Trigonometry GDC
1a.[2marks]
Let
.Thediagrambelowshowspartofthegraphoff,for
.
ThegraphhasalocalmaximumatP(3,5),alocalminimumatQ(7,−5),andcrossesthex-axisatR.
Writedownthevalueof
(i) ;
(ii) .
1b.[2marks]
Findthevalueofb.
1c.[2marks]
Findthex-coordinateofR.
1d.[2marks]
Findthex-coordinateofR.
1
2a.[1mark]
Thefollowingdiagramshows
,whereRQ=9cm,
and
Find
.
2b.[3marks]
FindPR.
2c.[2marks]
Findtheareaof
.
3a.[2marks]
ConsiderthefollowingcirclewithcentreOandradius6.8cm.
ThelengthofthearcPQRis8.5cm.
Findthevalueof .
3b.[4marks]
Findtheareaoftheshadedregion.
2
.
4a.[4marks]
ConsiderthetriangleABC,whereAB=10,BC=7and
Findthetwopossiblevaluesof
=
.
.
4b.[2marks]
Hence,find
,giventhatitisacute.
5a.[3marks]
ThefollowingdiagramshowsatriangleABC.
TheareaoftriangleABCis
cm,AB
cm,AC
cmand
.
Find .
5b.[3marks]
FindBC.
6a.[1mark]
ThefollowingdiagramshowsacirclewithcentreOandradius cm.
PointsAandBareonthecircumferenceofthecircleand
3
radians.
ThepointCison[OA]suchthat
Showthat
radians.
.
6b.[7marks]
ThepointCison[OA]suchthat
Theareaoftheshadedregionis
radians.
cm.Findthevalueof .
7a.[3marks]
ThefollowingdiagramshowstriangleABC.
FindAC.
7b.[3marks]
Find
.
8a.[3marks]
Thepopulationofdeerinanenclosedgamereserveismodelledbythefunction
,where isinmonths,and
correspondsto1January2014.
Findthenumberofdeerinthereserveon1May2014.
8b.[2marks]
Findtherateofchangeofthedeerpopulationon1May2014.
8c.[1mark]
Interprettheanswertopart(i)withreferencetothedeerpopulationsizeon1May2014.
4
9a.[2marks]
Thefollowingdiagramshowsacirclewithcentre andradius
.
Thepoints ,
and
Findthelengthofthearc
lieonthecircumferenceofthecircle,and
radians.
.
9b.[2marks]
Findtheperimeteroftheshadedsector.
9c.[2marks]
Findtheareaoftheshadedsector.
10a.[4marks]
Intriangle
,
and
.Theareaofthetriangleis
.
Findthetwopossiblevaluesfor .
10b.[3marks]
Giventhat isobtuse,find
.
11a.[2marks]
Let
,for
.Thefollowingdiagramshowsthegraphof .
5
Thegraphhasamaximumat
andaminimumat
.
Writedownthevalueof .
11b.[2marks]
Find .
11c.[2marks]
Find .
11d.[2marks]
Solve
.
12a.[2marks]
Thefollowingdiagramshowsacirclewithcentre andradius cm.
6
Thepoints , and areonthecircumferenceofthecircle,and
Findthelengthofarc
radians.
.
12b.[3marks]
Find
.
12c[2marks]
Hence,findtheperimeteroftheshadedsegment
.
13a.[2marks]
Thefollowingdiagramshowspartofthegraphof
.
Thepoint
isamaximumpointandthepoint
Findthevalueof
;
13b.[2marks]
;
13c.[2marks]
.
14a.[2marks]
ThediagrambelowshowsacirclewithcentreOandradius8cm.
7
isaminimumpoint.
ThepointsA,B,C,D,EandFareonthecircle,and[AF]isadiameter.ThelengthofarcABCis6cm.
FindthesizeofangleAOC.
14b.[6marks]
Hencefindtheareaoftheshadedregion.
14c.[2marks]
TheareaofsectorOCDEis
.
FindthesizeofangleCOE.
14d.[5marks]
FindEF.
15a.[5marks]
Thediagrambelowshowsaplanforawindowintheshapeofatrapezium.
Threesidesofthewindoware
baseis ,where
long.Theanglebetweentheslopingsidesofthewindowandthe
.
8
Showthattheareaofthewindowisgivenby
.
15b.[4marks]
Zoewantsawindowtohaveanareaof
.Findthetwopossiblevaluesof .
15c.[7marks]
JohnwantstwowindowswhichhavethesameareaAbutdifferentvaluesof .
FindallpossiblevaluesforA.
16a.[3marks]
Thefollowingdiagramshowsacircularplayareaforchildren.
ThecirclehascentreOandaradiusof20m,andthepointsA,B,CandDlieonthecircle.AngleAOBis
1.5radians.
Findthelengthofthechord[AB].
16b.[2marks]
FindtheareaoftriangleAOB.
16c.[3marks]
AngleBOCis2.4radians.
FindthelengthofarcADC.
16d.[3marks]
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AngleBOCis2.4radians.
Findtheareaoftheshadedregion.
16e.[4marks]
AngleBOCis2.4radians.
Theshadedregionistobepaintedred.Redpaintissoldincanswhichcost
each.Onecancovers
.Howmuchdoesitcosttobuythepaint?
17a.[4marks]
ConsiderthefollowingcirclewithcentreOandradiusr.
ThepointsP,RandQareonthecircumference,
Usethecosineruletoshowthat
,for
.
17b.[5marks]
LetlbethelengthofthearcPRQ.
Giventhat
,findthevalueof .
17c.[4marks]
Considerthefunction
,for
.
(i)Sketchthegraphoff.
(ii)Writedowntherootof
.
17d.[3marks]
10
.
Usethegraphofftofindthevaluesof forwhich
.
18a.[3marks]
Thediagramshowsacircleofradius metres.ThepointsABCDlieonthecircumferenceofthecircle.
BC=
m,CD=
m,AD= m,
,and
FindAC.
18b.[5marks]
(i)Find
.
(ii)Hence,find
.
18c.[2marks]
FindtheareaoftriangleADC.
18d.[6marks]
(c)FindtheareaoftriangleADC.
(d)Henceorotherwise,findthetotalareaoftheshadedregions.
18e.[4marks]
Henceorotherwise,findthetotalareaoftheshadedregions.
19a.[2marks]
11
.
AFerriswheelwithdiameter
metresrotatesclockwiseataconstantspeed.Thewheelcompletes
rotationseveryhour.Thebottomofthewheelis
metresabovetheground.
Aseatstartsatthebottomofthewheel.
Findthemaximumheightabovethegroundoftheseat.
19b.[2marks]
Aftertminutes,theheight metresabovethegroundoftheseatisgivenby
(i)Showthattheperiodof is
minutes.
(ii)Writedowntheexactvalueof .
19c.[9marks]
(b)(i)Showthattheperiodof is
minutes.
(ii)Writedowntheexactvalueof .
(c)Findthevalueof .
(d)Sketchthegraphof ,for
.
19d.[3marks]
Findthevalueof .
19e.[4marks]
Sketchthegraphof ,for
.
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19f.[5marks]
Inonerotationofthewheel,findtheprobabilitythatarandomlyselectedseatisatleast
metres
abovetheground.
20a.[3marks]
Let
Sketchthegraphof .
20b.[5marks]
Findthevaluesof wherethefunctionisdecreasing.
20c.[3marks]
Thefunction canalsobewrittenintheform
,where
,and
,where
,and
.Findthevalueof ;
20d.[4marks]
Thefunction canalsobewrittenintheform
.Findthevalueof .
21a.[3marks]
Consideracirclewithcentre andradius cm.Triangle
isdrawnsuchthatitsverticesareon
thecircumferenceofthecircle.
13
cm,
Find
cmand
radians.
.
21b.[5marks]
Find
.
21c.[6marks]
Henceorotherwise,findthelengthofarc
.
14