mc-TY-radians-2009-1 - PDF document

Radians mc-TY-radians-2009-1 Atschoolweusuallylearntomeasureanangleindegrees.However,thereareotherwaysofmeasuringanangle.Onethatwearegoingtohavealookathereismeasuringanglesinunitscalledradians.Inmanyscienticandengineeringcalculationsradiansareusedinpreferencetodegrees.Inordertomasterthetechniquesexplainedhereitisvitalthatyouundertakeplentyofpracticeexercisessothattheybecomesecondnature.Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto:useradianstomeasureanglesconvertanglesinradianstoanglesindegreesandviceversandthelengthofanarcofacirclendtheareaofasectorofacirclendtheareaofasegmentofacircleContents1.Introduction22.Denitionofaradian23.Arclength34.Equivalentanglesindegreesandradians45.Findinganarclengthwhentheangleisgivenindegrees56.Theareaofasectorofacircle67.Miscellaneousexamples6 www.mathcentre.ac.uk1c\rmathcentre2009 1.IntroductionAtschoolweusuallylearntomeasureanangleindegrees.Wearewellawarethatafullrotationis360asshowninFigure1. 360 Figure1.Afullrotationis360.However,thereareotherwaysofmeasuringanangle.Onewaythatwearegoingtohavealookathereismeasuringanglesinunitscalledradians.Inmanyscienticandengineeringcalculationsradiansareusedinpreferencetodegrees.2.DenitionofaradianConsideracircleofradiusrasshowninFigure2. rr 1 rad Figure2.Thearcshownhasalengthchosentoequaltheradius;theangleisthen1radian.InFigure2wehavehighlightedpartofthecircumferenceofthecirclechosentohavethesamelengthastheradius.Theangleatthecentre,soformed,is1radian. KeyPointAnangleofoneradianissubtendedbyanarchavingthesamelengthastheradiusasshowninFigure2. www.mathcentre.ac.uk2c\rmathcentre2009 3.ArclengthWewillnowusethisdenitiontondaformulaforthelengthofanarbitraryarc.Wehaveseenthatanangleof1radianissubtendedbyanarcoflengthrasillustratedintheleft-mostdiagraminFigure3.Byextensionanangleof2radianswillbesubtendedbyanarcoflength2r,asshown. rr r 1r11 1 2rr Figure3.Anangleof2radiansissubtendedbyanarcoflength2r.NotefromthesediagramsthatthelengthofthearcisalwaysgivenbytheangleinradianstheradiusInthegeneralcase,thelengths,ofanarbitraryarcwhichsubtendsanangleisrasillustratedinFigure4. rrs µ Figure4.Thearclengths,isgivenbyrThisgivesusawayofcalculatingthearclengthwhenweknowtheangleatthecentreofthecircleandweknowitsradius. KeyPointarclengths=r(note:mustbemeasuredinradians) Exercise1Determinetheangle(inradians)subtendedatthecentreofacircleofradius3cmbyeachofthefollowingarcs:a)arcoflength6cmb)arcoflength3cmc)arcoflength1.5cmd)arcoflength6cm www.mathcentre.ac.uk3c\rmathcentre2009 4.EquivalentanglesindegreesandinradiansWeknowthatthearclengthforafullcircleisthesameasitscircumference,2r.Wealsoknowthatthearclength=r.Soforafullcircle2r=rthatis=2Inotherwords,whenweareworkinginradians,theangleinafullcircleis2radians,inotherwords360=2radiansThisenablesustohaveasetofequivalencesbetweendegreesandradians. KeyPoint360=2radiansfromwhichitfollowsthat180=radians90= 2radians45= 4radians60= 3radians30= 6radians TheKeyPointgivesalistofanglesmeasuredindegreesontheleftandtheequivalentlistinradiansontheright.Itisimportantinmathematicalworkthatyourecordcorrectlytheunitofmeasureyouareusing.Anotherusefulrelationshipisgivenasfollows:radians=180so1radian=180 degrees=57:296(3d.p.)So1radianisjustover57.Somenotation.Therearevariousconventionsusedtodenoteradians.Somebooksandsometeachersuse`rads'asin2rads.Othersuseasmallcasin2c.Someothersusenosymbolatallandassumethatradiansarebeingused.Whenanangleisexpressedasamultipleof,forexampleasintheexpressionsin3 2,itistakenasreadthattheangleisbeingmeasuredinradians. www.mathcentre.ac.uk4c\rmathcentre2009 Exercise21.Wheneachofthefollowinganglesisconvertedfromdegreestoradianstheanswercanbeexpressedasamultipleof(notethatitmaybeafractionalmultiple).Ineachcasestatethemultiple(e.gforananswerof4 5themultipleis4 5).a)90ob)360oc)60od)45oe)120of)15og)135oh)270o2.Converteachofthefollowinganglesfromradianstodegrees.a) 2radiansb)3 4radiansc)radiansd) 6radianse)5radiansf)4 5radiansg)7 4radiansh) 10radians3.Converteachofthefollowinganglesfromdegreestoradiansgivingyouranswerto2decimalplaces.a)17ob)49oc)124od)200o4.Converteachofthefollowinganglesfromradianstodegrees,givingyouranswerto1decimalplace.a)0.6radiansb)2.1radiansc)3.14radiansd)1radian5.FindinganarclengthwhentheangleisgivenindegreesWeknowthatifismeasuredinradians,thenthelengthofanarcisgivenbys=r.Supposeismeasuredindegrees.Weshallderiveanewformulaforthearclength. rrso µ Figure5.Inthiscircletheangleismeasuredindegrees.ReferringtoFigure5,theratioofthearclengthtothefullcircumferencewillbethesameastheratiooftheanglesubtendedbythearc,totheangleinafullcircle;thatiss 2r= 360So,whenismeasuredindegreeswecanusethefollowingformulaforarclength:s=2r 360Noticehowtheearlierformula,usedwhentheangleismeasuredinradians,ismuchsimpler. www.mathcentre.ac.uk5c\rmathcentre2009 6.TheareaofasectorofacircleAsectorofacirclewithangleisshownshadedinFigure6. rr µ Figure6.Theshadedareaisasectorofthecircle.Theratiooftheareaofthesectortotheareaofthefullcirclewillbethesameastheratiooftheangletotheangleinafullcircle.Thefullcirclehasarear2.Thereforeareaofsector areaoffullcircle= 2andsoareaofsector= 2r2=1 2r2 KeyPointareaofsector=1 2r2whenismeasuredinradians 7.MiscellaneousExamplesExampleConsiderthecircleshowninFigure7.Supposewewishtocalculatetheangle. 101025 µ Figure7.Calculatetheangle. www.mathcentre.ac.uk6c\rmathcentre2009 Weknowthearclengthandradius.Wecanusetheformulas=r.Substitutingthegivenvalues25=10andso=25 10=2:5radsWhatisthisangleindegrees?Weknowrads=180andso1rad=180 Itfollowsthat2:5rads=2:5180 =143:2ExampleRefertoFigure8.Supposewehaveacircleofradius10cmandanarcoflength15cm.Supposewewanttond(a)theangle,(b)theareaofthesectorOAB,(c)theareaoftheminorsegment(shaded). O AB 1010 15 µ Figure8.Theshadedareaiscalledtheminorsegment.(a)Usings=rwehave15=10andso=15 10=1:5c.(b)Usingtheformulafortheareaofthesector,A=1 2r2,wendarea=1 2r2=1 2(102)(1:5)=75cm2(c)WealreadyknowthattheareaofthesectorOABis75cm2.IfwecanworkouttheareaofthetriangleAOBwecanthendeterminetheareaoftheminorsegment.(Recalltheformulaefortheareaoftriangle,A=1 2absinC.)areaoftriangle=1 2r2sin=1 2102sin1:5=49:87cm2 www.mathcentre.ac.uk7c\rmathcentre2009 Thereforetheareaoftheminorsegmentis7549:87=25:13cm2(to2dp.)ExampleSupposewehaveanangleof120.Whatisthisangleinradians?Weknowthatrads=180andso 180rads=1then120= 180120radsThiscanbewrittenas2 3radians(=2.094radians).Exercise3Asectorofacircleisanareaboundedbytworadiiandanarc.Asectorhasanangleatthecentreofthecircle.Allthequestionsbelowrelatetoacirclewithradius5cm.1.Determinethelengthofthearc(correctto2decimalplaces)whentheangleatthecentreisa)1.2radiansb) 2radiansc)45o2.Calculatethearea(correctto2decimalplaces)ofeachofthethreesectorsinQuestion1.3.Asectorofthiscirclehasarea50cm2.Whatistheangle(inradians)atthecentreofthissector?AnswersExercise1a)2b)c)0.5d)2Exercise21.a)1 2b)2c)1 3d)1 4e)2 3f)1 12g)3 4h)3 22.a)90ob)135oc)180od)30oe)900of)144og)315oh)18o3.a)0.30radiansb)0.86radiansc)2.16radiansd)3.49radians4.a)15.3ob)120.3oc)179.9od)57.3oExercise31.a)6cmb)7.85cmc)3.93cm2.a)15cm2b)19.63cm2c)9.82cm23.4radians www.mathcentre.ac.uk8c\rmathcentre2009