4. Circular measure

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1、4.Circularmeasure•understandthedefinitionofaradian,andusetherelationshipbetweenradiansanddegrees;•usetheformulaes=rθandA=r2θ/2insolvingproblemsconcerningthearclengthandsectorareaofacircle.1.Thediagramshowsthecircularcross-sectionofauniformcylindricallogwithcentreO

2、andradius20cm.ThepointsA,XandBlieonthecircumferenceofthecross-sectionandAB=32cm.(i)ShowthatangleAOB=1.855radians,correctto3decimalplaces.[2](ii)FindtheareaofthesectorAXBO.[2]ThesectionAXBCD,whereABCDisarectanglewithAD=18cm,isremoved.(iii)Findtheareaofthenewcross-s

3、ection(shownshadedinthediagram).[3](02may7)2.Inthediagram,OPQisasectorofacircle,centreOandradiusrcm.AngleQOP=θradians.ThetangenttothecircleatQmeetsOPextendedatR.12(i)Showthatthearea,Acm2,oftheshadedregionisgivenbyAr(tan).[2]2(ii)Inthecasewhereθ=0.8andr=15,eval

4、uatethelengthoftheperimeteroftheshadedregion.[4](02nov3)3ThediagramshowsasemicircleABCwithcentreOandradius8cm.AngleAOB=θradians.(i)Inthecasewhereθ=1,calculatetheareaofthesectorBOC.[3](ii)FindthevalueofθforwhichtheperimeterofsectorAOBisonehalfoftheperimeterofsector

5、BOC.[3](iii)Inthecasewhereθ=π/3,showthattheexactlengthoftheperimeteroftriangleABCis(24+8√3)cm.[3](03may9)4.ThediagramshowsthesectorOPQofacirclewithcentreOandradiusrcm.TheanglePOQisθradiansandtheperimeterofthesectoris20cm.20(i)Showthat2.[2]r(ii)Henceexpressthear

6、eaofthesectorintermsofr.[2](iii)Inthecasewherer=8,findthelengthofthechordPQ.[3](03nov6)5.Inthediagram,OCDisanisoscelestrianglewithOC=OD=10cmandangleCOD=0.8radians.ThepointsAandB,onOCandODrespectively,arejoinedbyanarcofacirclewithcentreOandradius6cm.Find(i)theareao

7、ftheshadedregion,[3](ii)theperimeteroftheshadedregion.[4](04may5)6.Inthediagram,ACisanarcofacircle,centreOandradius6cm.ThelineBCisperpendiculartoOCandOABisastraightline.AngleAOC=π/3radians.Findtheareaoftheshadedregion,givingyouranswerintermsofπand√3.[5](04nov3)7.I

8、nthediagram,ABCisasemicircle,centreOandradius9cm.ThelineBDisperpendiculartothediameterACandangleAOB=2.4radians.(i)ShowthatBD=6.08cm,correctto3significan