LESSON 1: Trigonometry Pre-test · Things to know about GeoGebra: On the toolbar on the top, you can construct segments, circles, points, polygons, perpendicular

TrigonometryBridgeCurriculum—TeacherEdition

©JennaVanSickle ClevelandStateUniversity

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?



7. Are𝑦 = sin 𝑥 and𝑦 = cos(𝑥)functions?Ifso,whatistheirdomainandrange?

8. Theheightofabuilding’sshadowis56ftwhenthesunisshiningata35˚angletothehorizon.Whatistheheightofthebuilding?Explainhowyoufoundyouranswer. h

35˚56ft



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



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.



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?



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?



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.)



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?



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?



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.



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



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)



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



Thewordcosinemeans“sineofthecomplement”becausethecosineisactuallythesegmentthatisequaltothesineof(90-x)˚.Inthediagrambelow,wethinkofcosineasOM,buttheAncientGreeksthoughtofitasNP(whichiscongruenttoOM). Rememberthatcomplementaryanglesaddto90˚.Sinehasthesamerelationshiptotheoriginalangleascosinedoestothecomplementaryangle(90-x)˚.Tryturningthecirclesidewaystoseethisbetter.

3. Eventhoughourtrigonometrytabledoesnotincludecosine,howcanweusethisinformationtofindcos(x)usingourtrigtable?

sine

cosine



4. Usingyourtrigtable,findthefollowing: Checkusingcalculator:

a. cos(7˚)

b. cos(88˚)

c. cos(135˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)

d. cos(111˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)

e. cos(-59˚)(sketchadiagramthatshowshowyoufiguredoutwhichangletouseinyourtrigtable)



5. Usingyourtrigtable,solveforxinthefollowingtriangle:


6. Nowthatyouknowhowtofindsin(x)andcos(x)usingyourtrigonometrytable,howcanyoufindvaluesfortan(x)usingthetrigonometrytable?

7. Findthefollowingusingyourtrigtable:a. tan(32˚)


1.5in

0.6in

x



b. tan(83˚)


8. Whenyouhaveusedyourtrigtableandcheckedusingyourcalculator,howclosehaveyourcalculationsbeen?Whenhavetheybeenoff,andbyhowmuch?Whatdoyouattributethisdifferenceto?Isthisalotoferror,orjustalittlebit?

9. Iamsureyouwillbehappytogobacktousingyourcalculatorafterthis,butwhathaveyoulearnedfromcreatingandusingthistrigonometrytable?

Asyoucontinuetouseyourcalculatortosolveatrigonometricequation,trytorememberwhatishappeninginsideofyourcalculator.



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?



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.



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?



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?


b. Whereissec(x)positive,andwhereisitnegative?Isitever0?Isiteverundefined?



22. Noweraseallprevioustraces,andgraphcsc(x).Whatsegmentonthecirclecorresponds

withtheheightofthegraphinthiscase?


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?



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)?



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?



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?



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.



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?



15. Sincethisisarighttriangle,canyouapplythePythagoreantheoremtothosesidelengths?Whatdoyouget?

16. Nowlet’sconsiderthisrelationshipinanotherway.GototheCASandcalculatetheleft-handsideoftheequationyoufound.Inaseparatecalculation,entertheright-handsideoftheequationyoufound.Seeifthetwosidesareequal.Useapproximatecalculations.

17. Onceyouhavethetwocalculations,movepointCaround.Whathappenstothetwocalculations?

18. ThesethreeequationsareknownintrigonometryasthePythagoreanIdentities.ThefirstoneisoftencalledtheprincipalPythagoreanidentity.Doesthenamemakesense?Whyorwhynot?



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



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?



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?



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?



Challenge/Homework:ThinkbacktoLesson7,whereweinvestigatedthePythagoreanidentities.

1. ManypeopleusealgebratogetfromtheprincipalPythagoreanidentitytotheothertwoPythagoreanidentites.TrytakingtheprincipalPythagoreanidentity,anddividingthroughbysin2(x).Whatdoyouget?

2. WhatcanyoudividebytogetthefinalPythagoreanidentity?



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.



5. Whatarethesemeasuresinradians?(Keepintermsofπ.)

6. Findadecimalapproximationforyouranswerabove.DoesthatmatchwiththedecimalmeasureofxgiveninGeoGebra?

7. UsingthePythagoreantheorem,findthelengthsofthetwomissingsidesofthistriangle.Sinceyouknowthattheyarethesamelength,youcancallthembothx.Keepthisanswerexact.

x x

8. ConverttheansweryougotabovetoadecimalandcompareittothevaluethatGeoGebraisgivingyouforsin(x)andcos(x).Isitthesame?

1



9. Thisisatrianglethatisknownasaspecialrighttriangle.Triangleswiththeseanglesalwayshavethesameproportions,evenwhenyouscaleuporscaledownthesizeofthetriangle.Usingproportionality,findthefollowingmissingsidelengths,assumingtheseareisoscelesrighttriangles.

10. NowgobacktoGeoGebraandmovepointCsothatsin(x)=.5

11. UsethetrigonometrytableyoucreatedinLesson3tofindthedegreemeasureofxatthistime.



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



16. Thisisanotherspecialrighttriangle.Triangleswiththeseanglesalwayshavethesameproportions,evenwhenyouscaleuporscaledownthesizeofthetriangle.Usingproportionality,findthefollowingmissingsidelengths,assumingtheseare30-60-90righttriangles.



Extension:Usingwhatyouknowaboutspecialrighttriangles,canyoufindthefollowingpointsontheunitcirclebelow(useexactvalues,notdecimals).GobacktoGeoGebraandexamineittoseewhetherthesin(x)valuewouldbethex-coordinateorthey-coordinate,andwhetherthecos(x)valuewouldbethex-coordinateorthey-coordinate.Teachernote:Studentsmayneedhelp,especiallygettingstartedonthis.Youmaywanttodothefirstfewpointswiththem.



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?



7. Are𝑦 = sin 𝑥 and𝑦 = cos(𝑥)functions?Ifso,whatistheirdomainandrange?

8. Theheightofabuilding’sshadowis56ftwhenthesunisshiningata35˚angletothehorizon.Whatistheheightofthebuilding?Explainhowyoufoundyouranswer. h

9. Onthefollowingdiagram,labelanythingthatyoucanthatisrelevanttotrigonometry,andexplainhowitisrelevant.

35˚56ft



10. IfyouapplythePythagoreantheoremtothetriangleshighlightedineachofthediagramsshownbelow,whattrigonometricidentitywillyouget?

a.

b.



11. Identifythefollowinggraphs,andexplainhowyouknow.c.

d.

e.



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



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].

Documents

LESSON 1: Trigonometry Pre-test · Things to know about GeoGebra: On the toolbar on the top, you can construct segments, circles, points, polygons, perpendicular