Cosecant,Secant&Cotangent - PDF document

Cosecant,Secant&Cotangent mc-TY-cosecseccot-2009-1 Inthisunitweexplainwhatismeantbythethreetrigonometricratioscosecant,secantandcotangent.Weseehowtheycanappearintrigonometricidentitiesandinthesolutionoftrigonometricalequations.Finally,weobtaingraphsofthefunctionscosec,secandcotfromknowledgeoftherelatedfunctionssin,cosandtan.Inordertomasterthetechniquesexplainedhereitisvitalthatyouundertakethepracticeexercisesprovided.Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto:denetheratioscosecant,secantandcotangentplotgraphsofcosec,secandcotContents1.Introduction22.Denitionsofcosecant,secantandcotangent23.Thegraphofcosec44.Thegraphofsec55.Thegraphofcot6 www.mathcentre.ac.uk1c\rmathcentre2009 1.IntroductionThisunitlooksatthreenewtrigonometricfunctionscosecant(cosec),secant(sec)andcotan-gent(cot).Thesearenotentirelynewbecausetheyarederivedfromthethreefunctionssine,cosineandtangent.2.Denitionsofcosecant,secantandcotangentThesefunctionsaredenedasfollows: KeyPointcosec=1 sinsec=1 coscot=1 tan Thesefunctionsareusefulinthesolutionoftrigonometricalequations,theycanappearintrigono-metricidentities,andtheycanariseincalculusproblems,particularlyinintegration.ExampleConsiderthetrigonometricidentitysin2+cos2=1Supposewedivideeverythingonbothsidesbycos2.Doingthisproducessin2 cos2+cos2 cos2=1 cos2Thiscanberewrittenassin cos2+1=1 cos2thatisastan2+1=sec2This,incaseyouarenotalreadyaware,isacommontrigonometricalidentityinvolvingsec.ExampleConsideragainthetrigonometricidentitysin2+cos2=1Supposethistimewedivideeverythingonbothsidesbysin2;thisproducessin2 sin2+cos2 sin2=1 sin2 www.mathcentre.ac.uk2c\rmathcentre2009 Thiscanberewrittenas1+cos sin2=1 sin2thatisas1+cot2=cosec2Again,weseeoneofournewtrigonometricfunctions,cosec,appearinginanidentity.ExampleSupposewewishtosolvethetrigonometricalequationcot2=3for0360Webeginthesolutionbytakingthesquareroot:cot=p 3orp 3Itthenfollowsthat1 tan=p 3orp 3Invertingwendtan=1 p 3or1 p 3Theanglewhosetangentis1 p 3isoneofthespecialanglesdescribedintheunitTrigonometricalratiosinaright-angledtriangle.Infact1 p 3isthetangentof30.Sothisisonesolutionoftheequationtan=1 p 3.Whataboutothersolutions?WerefertoagraphofthefunctiontanasshowninFigure1. tan p 9018027036030210150330 p 3 µ µ -oooooooo Figure1.Agraphoftan.Fromthegraphweseethatthenextsolutionoftan=1 p 3is210(thatis180furtheralong).Fromthesamegraphwecanalsodeduce,byconsiderationofsymmetry,thattheangleswhosetangentis1 p 3are150and330. www.mathcentre.ac.uk3c\rmathcentre2009 Insummary,theequationcot2=3hassolutions=30;150;210;330So,solvingequationsinvolvingcosec,secandcotcanoftenbesolvedbysimplyturningthemintoequationsinvolvingthemorefamiliarfunctionssin,cosandtan.3.ThegraphofcosecWestudythegraphofcosecbyrststudyingthegraphofthecloselyrelatedfunctionsin,onecycleofthegraphofwhichisshowninFigure2. sin90180270360 ABCDoooo- µ µ Figure2.Agraphofsin.Thegraphofcoseccanbededucedfromthegraphofsinbecausecosec=1 sin.Notethatwhen=90,sin=1andhencecosec=1aswell.Similarlywhen=270,sin=1andhencecosec=1aswell.Theseobservationsenableustoplottwopointsonthegraphofcosec.ThecorrespondingpointsaremarkedAandBinbothFigures2and3.When=0,sin=0,butbecausewecanneverdivideby0wecannotevaluatecosecinthisway.However,notethatifisverysmallandpositive(i.e.closeto,butnotequaltozero)sinwillbesmallandpositive,andhence1 sinwillbelargeandpositive.PointsmarkedConthegraphsrepresentthis.Similarlywhen=180,sin=0andagainwecannotdividebyzerotondcosec180.Supposewelookatvaluesofjustbelow180.Here,sinissmallandpositive,soonceagaincosecwillbelargeandpositive(pointsD).TheseobservationsenableustograduallybuildupthegraphasshowninFigure3.Thevertical www.mathcentre.ac.uk4c\rmathcentre2009 dottedlinesonthegrapharecalledasymptotes. cosec90180270360 CD µ µ -oooo Figure3.Agraphofcosec.Notethatwhenisjustslightlygreaterthan180thensinissmallandnegative,sothatcosecislargeandnegativeasshowninFigure3.Continuinginthiswaythefullgraphofcoseccanbeconstructed.InFigure2weshowedjustonecycleofthesinegraph.Thisgeneratedonecycleofthegraphofcosec.Clearly,iffurthercyclesofthesinegrapharedrawnthesewillgeneratefurthercyclesofthecosecantgraph.Weconcludethatthegraphofcosecisperiodicwithperiod2.4.ThegraphofsecWecandrawthegraphofsecbyrststudyingthegraphoftherelatedfunctioncosonecycleofwhichisshowninFigure4. cos 90180270360 µ µ Figure4.Agraphofcos.Notethatwhen=0,cos=1andsosec0=1.Thisgivesusapoint(A)onthegraph.Similarlywhen=180,cos=1andsosec180=1(PointB).When=90,cos=0andsowecannotevaluatesec90.Weproceedasbeforeandlookalittletotheleftandright.Whencosissmallandpositive,1 coswillbelargeandpositive.ThisgivespointC.Whencosissmallandnegative,1 coswillbelargeandnegative.ThisgivespointD.ContinuinginthiswaywecanproducethegraphshowninFigure5. www.mathcentre.ac.uk5c\rmathcentre2009 RecallthatwehaveonlyshownonecycleofthecosinegraphinFigure4.Howeverbecausethisrepeatswithaperiodof2itfollowsthatthegraphofsecisalsoperiodicwithperiod2. sec90180270360 ABCD µ µ -oooo Figure5.Agraphofsec.5.ThegraphofcotWecandrawthegraphofcotbyrststudyingthegraphoftantwocyclesofwhichareshowninFigure6. tan90180270360 AB Figure6.Agraphoftan.Weproceedasbefore.Whenissmallandpositive(justabovezero),sotooistan.Socotwillbelargeandpositive(pointA).Wheniscloseto90thevalueoftanisverylargeandpositive,andsocotwillbeverysmall(pointB).Inthiswaywecanobtainthegraphshownin www.mathcentre.ac.uk6c\rmathcentre2009 Figure7.Becausethetangentgraphisperiodicwithperiod,sotooisthegraphofcot. cot90180270360 µ µ Figure7.Agraphofcot.Insummary,wehavenowmetthethreenewtrigonometricfunctionscosec,secandcotandobtainedtheirgraphsfromknowledgeoftherelatedfunctionssin,cosandtan.Exercises1.Usethevaluesofthetrigonometricrationsofthespecialangles30o,45oand60otode-terminethefollowingwithoutusingacalculatora)cot45ob)cosec30oc)sec60od)cosec245oe)cot260of)sec230og)cot315oh)cosec(30o)i)sec240o2.Findallthesolutionsofeachofthefollowingequationsintherangestated(giveyouranswersto1decimalplace)(a)cot=0:2with0o360o(b)cosec=4with0o180o(c)cosec=4with0o360o(d)cosec=4with180o180o(e)sec=4with0o180o(f)sec=4with0o360o(g)sec=4with180o180o(h)cot=0:5with0o360o(i)cosec=0:5with0o360o(j)sec=0:5with0o360o3.Determinewhethereachofthefollowingstatementsistrueorfalse(a)cotisperiodicwithperiod180o.(b)cosecisperiodicwithperiod180o. www.mathcentre.ac.uk7c\rmathcentre2009 (c)Sincethegraphofcosiscontinuous,thegraphofseciscontinuous.(d)cosecnevertakesavaluelessthan1inmagnitude.(e)cottakesallvalues,Answers1.a)1b)2c)2d)2e)1 3f)4 3g)-1h)-2i)-22.a)78.7o,258.7ob)14.5o,165.5oc)14.5o,165.5od)14.5o,165.5oe)75.5of)75.5o,284.5og)75.5o,-75.5oh)63.4o,243.4oi)Nosolutionsj)Nosolutions3.a)Trueb)Falsec)Falsed)Truee)True www.mathcentre.ac.uk8c\rmathcentre2009