Teacher-Notes TRIG LENGTHS - aiminghigh.aimssec.ac.za€¦ · This activity provides practice in...
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AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE TEACHER NETWORK TRIG LENGTHS In the diagram the line OS is perpendicular to the lines OR and PQ. The line RS is a tangent at P to the circle centre O and radius 1 unit. Find the lengths OQ, PQ, PS, OS, OR and RP. If OS and OR lie on the coordinate axes, what are the coordinates of the point P? Solution As OP=1 and it is the hypotenuse of ΔOPQ, by definition OQ=cosθ and PQ=sinθ PS/OP=PS/1=tanθ so PS=tanθ ∠OPR=∠OPS= 90 o (radius, tangent) OS/OP=OS/1=secθ so OS=secθ In ΔOPR, ∠ROP=90-θ and ∠ORP=θ, hence OR/OP=OR/1 so OR=cosecθ RP/OP=RP/1 so RP=cot θ Notes for teachers Diagnostic Assessment This should take about 5–10 minutes. 1. Write the question on the board, say to the class: “Put up 1 finger if you think the answer is A, 2 fingers for B, 3 fingers for C and 4 fingers for D”. 2. Notice how the learners responded. Ask a learner who gave answer A to explain why he or she gave that answer and DO NOT say whether it is right or wrong but simply thank the learner for giving the answer. 3. Then do the same for answers B, C and D. Try to make sure that learners listen to these reasons and try to decide if their own answer was right or wrong. 4. Ask the class again to vote for the right answer by putting up 1, 2, 3 or 4 fingers. Notice if there is a change and who gave right and wrong answers. It is important for learners to explain the reason for their answer otherwise many learners will just make a guess. 5. If the concept is needed for the lesson to follow, explain the right answer or give a remedial task. C. is the correct answer. Common Misconceptions Learners giving the answers A.,B. and D do not know and understand the fundamental Pythagorean trig identity sin 2 θ + cos 2 θ=1 https://diagnosticquestions.com
Transcript of 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
