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LESSON1:TrigonometryPre-testInstructions.Answereachquestiontothebestofyourability.Ifthereismorethanoneanswer,putboth/allanswersdown.Trytoanswereachquestion,butifthereisaquestionyoudonotknowanythingabout,itisokaytowritethatyoudon’tknow.
1. Whatissin(𝑥)?
2. Whatiscos(𝑥)?
3. Whatistan(𝑥)?
4. Whatisthepurposeofthetrigonometricfunctions?Inotherwords,whattypesofproblemscantheyhelpyousolve?
5. Dosin(𝑥)andcos(𝑥)haveanyrelationship?
6. Howdoessin 𝑥 changeasxgoesfrom0to90degrees?
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7. Are𝑦 = sin 𝑥 and𝑦 = cos(𝑥)functions?Ifso,whatistheirdomainandrange?
8. Theheightofabuilding’sshadowis56ftwhenthesunisshiningata35˚angletothehorizon.Whatistheheightofthebuilding?Explainhowyoufoundyouranswer. h
35˚56ft
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Notestotheteacher:Donottrytopreparestudentsforthepre-test.Itisokayiftheydon’tknowmuch.Thisisjusttogetabaselinetohelpusknowwhattheyunderstandgoingintothiscurriculum.TheremainderofthedaywillbespentreviewingSOHCAHTOAandwhattheyalreadyshouldknowabouttrigonometry,aswellasintroducingthebasicsofgeometer’ssketchpad.ThingstoknowaboutGeoGebra:Onthetoolbaronthetop,youcanconstructsegments,circles,points,polygons,perpendicularbisectors,intersections,andtakemeasurements.Eachtoolhasadrop-downmenutododifferent,butsimilaractions.StudentsshouldspendsometimeexploringandgettingcomfortablewithGeoGebrasothattheycanhitthegroundrunningtomorrow.Makesurewhentheyconstructtheirshapesthattheynotonlylookright,butalsothattheycandragandmovetheapplicablepointsintheappropriatemanner.Sometimestheycanmakethingslookright,buttheyhaveconstructedthemincorrectly,andwhentheydragthepointaround,youwillbeabletotell.Studentsshouldconstructthefollowing:
• apoint• asegment• aline• aray• atriangle• perpendicularlines• theintersectionpointoftwolines• parallellines• acircle
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LESSON2:Thegoalofthefirstactivityistocreatearighttrianglewithhypotenuse1inGeoGebra,wherewecanmovethetrianglearoundanywaywewant,andfindwherethetrigonometricfunctionsarelocatedonthattriangle.
1. OpenanewGeoGebrageometrywindow.2. Usethesegmenttooltocreateasegment.Thenusethemeasuringtooltomeasurethat
segment.Clickonthemeasuringtool,selectdistance,andthenclickonthesegmentyouwanttomeasure.Adjustthemeasurementuntilitis1cm.Trytomakethissegmentashorizontalaspossible.
3. Tip:Makesureyouselecttheselectiontool(lookslikeapointer)inordertoadjustthemeasurement.
4. Thenusethecircletooltodrawacircleusingthissegmentastheradius.Atthistime,youmightwanttozoominsomewhat.
5. Constructanadditionalradiusatsomethinglessthana90°anglecounterclockwisefromtheradiusyoualreadyhave.
6. Constructaperpendicularbetweentheendpointofyournewradiusandyourhorizontalsegment.Clickontheperpendicularlinetoolandtheclicktheendpointofthenewradiusaswellasthehorizontalsegment.
7. Thenconstructtheintersectionpointoftheperpendicularlineandthehorizontalline.Clickonthepointtoolandthenhoverovertheintersectionsothatbothlinesarehighlighted.Atthispoint,click,andthenewpointwillbeconstructed.
8. Highlighttheperpendicularline,andhideitbygoingtoEdit,andthen“Show/hideobjects.”
9. Drawasegmentwheretheperpendicularlinewas,betweenthepointthatwasonthecircumferenceofthecircleandthepointthatwasfoundusingtheperpendicularline.
10. Drawasegmentthatgoesalongthebaseofthetriangle,justuptowheretheperpendicularlineintersectedthex-axis.
11. Highlightthecircle,rightclick,andselect“Hidecircle.”Highlightthelinesegmentthatishorizontalthatislongerthanthetriangle,rightclick,andselect,“Hidesegment.”
12. Nowyouhavearighttrianglewithradius1,andasyouchangetheangle(throughthefirstquadrant,therighttriangleisalwaysarighttriangle,andthehypotenuseisalways1.
13. Highlightthepointsofthetriangle,beginningwiththeonethatisintheorigin,andcontinuinginacounter-clockwisefashion.GotoEdit,then“Show/HideLabels”andthenbeginwithA,thenB,thenC.Then,clickonthefourthpoint,notinthetriangle,andlabelitD.Itisimportantthatwealllabelourpointsthesametoavoidconfusioninthefuture.Thisiswhatyourdocumentshouldlooklikeatthispoint.
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14. Let’scallthecentralanglex(thisisangleCAB).
15. Whatissin(x)?UsingSOHCAHTOA,whatwouldtheratiobe?Knowingthatthehypotenuseis1,whatdoesthattellyou?
16. Forwhichlinesegmentisthelengthequaltosin(x)?Usingthemeasurementtool,measurethatlinesegment.Ifthemeasurementcomesupinaninconvenientlocation,youcanmoveit,butfirstyoumustclickbacktotheselectiontool.
17. Whatiscos(x)?UsingSOHCAHTOA,whatwouldtheratiobe?Knowingthatthehypotenuseis1,whatdoesthattellyou?
18. Forwhichlinesegmentisthelengthequaltocos(x)?Usingthemeasurementtool,measurethatlinesegment.
19. Movethecentralanglesothatxisverycloseto0°.Whatissin(x)approximately?
20. Movethecentralanglesothatxiscloseto45°.Whatissin(x)approximately?
21. Movethecentralanglesothatxiscloseto90°.Whatissin(x)approximately?
22. Movethecentralanglesothatxisverycloseto0°.Whatiscos(x)approximately?
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23. Movethecentralanglesothatxiscloseto45°.Whatiscos(x)approximately?
24. Movethecentralanglesothatxiscloseto90°.Whatiscos(x)approximately?
25. Usethemeasurementtooltomeasureanglex.Selectangleinthemeasurementtool.ThenselectsegmentABandthenAC(inthatorder).Thiswillmeasureanglex.Remember,usingtheselectiontool,youcanmovethemeasurementifitshowsupinaninconvenientlocation.
26. Thisgivesusameasurementoftheheightofthetriangleinourcoordinateplane.Rememberthatthisheightofthetriangleisequaltosin(x)aswesawearlier.MovepointCandseehowthismeasurementchanges.Writedownapatternthatyousee.
27. Idon’twantyoutotakemywordforit.Putyourcalculatorintodegreemode,andtypesin(x),usingwhatevertheangleisforxintoyourcalculator.Findsin(x).Issin(x)equaltotheheightofthetriangle?
28. Movethecentralanglesothatxisverycloseto0°.Useyourcalculatortofindsin(x)
exactly.Howdoesthiscomparetothemeasurementofthesegmentthatisequaltosin(x)?
29. Movethecentralanglesothatxiscloseto45°.Whatissin(x)exactly?HowdoeswhatyourcalculatorsayscomparetothemeasurementfromGeoGebra?
30. Movethecentralanglesothatxiscloseto90°.Whatissin(x)exactly?HowdoeswhatyourcalculatorsayscomparetothemeasurementfromGeoGebra?
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31. Next,movetheangletoananglearound60.Calculatecos(x)usingyourcalculator.
Iscos(x)equaltothelengthofthetriangle?
32. Movethecentralanglesothatxisverycloseto0°.Whatiscos(x)exactly?HowdoeswhatyourcalculatorsayscomparetothemeasurementfromGeoGebra?
33. Movethecentralanglesothatxiscloseto45°.Whatiscos(x)exactly?HowdoeswhatyourcalculatorsayscomparetothemeasurementfromGeoGebra?
34. Movethecentralanglesothatxiscloseto90°.Whatiscos(x)exactly?HowdoeswhatyourcalculatorsayscomparetothemeasurementfromGeoGebra?
35. Now,gotoEditandthen“Show/HideObjects.”HighlighttheperpendicularlinethroughCBandhideitagain.
36. HighlightADandthepointD,andgottoConstruct“PerpendicularLine.”Doyouknowwhatthislineiscalledwithrespecttothecircle?
37. Asyoumayhaveremembered,thatlineiscalledatangentline.38. ConstructrayAC(todothis,youcanusethesegmenttoolonthetoolbar,butholdit
downuntilyoucanselecttheraytool).ThenconstructtheintersectionpointofthetangentlineandrayACandlabeltheintersectionpointE.
39. Atthispoint,yourdocumentshouldlooklikethis.(Notethatyoucandragyourlabelssothattheyarenotcoveredupbyyourlines.)
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40. Next,hidethetangentlineandtherayAC,andaftertheyarehidden,drawsegmentsAEandDE.
41. TheAncientGreekswerethefirsttodiscovertrigonometry,andtheyconsideredsegmentDEtobethetangentofanglex.Knowingthattangentisequaltoopposite/adjacentintriangleABC,canyoushowthatsegmentDEisequaltotan(x)usingthepropertiesofsimilartriangles?
42. Usingsimilartriangles,findtherelationshipbetweensin(x),cos(x),andtan(x).
43. AnotherwaytolookatthesegmentDEisbylookingattriangleADE,andconsiderfindingthetan(x).Whatisthelengthoftheadjacentsegmentinthattriangle?Whatdoesthatmaketan(x)equalto?
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44. Movethecentralanglesothatxisverycloseto0°.Whatistan(x)approximately?
45. Movethecentralanglesothatxiscloseto45°.Whatistan(x)approximately?MeasurethesegmentDE.
46. Movethecentralanglesothatxiscloseto90°.Whatistan(x)?
47. Nowuseyourcalculatortofindavaluefortan(x).Howdotheycompare?48. Movethecentralanglesothatxisverycloseto0°.Findtan(x)exactlyinyour
calculator?Howdotheycompare?
49. Movethecentralanglesothatxiscloseto45°.WhatisthemeasurementofsegmentDE?Whatistan(x)inyourcalculator?Howdotheycompare?
50. Movethecentralanglesothatxiscloseto90°.WhatisthemeasurementofsegmentDE?Whatistan(x)inyourcalculator?Howdotheycompare?
51. Saveyourdocument.Wewillkeepusingthisaswecontinuetoexplorethetrigonometricfunctions.
HOMEWORK:Gohomeandaskyourparentsorgrandparentshowtheydidtrigonometry.Didtheyusetrigonometrytables?Ifso,dotheyrememberwhattheywereandhowtheyworked?
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LESSON3:Thegoalofthisactivityistocreateatrigonometrytableforsin(x).Introduction:Ifyouaskedyourparentsorgrandparentsforhomeworklastnight,whentheysolvedtrigonometricequations,theyprobablyusedatableofvalues,ratherthanacalculator.Infact,whenacalculatorgivesavalueforatrigonometricequation,itisusingatableofvaluesthathasbeenenteredintoitsharddiskmemory,inthesamewaythatyourcalculatorhasamemoryofthevalueofπtoacertainnumberofdigits.Yourcalculatordoesnotknowthatπistheratioofacircle’scircumferencetoitsdiameter,andneitherdoesitknowanythingabouttrigonometry,butrather,itsimplyhasatrigonometrytableinitsmemory.
Inancienttimes,trigonometrytableswerecreatedbydrawingalargeandextremelyprecisecircle,andmeasuringthelengthsofthesegmentsofsineatdifferentangles.Itwasextremelytime-consuming,difficult,andtedious.
WearegoingtoworktogetherandusethemeasurementtoolofGeoGebratocreateatrigonometrytableofourown.TeacherNote:Assigneachstudentorpairofstudentstoeachwhole-numberdegreevaluebetween0˚and90˚,sothatintheend,eachvalueiscoveredtwice.Inyourfinaltrigonometrytable,iftwovaluesdisagreeslightlybecausestudentsusedslightlydifferentapproximationsofanangle,averagethemforthefinaltable.Iftwovaluesdisagreesignificantly,investigatewhetheronestudentmayhaveanerror.YoumaywanttosetupasharedGooglespreadsheetsothatstudentscanputtheirvaluesintothespreadsheet,whichcanautomaticallyaveragethevalues.Keepinmind,youwillneedtolookoverthestudentvaluestobesuretherearen’terrors.
1. InyourGeoGebradocument,gotoOptions,thenRounding,andchangeto4decimalplaces.
2. Zoominasmuchaspossible.3. Yourteacherwillassignyouseveralwhole-numberanglemeasurements.
MovepointCsothatxisasclosetoeachanglemeasurementaspossible.Ifyoucan’tgetitexactly,getascloseaspossible,andrecordthemeasurementGeoGebragivesyouforthelengthof𝐶𝐵.
4. Allstudentsshouldputtheirmeasurementsupontheboard,averageswillbecalculated,andeveryonewillrecordthefinaltrigtableontheirownpaper.
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Angle(deg)
Student1 Student2 Sin(x)Avg
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Angle(deg)
Student1 Student2 Sin(x)Avg
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
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LESSON4:Thegoalofthisactivityistouseyourtrigonometrytabletosolvedifferentproblemsinvolvingtrigonometry.Notethatbecausetrigtablesaresodifficulttocreate,youhavetobecreativewithhowyouusethemtomakethemostofthevaluestheygiveyou.Ifavalueyouarelookingforisnotgiventoyoudirectlybyyourtrigtable,lookatyourGeoGebradocumentorgotohttps://www.geogebra.org/m/Cb2jWUPS,andseeifyoucanfigureoutavaluethatwouldbethesamethatisinyourtrigtable.Teachernote:Fornumbers1c-eand4c-e,studentswilllikelyneedhelp.Additionally,sinceGeoGebrainsomecasesistakingameasurementofdistance,itwon’tmeasurewhenthevaluewouldbenegative,whilethecalculatorandcommonsensemighttellusthatweoughttohaveanegativeanswer.Thesewillbegoodopportunitiesfordiscussion.ThisGeoGebradocument:https://www.geogebra.org/m/Cb2jWUPSshowsasimilarpicturetowhatstudentshavecreatedontheirown,butwiththetrigonometricfunctionsallappearingontheunitcircle.Itmaybeadditionallyhelpful.
1. Usingyourtrigtable,findthefollowing: Checkusingcalculator:
a. sin(47˚)
b. sin(28˚)
c. sin(150˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)
d. sin(97˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)
e. sin(-22˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)
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2. Usingyourtrigtable,solveforxinthefollowingtriangles:
a.
Checkusingcalculator:
b.
4.3cm
5.6cm Checkusingcalculator:Sidenote:Wealreadytalkedaboutthenameoftangent.ThewordtheAncientGreeksusedforsinemeantchord,becausesineisequaltohalfofthechord,butwhenitwastranslatedfromGreektoArabictoLatin,therewasamis-translationthatledtothewordsinusinLatin,whichmeans“inlet”beingused,ratherthanthewordthatmeanschord.
x
17.5in4.6in x
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Thewordcosinemeans“sineofthecomplement”becausethecosineisactuallythesegmentthatisequaltothesineof(90-x)˚.Inthediagrambelow,wethinkofcosineasOM,buttheAncientGreeksthoughtofitasNP(whichiscongruenttoOM). Rememberthatcomplementaryanglesaddto90˚.Sinehasthesamerelationshiptotheoriginalangleascosinedoestothecomplementaryangle(90-x)˚.Tryturningthecirclesidewaystoseethisbetter.
3. Eventhoughourtrigonometrytabledoesnotincludecosine,howcanweusethisinformationtofindcos(x)usingourtrigtable?
sine
cosine
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4. Usingyourtrigtable,findthefollowing: Checkusingcalculator:
a. cos(7˚)
b. cos(88˚)
c. cos(135˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)
d. cos(111˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)
e. cos(-59˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)
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5. Usingyourtrigtable,solveforxinthefollowingtriangle:
Checkusingcalculator:
6. Nowthatyouknowhowtofindsin(x)andcos(x)usingyourtrigonometrytable,howcanyoufindvaluesfortan(x)usingthetrigonometrytable?
7. Findthefollowingusingyourtrigtable:a. tan(32˚)
Checkusingcalculator:
1.5in
0.6in
x
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b. tan(83˚)
Checkusingcalculator:
8. Whenyouhaveusedyourtrigtableandcheckedusingyourcalculator,howclosehaveyourcalculationsbeen?Whenhavetheybeenoff,andbyhowmuch?Whatdoyouattributethisdifferenceto?Isthisalotoferror,orjustalittlebit?
9. Iamsureyouwillbehappytogobacktousingyourcalculatorafterthis,butwhathaveyoulearnedfromcreatingandusingthistrigonometrytable?
Asyoucontinuetouseyourcalculatortosolveatrigonometricequation,trytorememberwhatishappeninginsideofyourcalculator.
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LESSON5:Thegoalofthisactivityistodefinethreenewtrigonometricfunctions,anduseGeoGebratodrawthegraphsofthefunctionsastheanglexmoves.
1. Secantisatrigonometricfunctionthatislesscommonlyusedthansine,cosineandtangent.Youmayormaynothaveheardofitbefore.TheAncientGreeksconsideredsec(x)tobethesegmentAE.(Note:Ingeometry,asecantlinereferstoalinethatintersectsacircleintwoplaces.IfyouextendthesegmentAEthroughbothsidesofthecircle,itwouldbeasecantline.)
2. UsingthePerpendicularlinetool,constructalineperpendiculartoABthroughA.Dothisbyclickingthetool,thenclicksegmentAB,thenclickpointA.Then,usethepointtooltoconstructtheintersectionpointbetweenthecircleandtheperpendicularlineyoujustconstructed.LabelthispointF.
3. NextselecttheverticallineyoujustcreatedandpointF,andconstructalineperpendiculartotheverticallinethatgoesthroughpointF.
4. Then,drawrayAE,andconstructtheintersectionbetweenthemostrecentperpendicularlineyoucreatedandrayAE.LabelthisintersectionpointG,andthenhidethetwoperpendicularlinesandrayAE.
5. Finally,constructsegmentAGandFG.Yourdocumentshouldlooklikethis.
6. NowconsiderthesegmentFG.Thisisoneofthetrigonometricfunctionsofthecomplementaryangle.Whichtrigonometricfunctionisit?
7. Sincethesineofthecomplementiscalledcosine,whatdoyouthinkFGshouldbecalled?
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8. NowconsiderthesegmentAG.Thisisoneofthetrigonometricfunctionsofthe
complementaryangle.Whichtrigonometricfunctionisit?
9. Sincethesineofthecomplementiscalledcosine,whatdoyouthinkAGshouldbecalled?
Bignewidea:Untilthispoint,wehavebeenusingdegreestomeasuretheanglex,butnowwearegoingtoswitchandusesomethingcalledradians.Aradianmeasuresananglebyhowmanyradiusdistancesthearcoftheanglepassesthrough.Sincethedistancearoundawholecircleis2π(lengthoftheradius),360˚=2π≈6.28radians.Thismeansthat180˚=π≈3.14radians,and90˚=π/2≈1.57radians.
10. NowwearegoingtoconvertourGeoGebradocumenttoradians.GotoOptions“Advanced”inAngleUnits,changeitto“radians.”
11. Lookatyourmeasurements.Draganglexaroundthecircleandnoticewhattheradianmeasureisatdifferentlocations.Doesthismakesenseusingtheconversionsabove?
12. Ifyoueverwanttoreferbacktothisdocument,hereiswebpagewithasimilardocumentthathassomeadditionalfeatures:https://www.geogebra.org/m/Cb2jWUPS.Itwillbehandyforfuturereference.
13. Next,wearegoingtographthetrigonometricfunctions.Inordertocreateagraph,youaregoingtouseaGeoGebraworksheetthatisverysimilartowhatyoucreated,buthassomeextrafeatures.
14. Gotothefollowinglink:https://www.geogebra.org/m/G9mjcC7D.15. Tographsin(x),checktheboxmarkedsine,andmovethepointaroundthecircleto
changethedegreemeasurementofx(herecalleda).16. Youcanalsocreatethegraphbyclicking“StartAnimation.”.Youcanclick“EraseTraces”
ifyouwanttostartoverandmakeanewgraph.
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17. Usethegraphofsin(x)toanswerthefollowingquestions:a. Whatisthedomainofsin(x)?Arethereanglesbeyondwhatisshownin
ourgraph?Aretheseacceptableanglesforthedomain?
b. Whatistherangeofsin(x)?
c. Whereissin(x)positive,andwhereisitnegative?
18. Nowgraphcos(x),afterdeletingthetracesofsin(x).Tographcos(x),unchecktheboxthatsayssineandchecktheboxthatsayscosine.Eitheranimateormovethepointtographcos(x).Whatsegmentonthecirclecorrespondswiththeheightofthegraphinthiscase?
a. Whatistherangeofcos(x)?
b. Whereiscos(x)positive,andwhereisitnegative?
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19. Gothefollowingworksheethttps://www.geogebra.org/m/shAV7BSBandgraphtan(x).Whatsegmentonthecirclecorrespondswiththeheightofthegraphinthiscase?
a.Whatistherangeoftan(x)?
b.Whereistan(x)positive,andwhereisitnegative?
c.Arethereanyangleswheretan(x)isundefined?
20. Noweraseallprevioustraces,andgraphcot(x).Whatsegmentonthecirclecorrespondswiththeheightofthegraphinthiscase?
a. Whatisitsdomain?Range?
b. Whereiscot(x)positive,andwhereisitnegative?Isitever0?Isiteverundefined?
21. Nowgotothefollowingworksheethttps://www.geogebra.org/m/UPWgJDrtandgraphsec(x).Whatsegmentonthecirclecorrespondswiththeheightofthegraphinthiscase?
a. Whatisitsdomain?Range?
b. Whereissec(x)positive,andwhereisitnegative?Isitever0?Isiteverundefined?
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22. Noweraseallprevioustraces,andgraphcsc(x).Whatsegmentonthecirclecorresponds
withtheheightofthegraphinthiscase?
a. Whatisitsdomain?Range?
b. Whereiscsc(x)positive,andwhereisitnegative?Isitever0?Isiteverundefined?
Challenge/Extension:Inthecomingdays,wewillspendmoretimeexploringhowthe6trigonometricfunctionsarerelatedtoeachother.Untilthen,seeifyoucanansweranyofthesechallengequestions(youmaywanttouseGeoGebratoseeifyoucandeterminesomeoftheanswers):1.Arethereanypairsoftrigonometricfunctionsthatareinverselyrelated(thatis,whenonegetsbigger,theotheronegetssmaller)?2.Arethereanypairsoftrigonometricfunctionsthataredirectlyrelated(thatis,theybothgetbiggertogetherandsmallertogether)?3.Threespecialcasesoftrigonometricfunctionsarewhentheyareequalto0,equalto1,orundefined.Isthereanyrelationshipamongthetrigonometricfunctionsastowhenthathappenstowhichones?
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LESSON6:Thegoalofthisactivityistoinvestigatetherelationshipsbetweensineandcosine,tangentandcotangent,andsecantandcosecant.Youwillgraphtheminpairsanddeterminehowtheyarerelatedtoeachother.Teachernote:Itwillbeimportanttoteaseoutwhysin(x)andcos(x)andcsc(x)andsec(x)arehorizontalshiftsofeachotherwhiletan(x)andcot(x)alsorequireareflection
1. Gotothefollowinglink:https://www.geogebra.org/m/G9mjcC7D.2. Clickbothcosineandsinetographbothgraphsatthesametime.3. Click“StartAnimation”tocreatethegraphs.4. Whichcolorrepresentssin(x)andwhichonerepresentscos(x)?Howdoyouknow?
5. Whatistherelationshipbetweenthegraphsofsin(x)andcos(x)?
6. Howcanyouincorporateahorizontalshiftintoafunction?
7. Canyouwritecos(x)asasin(x)functionwithahorizontalshift?
8. Howdoesthatmakesensewithwhatyouknowabouttherelationshipbetweensin(x)andcos(x)?Whataboutthewordssineandcosine?Howarethosewordsrelated?Doesthatrelatetothefunctionyouwroteinnumber8?
9. Nowgotohttps://www.geogebra.org/m/shAV7BSBgraphtan(x)andcot(x)atthesametime.
10. Whichcolorrepresentstan(x)andwhichcolorrepresentscot(x)?
11. Whatistherelationshipoftan(x)tocot(x)?
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12. Canyouwritecot(x)astan(x)withahorizontalshift?Whatelseneedstohappenbesidesahorizontalshiftinthiscase?
13. Howdoesthatmakesensewithwhatyouknowabouttherelationshipbetweentan(x)andcot(x)?Whataboutthewordstangentandcotangent?Howarethosewordsrelated?Doesthatrelatetothefunctionyouwroteinnumber12?
14. Gotohttps://www.geogebra.org/m/UPWgJDrtandgraphsec(x)andcsc(x)atthesame
time.
15. Whichcolorrepresentssec(x)andwhichcolorrepresentscsc(x)?Howdoyouknow?
16. Whatistherelationshipofsec(x)tocsc(x)?
17. Canyouwritecsc(x)assec(x)withahorizontalshift?
18. Howdoesthatmakesensewithwhatyouknowabouttherelationshipbetweensec(x)andcsc(x)?Whataboutthewordssecantandcosecant?Howarethosewordsrelated?Doesthatrelatetothefunctionyouwroteinnumber16?
19. Ifthesefunctionsarejusthorizontalshiftsofeachother,dowereallyneedseparatefunctions,orwoulditbesufficienttojusthavesin(x),sec(x)andtan(x)?
20. Whenisithelpfultohavecos(x),cot(x),andcsc(x)?
21. Arethereevercaseswhereitseemsredundanttohavetheseadditionalfunctions?
22. Seeifyourgraphingcalculatorhasasineregression.Doesitalsohaveacosineregression?Whydoyouthinkthiswouldbe?
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LESSON7:Thegoalofthisactivityistoinvestigatetherelationshipsofthetrigonometricfunctionsonthecircle.Introduction:Forseveraldaysnow,wehavebeenworkingwithacircle,whoseradiusis1unit.Thiscircleisoftencalledtheunitcircle,becauseitisacirclewithaunitradius.Ontheunitcircle,wecanfindseveraldifferentrighttriangles.
1. Seewhatrighttrianglesyoucanfind.Youshouldbeabletofindthreedifferentrighttriangles(notethattherearetworighttrianglesthatarecongruent,wecanjustconsideroneofthose).
2. GobacktoyoursavedGeoGebradocument.StartwiththerighttriangleABC.ConstructthisrighttriangleinGeoGebrausingthePolygontoolonthetoolbar.Notethatyouwillhavetohighlightall3pointsandthenhighlightthefirstpointagaintoconstructthepolygon(A-B-C-A,forexample).Note:youcangetridofanyunwantedmeasurementsbyright-clickingandselecting“hidelabel.”
3. Considerthissidesofthistriangle.Whattrigonometricfunctionrepresentsthelengthof
AB?BC?WhatisthelengthofAC?
4. GotoOptions,thenRounding,andselect2decimalplaces.5. Sincethisisarighttriangle,canyouapplythePythagoreantheoremtothoseside
lengths?Whatdoyouget?
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6. Nowlet’sconsiderthisrelationshipinanotherway.GotoViewand“CAS.”Nowyouhaveacalculatoronthesideofyourdocument.Now,findthenamesofthesegmentsABandBCbyrightclickingtheirmeasurements.Minearecallediandn.Puttheleft-handsideofthePythagoreantheoremintothecalculator.Seeifitequalstheright-handsideoftheequationyoufound.Notethatifyouhittheequalssign,itwillgiveyouanexactvalue(withalarmingaccuracy!)butifyouhittheapproximatelyequalsign,itwillgiveyousomethingmorereasonable.Yourscreenshouldlooksomethinglikethis:
7. Onceyouhavethecalculation,movepointCaround,andseeifthecalculationchangesorstaysthesame.(Hint:theapproximatecalculationshouldstaythesame,buttheexactcalculationshouldchange—thisisbecausewhatwearedoinghereisnotperfect.Ourradiusof1isnotexactly1ifyougoouttoenoughdecimalplaces.Thatistheerroryouareseeingintheexactcalculations.)
TEACHERNOTE:Thiswouldbeagoodtimetostopandcometogetherasagrouptomakesureeveryonehascreatedthecorrectidentitysin2(x)+cos2(x)=1,andthattheyhavebeenabletocorrectlyenterthatcalculationintoGeoGebra.
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8. DeletetriangleABC,andnowconsiderthetriangleADE.Constructthetrianglewiththepolygontool.
9. Considerthissidesofthistriangle.WhattrigonometricfunctionrepresentsthelengthofAE?DE?WhatisthelengthofAD?
10. Sincethisisarighttriangle,canyouapplythePythagoreantheoremtothosesidelengths?Whatdoyouget?
11. Nowlet’sconsiderthisrelationshipinanotherway.GototheCASandcalculatetheleft-handsideoftheequationyoufound.Inaseparatecalculation,entertheright-handsideoftheequationyoufound.Seeifthetwosidesareequal.Useapproximatecalculations.
12. Onceyouhavethetwocalculations,movepointCaround.Whathappenstothetwocalculations?
13. DeletetriangleADE,andnowconsiderthetriangleAGF.Constructthetrianglewiththepolygontool.
14. Considerthissidesofthistriangle.WhattrigonometricfunctionrepresentsthelengthofAG?FG?WhatisthelengthofAF?
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15. Sincethisisarighttriangle,canyouapplythePythagoreantheoremtothosesidelengths?Whatdoyouget?
16. Nowlet’sconsiderthisrelationshipinanotherway.GototheCASandcalculatetheleft-handsideoftheequationyoufound.Inaseparatecalculation,entertheright-handsideoftheequationyoufound.Seeifthetwosidesareequal.Useapproximatecalculations.
17. Onceyouhavethetwocalculations,movepointCaround.Whathappenstothetwocalculations?
18. ThesethreeequationsareknownintrigonometryasthePythagoreanIdentities.ThefirstoneisoftencalledtheprincipalPythagoreanidentity.Doesthenamemakesense?Whyorwhynot?
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LESSON8:Thegoalofthisactivityistoinvestigatetherelationshipsbetweensineandcosecant,cosineandsecant,tangentandcotangent.Itisalsotoinvestigatetherelationshipbetweenamongsine,cosine,andtangent.Youwillgraphtheminpairs/groupsanddeterminehowtheyarerelatedtoeachother.
1. Gotohttps://www.geogebra.org/m/UPWgJDrt.Graphsin(x)andcsc(x)atthesametime.Whichcolorrepresentssin(x)andwhichonerepresentscsc(x)?
2. Whenissin(x)=0?Wheniscsc(x)undefined?
3. Whensin(x)getscloseto0,whathappenstocsc(x)?
4. Trymovingthesliderforqandwatchingwhathappenstosine(BC)andcosecant(AG).Lookatwhathappenswhenonegetsverysmall,whathappenstotheotherone.Whenonegetscloseto1,whathappenstotheotherone?
5. MakeatableofvaluesbyslidingpointCsothatsin(x)isthefollowing(roundcsc(x)to
onedecimalplace):sin(x) csc(x)-1 -0.5 0 0.1 0.2 0.5 1
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6. Canyoumakeanyguessesastotherelationshipbetweenthegraphsofsin(x)and
csc(x)?
TEACHERNOTE:Youwillhavetostophereanddiscusstomakesureallstudents/groupshavediscoveredtherelationship.Youmayhavetoguidetheclassorhaveagroupdiscussiontoleadtheclasstotheideathattheyarereciprocalfunctions.
7. Erasethetracesofsin(x)andcsc(x),andgraphcos(x)andsec(x)indifferentcolors.
Whichcolorrepresentscos(x)andwhichonerepresentssec(x)?
8. Wheniscos(x)=0?Whenissec(x)undefined?
9. Whencos(x)getscloseto0,whathappenstosec(x)?
10. Trymovingthesliderforqandwatchingwhathappenstocosineandsecant.Lookatwhathappenswhenonegetsverysmall,whathappenstotheotherone.Whenonegetscloseto1,whathappenstotheotherone?
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11. MakeatableofvaluesbyslidingpointCsothatcos(x)isthefollowing(roundsec(x)toonedecimalplace):cosx) sec(x)-1 -0.5 0 0.1 0.2 0.5 1
12. Canyoumakeanyguessesastotherelationshipbetweenthegraphsofcos(x)andsec(x)?
13. Doesthismakesensebasedontherelationshipyoudiscoveredearlierbetweensin(x)andcsc(x)?
14. Gotohttps://www.geogebra.org/m/shAV7BSBandgraphtan(x)andcot(x)atthesametime.Whichcolorrepresentstan(x)andwhichonerepresentscot(x)?
15. Whenistan(x)=0?Wheniscot(x)undefined?
16. Whentan(x)getscloseto0,whathappenstocot(x)?
17. Trymovingthesliderforqandwatchingwhathappenstotangentandcotangent.Lookatwhathappenswhenonegetsverysmall,whathappenstotheotherone.Whenonegetscloseto1,whathappenstotheotherone?
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18. MakeatableofvaluesbyslidingpointCsothattan(x)isthefollowing(roundcot(x)toonedecimalplace):tan(x) cot(x)-1 -0.5 0 0.1 0.2 0.5 1
19. Canyoumakeanyguessesastotherelationshipbetweenthegraphsoftan(x)andcot(x)?
20. Doesthismakesensebasedontherelationshipyoudiscoveredearlierbetweensin(x)andcsc(x)andcos(x)andsec(x)?
21. YoualreadysawinLesson6thattan(x)andcot(x)arehorizontalshiftsofeachother.Cantheyhaveanotherrelationshipatthesametime?
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Challenge/Homework:ThinkbacktoLesson7,whereweinvestigatedthePythagoreanidentities.
1. ManypeopleusealgebratogetfromtheprincipalPythagoreanidentitytotheothertwoPythagoreanidentites.TrytakingtheprincipalPythagoreanidentity,anddividingthroughbysin2(x).Whatdoyouget?
2. WhatcanyoudividebytogetthefinalPythagoreanidentity?
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LESSON9:Inthislesson,thegoalistoinvestigatewhathappenstothetrigonometricfunctionswhenxarrivesatcertainspecialangles.
1. Gotohttps://www.geogebra.org/m/Cb2jWUPS.ClickonSpecialandSnap.Thiswillallowyoutoseepointswhere“special”trigonometricfunctionsoccur,andtheanglewillautomaticallysnaptotheseangles.Noticethatyoucantogglebetweenanglesandradians.
Now,GeoGebraisgivingusalldecimalmeasurements.Let’sseeifwecanfigureouttheexactmeasurementsforthistriangle.
2. Ifsin(x)andcos(x)arethesame,inotherwords,AB=BC.Whatisitcalledwhenatrianglehastwosideswithequalmeasures?
3. Whatdoesitmeanfortheangleswhentwosidesofatrianglehaveequalmeasures?
4. Ifyouknowthatthelargestanglemeasureis90˚andtheothertwoanglesarethesamedegreemeasures,whatwouldthatmakethedegreemeasuresoftheotheranglesofthistriangle?Drawitbelow.
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5. Whatarethesemeasuresinradians?(Keepintermsofπ.)
6. Findadecimalapproximationforyouranswerabove.DoesthatmatchwiththedecimalmeasureofxgiveninGeoGebra?
7. UsingthePythagoreantheorem,findthelengthsofthetwomissingsidesofthistriangle.Sinceyouknowthattheyarethesamelength,youcancallthembothx.Keepthisanswerexact.
x x
8. ConverttheansweryougotabovetoadecimalandcompareittothevaluethatGeoGebraisgivingyouforsin(x)andcos(x).Isitthesame?
1
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9. Thisisatrianglethatisknownasaspecialrighttriangle.Triangleswiththeseanglesalwayshavethesameproportions,evenwhenyouscaleuporscaledownthesizeofthetriangle.Usingproportionality,findthefollowingmissingsidelengths,assumingtheseareisoscelesrighttriangles.
10. NowgobacktoGeoGebraandmovepointCsothatsin(x)=.5
11. UsethetrigonometrytableyoucreatedinLesson3tofindthedegreemeasureofxatthistime.
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12. Usetheformulayouknow(thatπradians=180°)toconvertthisintoradianmeasures.ThenconvertthatintoadecimalandcompareittothexyouhaveinGeoGebra.Aretheythesame?Ifnot,aretheyclose?
13. Ifthisisarighttriangle,knowingoneofthenon-rightangles,findthethirdangle.
14. UsingthePythagoreanTheorem,knowingthatthehypotenuseis1andthesin(x)sideofthetriangleis½,findthethirdsideofthetriangle.
15. Fillinthediagrambelowwiththemissingsideandanglemeasurements(useexactvalues,notdecimals).
½
1
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16. Thisisanotherspecialrighttriangle.Triangleswiththeseanglesalwayshavethesameproportions,evenwhenyouscaleuporscaledownthesizeofthetriangle.Usingproportionality,findthefollowingmissingsidelengths,assumingtheseare30-60-90righttriangles.
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Extension:Usingwhatyouknowaboutspecialrighttriangles,canyoufindthefollowingpointsontheunitcirclebelow(useexactvalues,notdecimals).GobacktoGeoGebraandexamineittoseewhetherthesin(x)valuewouldbethex-coordinateorthey-coordinate,andwhetherthecos(x)valuewouldbethex-coordinateorthey-coordinate.Teachernote:Studentsmayneedhelp,especiallygettingstartedonthis.Youmaywanttodothefirstfewpointswiththem.
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LESSON10:TrigonometryPost-testInstructions.Answereachquestiontothebestofyourability.Ifthereismorethanoneanswer,putboth/allanswersdown.
1. Whatissin(𝑥)?
2. Whatiscos(𝑥)?
3. Whatistan(𝑥)?
4. Whatisthepurposeofthetrigonometricfunctions?Inotherwords,whattypesofproblemscantheyhelpyousolve?
5. Dosin(𝑥)andcos(𝑥)haveanyrelationship?
6. Howdoessin 𝑥 changeasxgoesfrom0to90degrees?
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7. Are𝑦 = sin 𝑥 and𝑦 = cos(𝑥)functions?Ifso,whatistheirdomainandrange?
8. Theheightofabuilding’sshadowis56ftwhenthesunisshiningata35˚angletothehorizon.Whatistheheightofthebuilding?Explainhowyoufoundyouranswer. h
9. Onthefollowingdiagram,labelanythingthatyoucanthatisrelevanttotrigonometry,andexplainhowitisrelevant.
35˚56ft
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10. IfyouapplythePythagoreantheoremtothetriangleshighlightedineachofthediagramsshownbelow,whattrigonometricidentitywillyouget?
a.
b.
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11. Identifythefollowinggraphs,andexplainhowyouknow.c.
d.
e.
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12. Writetrigonometricfunctionsthatareequivalenttothefollowingf. 0
123 4=
g. 0
567 4=
h. 0
189(4)=
i. 123(4)
561(4)=
j. 561(4)
123(4)=
13. Inthefollowingtriangles,findthesidelengthswithexactmeasurements(donotusedecimals).
k.
4.5
l.
30
6090
4590
8
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Notestotheteacher:Forthepost-test,youwillnoticethatsomequestionsareidenticaltothepre-test.Thisissothatknowledgegainscanbemeasuredinahelpfulway.Otherquestionshavebeenaddedtothepost-testtotestonadditionalconceptstheyhavelearnedduringthisunit.Youmayalsonoticethatthetestisveryconceptual,anddoesnotfocusontestingskillsmuch.Tosomeextent,thatisareflectionofthebridgecurriculum’sfocusonconcepts.Still,youmaywishtoaddsomequestionsthatfocusontestingskills.Afterthepost-test,youmaywanttoconnectwhatyouhavebeendoinginthisunittowhatiscomingnext.Therearevariouswaysyoumightwanttodothis.YoumightuseLesson9asajumpingoffpointandmapoutthetraditionalunitcircle,withallthespecialanglesrepresented.YoumightuseLesson7asajumpingoffpointanddiscusstrigonometricidentities.YoumightuseLessons4,5,and8asajumpingoffpointanddiscussthegraphsofthetrigonometricfunctionsanddiscussallthepossibletransformationsofthosefunctions.YoumightuseLessons3and4asajumpingoffpointandsolveproblemsusingtrigonometry.Ifyouhaveusedsomeorallofthiscurriculum,andyouwouldliketoshareresults,thoughts,orfeedback,[email protected] .