecomplexnumbersinpolarform 17. 3

√ . ⋅ .

18.3 3

√ ⋅

19. 4 4

√ . ⋅ . 20.√3

⋅

Writethecomplexnumberinstandardform, .21.3 cos 120° sin 120°

√

22.5 cos 135° sin 135°

√ √

23.√8 cos sin

24.8 cos sin

GoTopic:ThearithmeticofcomplexnumbersPerformtheindicatedoperation.Writeyouranswersinstandardform, .25. 4 7 12 2

26. 11 8 4 3

27. 10 6 16 3

28. 7 9

29. 1 4 2

30. 5 6 5 6