teacher-notes trig lengths - aiminghigh.aimssec.ac.za€¦ · this activity provides practice in...

AFRICANINSTITUTEFORMATHEMATICALSCIENCESSCHOOLSENRICHMENTCENTRE

TEACHERNETWORK

TRIGLENGTHSInthediagramthelineOSisperpendiculartothelinesORandPQ.

ThelineRSisatangentatPtothecirclecentreOandradius1unit.

FindthelengthsOQ,PQ,PS,OS,ORandRP.

IfOSandORlieonthecoordinateaxes,whatarethecoordinatesofthepointP?

SolutionAsOP=1anditisthehypotenuseofΔOPQ,bydefinitionOQ=cosθandPQ=sinθPS/OP=PS/1=tanθsoPS=tanθ∠OPR=∠OPS=90o(radius,tangent)OS/OP=OS/1=secθsoOS=secθInΔOPR,∠ROP=90-θand∠ORP=θ,henceOR/OP=OR/1soOR=cosecθRP/OP=RP/1soRP=cotθ

NotesforteachersDiagnosticAssessmentThisshouldtakeabout5–10minutes.1. Writethequestionontheboard,saytotheclass:

“Putup1fingerifyouthinktheanswerisA,2fingersforB,3fingersforCand4fingersforD”.2. Noticehowthelearnersresponded.AskalearnerwhogaveanswerAtoexplainwhyheorshegavethat

answerandDONOTsaywhetheritisrightorwrongbutsimplythankthelearnerforgivingtheanswer.3. ThendothesameforanswersB,CandD.Trytomakesurethatlearnerslistentothesereasonsandtryto

decideiftheirownanswerwasrightorwrong.4. Asktheclassagaintovotefortherightanswerbyputtingup1,2,3or4fingers.Noticeifthereisa

changeandwhogaverightandwronganswers.Itisimportantforlearnerstoexplainthereasonfortheiranswerotherwisemanylearnerswilljustmakeaguess.

5. Iftheconceptisneededforthelessontofollow,explaintherightanswerorgivearemedialtask.

C.isthecorrectanswer.CommonMisconceptionsLearnersgivingtheanswersA.,B.andDdonotknowandunderstandthefundamentalPythagoreantrigidentitysin2θ+cos2θ=1https://diagnosticquestions.com

Whydothisactivity?Thisactivityprovidespracticeinworkingwithrightangledtrianglesandusingthesixbasictrigonometricratios.Learnersshouldbeintroducedtothegeneraltrigonometricfunctionsbeforemeetingthegraphsofthetrigonometricfunctions.Theconnectionwiththeformulaforthecircumferenceofthecircleintermsoftheanglemeasuredinradianscanbemade.Also,schoolstudentsshouldbeawareofradianmeasureasitappearsoncalculators,andasitisgenerallyusedinhighermathematics,evenifitisnotpartoftheschoolcurriculum.Teacherscantakethisopportunitytoexplainhowtrigonometryarosethousandsofyearsagointhestudyofastronomyandtoengagetheclassindiscussionoftheusesoftrigonometrytodaywhereweneedtoworkwithobtuseanglesandtrigonometricfunctionsofangleswheretheanglescantakeanyvalue.

IntendedLearningOutcomesTheactivityprepareslearnerstomeetthedefinitionsofthetrigonometricfunctionsforanglesofanysize,positiveornegative)intermsofthecoordinatesofpointsontheunitcircle.

PossibleapproachStartwiththediagnosticquestion.Theemphasisethatthetrigonometricidentitysin2θ+cos2θ=1issimplyPythagorasTheoremfortherightangledtrianglewithhypotenuseoflength1unit.Thisactivitycanbeusedasalessonstarterwhenlearnershavemetthedefinitionsofthesixtrigonometricratiosforanglesinarightangledtriangleandareabouttobeintroducedtothegeneraldefinitionforanglesgreaterthan90o.Buildingonwhatthelearnersknow,thisdrawsoutthefactthat,forapointPontheunitcircle,thecosineandsineof∠POQarerelatedtothecoordinatesofPifthepointOistheoriginandSandRareontheaxes.ThegeneralisationtoallanglesasPmovesaroundtheunitcirclefollowsnaturally.

TeacherscouldgiveoneofthesixlengthsOQ,PQ,PS,OS,ORandRPeachtodifferentpairsoflearnersandthenaskdifferentpairsinturntocometotheboardtoexplainwhattheyhadfound.Thiswaythelearnerswhostrugglecouldbegiventhechancetosucceedwhilethequickeronescouldbegivenabitmorechallenge.Keyquestions• WhatdoyounoticeaboutOPandRS?• Howmanyrightangledtrianglescanyousee?• WhataretheanglesoftriangleORP?

PossibleextensionPliesonthecirclecentreOradius1unitandTXisatangenttothecircleatX.

PQisperpendiculartoOX.VWisperpendiculartoOV.

FindthelengthsOQ,PQ.

WhatarethecoordinatesofP?

FindthelengthsTX,OT,OWandVWintermsoftrigonometricfunctionsoftheangleθ.PossiblesupportDrawalargecopyofthediagramandaskthelearnertomarkinalltheangles.Thendrawattentiontoeachrightangledtriangleonebyone(perhapsoutliningthemindifferentcolours)andaskthelearnertoworkoutthelengthsofthesidesofthetriangle.

Solutiontoextensionactivity

FURTHERSTUDYMATERIALThreearticlesontheHistoryofTrigonometrybyLeoRogersfromtheNRICHwebsitewhereyouwillfindTeachers’Notesgivingsuggestionsforusingthematerialinyourteaching.www.nrich.maths.org/6843andwww.nrich.maths.org/6853andwww.nrich.maths.org/6908TrigonometryisneededinnavigationincludingGPSsystems,insurveying,architecture,engineering,physics,computerscience,mathematicandastronomy.Intheseapplicationsweworkwithobtuseanglesandtrigonometricfunctionsofanglesofallvalues,measuredinradians.

Trigonometryfirstaroseinworkonastronomyandthestudyoftherotationsofthesun,moon,planetsandconstellationsacrosstheskyinwhichsphericaltrianglesareasimportantasplanetriangles.Trigonometryhasbeenusedforastronomicalcalculationsforthousandsofyears.ThestudyofastronomyandrelatedmathematicsgoesbacktotheBabylonian,Egyptian,Greek,Chinese,IndianandArabcultures.Theshadowstick(gnomon)wasusedinsundialstostudythemotionofthesunandtotellthetimeandalsotodrawrightangles.

Allovertheworldancientcivilisationsbuiltlargestandingcirclesofstonesorwoodenpostspreciselypositionedformakingaccurateobservationsofthemovementofthesun,moonandplanetsandpredictingastronomicaleventssuchaslunarcyclesandeclipses.

ThephotoshowsastonecircleinSenegal.Fromabout1900BCtheBayloniansworkedinabase60placevaluenumbersystemthatistheoriginofourmeasuresoftime(60minutesinanhouretc.)andourmeasuresofangles.TheGreekmathematicianHipparchusin140BCcompiledthefirstknowntableofvaluesofthesinefunction(atableofchords).

TheArabmathematicianAbulWafa(940-998AD)wasthefirsttostudytrigonometricidentitiessystematically.Moreefficientastronomicalcalculationscouldbemade,andmoreaccuratetablescouldbeestablished,usingtrigonometricidentities.AbulWafabroughttogetherandestablishedtherelationsbetweenthesixfundamentaltrigonometricfunctionsforthefirsttime.HealsousedR=1fortheradiusofthebasiccircle.

Thewordssine,cosine,tangent,secantandcosecantderivefromGreekwordsforthelengthsasshowninthediagrams.Note:TheGradesorSchoolYearsspecifiedontheAIMINGHIGHWebsitecorrespondtoGrades4to12inSouthAfricaandtheUSA,toYears4to12intheUKanduptoSecondary5inEastAfrica.Note:ThemathematicstaughtinYear13(UK)andSecondary6(EastAfrica)isnotincludedintheschoolcurriculumforGrade12SA. LowerPrimary

orFoundationPhaseAge5to9

UpperPrimaryAge9to11

LowerSecondaryAge11to14

UpperSecondaryAge15+

SouthAfrica GradesRand1to3 Grades4to6 Grades7to9 Grades10to12USA KindergartenandG1to3 Grades4to6 Grades7to9 Grades10to12UK ReceptionandYears1to3 Years4to6 Years7to9 Years10to13EastAfrica NurseryandPrimary1to3 Primary4to6 Secondary1to3 Secondary4to6