Trig. Past Papers Unit 2 Outcome 3

1 Outcome3 WrittenQuestions1.[SQA]Solvetheequation3 cos2x +cosx = 1 intheinterval0 x [SQA]Solvetheequationcos2x +5 cosx 2=0 , 0 x< [SQA]Findtheexactsolutionsoftheequation4 sin2x=1 , 0 x< [SQA]Solvetheequation2 cos2x=12, for0 x [SQA]Solvetheequationcos2x +cosx =0 , 0 x< [SQA]Solve2 sin3x 1=0 for0 x [SQA](a) Showthat2 cos2x cos2x =1 3 sin2x .2(b) Hencesolvetheequation2 cos2x cos2x =2 sinx intheinterval0 x< [SQA]Solvetheequationsin2x +sinx =0 , 0 x< [SQA]Find,correcttoonedecimalplace,theva lueofxbetween180and270whichsatis estheequation3 cos(2x 40 ) 1=0 .510.[SQA](a) Writetheequationcos2q+8 cosq+9=0 intermsofcosqandshowthat,forcosq, it (b) Showthatthere are PSfragreplacementsOxyPage 1 Questionsmarked `[SQA]'c SQAAll othersc HigherStill NotesHigherMathematicsPSfragreplacements Oxy11.

2 [SQA]Iff(a) =6 sin2a cosa, expressf(a)intheformpcos2a+qcosa+ ,correcttothreedecimalplaces,theequation 6 sin2a cosa=5 for0 a [SQA]Findthevaluesoft, where 0<t<2p, forwhich4 cos 2t p4 [SQA]Solvetheequation2 sin 2x p6 =1 , 0 x< (a)[SQA]Solvetheequationsin2x cosx =0 intheinterval0 x (b) Thediagramshowspartsoftwotrigonometricgr aphs,y=sin2x andy=cosx .Useyoursolutionsin(a) y=cosx P9018015.[SQA]PSfragreplacementsOxy16.[S QA]Functionsfandgare de nedonsuitabledomainsbyf(x) =sin(x )andg(x) =2x.(a) Findexpressionsfor:(i)f(g(x));(ii)g(f(x) ).2(b) Solve2f(g(x)) =g(f(x))for0 x PSfragreplacementsOxyPage2 Questionsmarked `[SQA]'c SQAA llothersc HigherStillNotesHigherMathematicsPSfragr eplacementsOxy17.[SQA]PSfragreplacements Oxy18.

3 [SQA]Thediagramshowsthegraphofacosinefun ctionfrom0 top.(a) (b) Thelinewithequationy= p3p2p2 219.[SQA] PSfragreplacementsOxyPage3 Questionsmarked `[SQA]'c SQAA llothersc HigherStillNotesHigherMathematicsPSfragr eplacementsOxy20.[SQA]PSfragreplacements Oxy21.[SQA]PSfragreplacementsOxy22.[SQA] PSfragreplacementsOxyPage4 Questionsmarked `[SQA]'c SQAA llothersc HigherStillNotesHigherMathematicsPSfragr eplacementsOxy23.[SQA]PSfragreplacements Oxy24.[SQA]PSfragreplacementsOxy25.[SQA] PSfragreplacementsOxy26.[SQA]Ifcosq=45, 0 q<p2, ndtheexactvalueof(a) sin2q2(b) PSfragreplacementsOxyPage5 Questionsmarked `[SQA]'c SQAA llothersc HigherStillNotesHigherMathematicsPSfragr eplacementsOxy27.[SQA]Giventhattana=p113 , 0<a<p2, [SQA]GiventhatcosD=2p5and0<D<p2, [SQA]GiventhatsinA=34, where 0<A<p2, [SQA]ForacuteanglesPandQ, sinP=1213andsinQ= (P+Q) [SQA]Findtheexactvalueofsinq +sin(q +120 ) +cos(q +150 ).

4 332.[SQA]Ifx isanacuteanglesuchthattanx =43, showthattheexactvalueofsin(x +30 )is4p3+ [SQA]PSfragreplacementsOxy34.[SQA]AandBa re acuteanglessuchthattanA=34andtanB= (a) sin2A2(b) cos2A1(c) sin(2A+B). PSfragreplacementsOxyPage6 Questionsmarked `[SQA]'c SQAA llothersc HigherStillNotesHigherMathematicsPSfragr eplacementsOxy35.[SQA]Functionsf(x) =sinx,g(x) =cosxandh(x) =x+p4are de nedona suitablesetofrealnumbers.(a) Findexpressionsfor:(i)f(h(x));(ii)g(h(x) ).2(b)(i)Showthatf(h(x)) =1p2sinx+1p2cosx.(ii)Finda similarexpressionforg(h(x))andhencesolve theequationf(h(x)) g(h(x)) =1 for0 x [SQA]Onthecoordinatediagramshown,Aisthep oint(6, 8)andB is thepoint(12, 5). AngleAOC=pandangleCOB= (p+q).4 PSfragreplacementsOxyA(6, 8)CB(12, 5)pq37.

5 [SQA]IntriangleABC,showthattheexactvalue ofsin(a+b) [SQA] PSfragreplacementsOxyPage7 Questionsmarked `[SQA]'c SQAA llothersc HigherStillNotesHigherMathematicsPSfragr eplacementsOxy39.[SQA]PSfragreplacements Oxy[ENDOFWRITTENQUESTIONS] PSfragreplacementsOxyPage8 Questionsmarked `[SQA]'c SQAA llothersc HigherStillNotes