Introduction to Trigonometry

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Transcript of Introduction to Trigonometry

1Assignment lIntrigonometry,wedealwithrelationsbetweenthesidesandanglesofa triangle.lRatiosofthesidesofarightangledtrianglewithrespecttoitsacuteangles,are calledtrigonometricratiosoftheangle.lForA,ACisthebase,BCtheperpendicularandABisthehypotenuse.ForB, BCisthebase,ACtheperpendicularandABisthehypotenuse.lSixtrigonometricalratios (i)Sine = PerpendicularHypotenuse

yr .

Sine is written as sin . (ii)Cosine = BaseHypotenuse xr .

Cosine is written as cos . (iii)Tangent = PerpendicularBase

yx.

Tangent is written as tan . (iv)Cotangent = BasePerpendicular xy. Cotangent is written as cot . (v)Secant = HypotenuseBase

rx.

Secant is written as sec . (vi)Cosecant = HypotenusePerpendicular ry. Cosecant is written as cosec .lRelationsbetweentrigonometricratios(a)Reciprocalrelations (i)cosec

1sin orsin

1cosec orsin cosec=1 (ii)sec

1cos orcos

1sec orcos sec=1 (iii)cot

1tan ortan

1cot ortan cot=1(b)Quotientrelations (i)tan

sin cos (ii)cot

cos sinlsinAisasymbolwhichdenotestheratio perpendicularhypotenuse. Itdoesnotmeantheproductofsinand A, i.e.,sinAsinA.InfactsinseparatedfromAhasnomeaning.Similarinterpretationsfollowforother trigonometricratios.lTableofvaluesofvarioustrigonometricratiosof0,30,45,60and90. INTRODUCTION TO TRIGONOMETRYIMPORTANTTERMS,DEFINITIONS ANDRESULTS2T-030456090ratios sin0 12 12 32 1lTrigonometricratiosofcomplementaryangles(i)sin(90)=cos,cos(90)=sin(ii)tan(90)=cot,cot(90)=tan(iii)sec(90 )=cosec,cosec(90)=seclTrigonometricIdentities (a)Anequationinvolvingtrigonometric ratiosofanangle(say)issaidtobe atrigonometricidentity,ifitissatisfed forallvaluesofforwhichthegiven trigonometricratiosaredefned. (b)Some i mpor t a nt t r i gonome t r i c identities:(i)sin2+cos2=1 orsin2=1cos2 orcos2=1sin2 cos1 32 12 12 0 tan0 13 1 3Notdefned cotNotdefned 31 13 0 sec1 23 22Notdefned cosecNotdefned2 2 23 1Studentsmayfndeasiertomemorizethefrstrow(valuesofsineratio)assin030456090 04 14 24 34 44 =0= 12 = 12 = 32=1 (ii)sec2tan2=1 or1+tan2=sec2 ortan2=sec21 (iii)cosec2cot2=1 orcosec2=1+cot2 orcot2=cosec21(c)Thefollowingstepsshouldbekeptinmindwhile provingtrigonometricidentities:(i)Start with more complicated side of the identity andproveitequaltotheotherside.(ii)Iftheidentitycontainssine,cosineandother trigonometricratios,thenexpressalltheratios intermsofsineandcosine. (iii)Ifonesideofanidentitycannotbeeasily reducedtotheothersidevalue,thensimplify bothsidesandprovethemidenticallyequal. (iv)Whileprovingidentities,nevertransferterms fromonesidetoanother.3 1.IfcosA= 45, thenthevalueoftanAis : (a) 35 (b) 34 (c) 43 (d) 53 2.Ifsin= ab, thencosisequalto: (a) bb a2 2 (b) ba (c) b ab2 2 (d) ab a2 23.ThevalueoftanAisalwayslessthan1.(a)false (b)true (c)sometimestrue,sometimesfalse (d)noneoftheabove4.Maximumvalueofsinis:(a)morethan1(b)lessthan1 (c)equalto1(d)noneofthese5.Minimumvalueofsin,whereisacute,is :(a)zero(b)morethan1(c)equalto1(d)lessthan16.If4tan=3,then 44sin cossin cos +j(,\,(isequal to: (a) 23 (b) 13 (c) 12 (d) 347.Ifisanacuteanglesuchthatsec2=3,then thevalueof tan costan cos2 22 2 ececis :(a) 47(b) 37(c) 27(d) 17 8.sin= 43 forsomeangle,is:(a)true (b)false(c)itisnotpossibletosayanythingaboutit defnitely(d)neither(a)nor(b)SUMMATIVEASSESSMENTMULTIPLECHOICEQUESTIONS[1Mark]A.ImportantQuestions 9.Ifcot= 43, thencos2sin2isequalto: (a) 725 (b)1(c) 725 (d) 425 10.Ifsin A= 12, thenthevalueofcot Ais: (a)3 (b) 13 (c) 32 (d)1 11.Ifa=btan,then a ba bsin cossin cos + isequalto: (a) a ba b2 22 2+ (b) a ba +b2 22 2- (c) a ba b+ (d) a ba b+ 12.Ifsin= 35 , thenthevalueof(tan+sec)2 is equalto: (a)1(b) 12 (c)2(d)2 13. 1 451 4522 + sinsin isequalto:

(a)cos60(b)sin60(c)tan30(d)sin30 14.Thevalueof(sin30+cos30)(sin60 +cos60)is:(a)1(b)0(c)1(d)2 15.Thevalueof(sin45+cos45)is: (a) 12 (b)2 (c) 32 (d)1 16.Ifxtan45.cos60=sin60.cot60,thenxis equalto:(a)1(b) 3(c) 12(d) 12 17.Thevalueof tancos3060is:(a) 12(b) 13(c)3 (d)1 18.Thevalueof sin 4545 cosecis:(a)1(b) 12 (c) 2(d)noneofthese4 19.Thevalueof(sin45cos30+cos45sin30) is: (a) 3 12+ (b) 32 (c) 3 12 2+ (d) 3 12 2 20.Thevalueof(sin30cos60+cos30sin60) is: (a)sin90(b)cos90(c)sin0(d)cos30 21. 1 602 sin isequalto: (a)sin60(b)sin30(c)sin90(d)sin0 22.Thevalueof3sin30 4sin330is:(a)1(b)0(c)2(d) 12 23.Thevalueof sin1872 cos is: (a) 1 (b) 0 (c) 1(d) 12 24. cos48sin42is: (a) 1 (b) 0 (c) 1(d) 12 25.Thevalueoftan80.tan75.tan15.tan10 is:(a)1(b)0 (c)1(d)noneofthese 26.Thevalueof tan 2664 cot is:(a)0(b)1 (c)1(d)noneofthese 27.cosec31sec59isequalto: (a)0(b)1(c)1(d) 12 28.Thevalueof(tan2tan4tan6...tan88)is :(a)1(b)0 (c)2(d)notdefned 29.tan(40+)cot(40)isequalto : (a)1(b)0(c)2(d) 1230. Thevalueofsin(50+)cos(40)is:(a)1(b)2(c) 12(d)0 31.Thevalueoftheexpressioncosec(75+)sec (15)tan(55+)+cot(35)is: (a)1(b)0(c)1(d) 32 32.sin(45+)cos(45)isequalto:(a)2cosec(b)0 (c)sin(d)1 33.9sec29tan2isequalto:(a)1(b)9(c)8(d)0 34.IfsinA= 817 andAisacute,thencotAisequal to: (a) 158 (b) 1517 (c) 815 (d) 178 35.(cosec272tan218)isequalto: (a)0(b)1 (c) 32 (d)noneofthese 36.Ifx=sec+tan,thentanisequalto: (a) xx21 + (b) xx21 (c) xx2142+ (d) xx212 37.tan2 sin2 isequalto:(a)tan2 sin2 (b)tan2 +sin2 (c) tansin22(d)noneofthese 38.Ifcossin=1,thenthevalueofcos+ sin isequalto:(a)4(b)3(c)2(d)1 39. 1122++tancot isequalto:(a)sec2(b)1(c)cot2(d)tan2 40.(sec210cot280)isequalto: (a)1(b)0(c)2(d) 12 41.Thevalueof 11+coscosis:(a)cotcosec(b)cosec+cot (c)cosec2+cot2 (d)cot+cosec2 42. sincos 1+ isequalto: (a) 1+ cossin (b) 1cotsin (c) 1cossin (d) 1sincos 43.Ifx=acosandy=bsin,thenb2x2+a2y2 isequalto: (a)a2b2(b)ab(c)a4b4(d)a2+b25 44.( sin )( sin ) 1 1 + isequalto:(a)sin(b)sin2 (c)cos2(d)cos 45.Thevalueoftheexpression sin sincos cossin cos sin2 22 2222 6822 6863 63 27 + + + + ,,]]]is:(a)2(b)1 (c)0(d)noneofthese 46.Ifcos sin 9 and9 90 < , thenthevalueof tan5 is:(a)0(b)1 (c) 3(d)cannotbedeterminedB.QuestionsFromCBSEExaminationPapers 1.Inthegivenfgure,ACB=90,BDC=90, CD4cm,BD=3cm,AC12cm,cos Asin A isequalto: [2010(T-I)]A CBD (a) 512 (b) 513 (c) 712 (d) 7132.IfcotA 125 , thenthevalueof (sinA+cosA)cosecAis: [2010(T-I)] (a) 135 (b) 175 (c) 145 (d)13.cos1,cos2,cos3,........cos180isequal to:[2010(T-I)](a)1(b)0(c)1/2(d)14.5cosec2-5cot2isequalto:[2010(T-I)](a)5(b)1(c)0(d)55.Ifsin=cos,thenvalueofis:[2010(T-I)](a)0(b)45(c)30(d)906.9sec29tan2isequalto:[2010(T-I)](a)1(b)1(c)9(d)97.Ifsin+sin2=1,thevalueof(cos2+cos4) is:[2010(T-I)](a)3(b)2(c)1(d)08.In the fgure, if D is the mid-point of BC, the value of cotcotyxis:[2010(T-I)]DBACxy (a)2(b) 14 (c) 13 (d) 129.Ifcosec 32, then2(cosec2+cot2)is: [2010(T-I)](a)3(b)7(c)9(d)5 10.Inthefgure,ifPS=14cm,thevalueoftanais equalto:[2010(T-I)]PR13cmQSa5 cmT (a) 43 (b) 143 (c) 53 (d) 133 11.Ifx=3sec21,y=tan22,thenx3yis equalto:[2010(T-I)](a)3(b)4(c)8(d)5 12.(secA+tanA)(1sinA)isequalto:[2010(T-I)](a)secA(b)tanA(c)sinA(d)cosA 13.Ifsectan= 13, thevalueof(sec+tan) is: [2010(T-I)](a)1(b)2(c)3(d)4 14.Thevalueof cotsin cos4530 60 + isequalto: [2010(T-I)] (a)1(b) 12 (c) 23 (d) 12 15.Ifcos ; 3320 20 < < , thenthevalueof is: [2010(T-I)](a)15(b)10(c)0(d)12 16.ABCisarightangledatA,thevalueof tanBtanCis:[2010(T-I)](a)0(b)1 (c)1(d)noneofthese6 17.Ifsin , 13 then the value of 2 cot2 + 2 is equal to:[2010(T-I)](a)6(b)9(c)4(d)18 18.Thevalueoftan1.tan2.tan3........tan89 is:[2010(T-I)] (a)0(b)1(c)2(d) 12 19.If sin( ) A B 12 and cos( ) , A B + 12 thenthe valueofBis: [2010(T-I)](a)45(b)60(c)15(d)0 20.Valueof(1+tan+sec)(1+cot-cosec) is:[2010(T-I)](a)1(b)1(c)2(d)4 21.Thevalueof[sin220+sin270tan245]is:[2010(T-I)](a)0(b)1(c)2(d)1 22.Giventhat sin , A 12 and cos , B 12 thenthe valueof(A+B)is:[2010(T-I)](a)30(b)45(c)75(d)15 23.Thevalueof coscotsinAAA +j(,\,(is: [2010(T-I)](a)cotA(b)2sinA(c)2cosA (d)secA 24.Iftan2A=cot(A18),thenthevalueofA is:[2010(T-I)](a)18(b)36(c)24(d)27 25.ExpressionofsinAintermsofcotAis:[2010(T-I)] (a) 12+ cotcotAA (b) 112 cot A (c) 112+ cot A (d) 12cotcotAA 26.If A is an acute angle in a right ABC, right angled atB,thenthevalueofsinA+cosAis:[2010(T-I)](a)equaltoone(b)greaterthanone (c)lessthanone(d)equaltotwo 27.If cos ( + ) = 0, then sin ( ) can be reduced to:[2010(T-I)](a)cos(b)cos2(c)sin(d)sin2 28.Inthefgure,ifDismidpointofBC,thenthe valueof tantanxyis:[2010(T-I)]DBACx y(a)4(b)3(c)2(d)1 29.If cosec cot ,13 the value of (cosec + cot ) is: [2010(T-I)](a)1(b)2(c)3(d)4 30.Ifsin=cos,thenthevalueofcosecis:[2010(T-I)] (a)2(b)1(c) 23 (d) 2 31.Insin3=cos(26),where3and(26) areacuteangles,thenvalueofis:[2010(T-I)](a)30(b)29(c)27(d)26 32.If sin 12 andisacute,then(3cos4cos3) isequalto: [2010(T-I)] (a)0(b) 12 (c) 16 (d)1 33.If sec A B cosec127, then(A+B)isequal to: [2010(T-I)](a)0(b)90(c)90 34.If cotcot, AA+ 11 thevalueof cotcot221AA+ is: [2010(T-I)](a)1(b)2(c)1(d)2 35.Ifsec+tan=x,thentanis:[2010(T-I)] (a) xx21 + (b) xx21 (c) xx212+ (d) xx212 36.If 2 2 3 sin , thenthevalueofis: [2010(T-I)](a)90(b)30(c)45(d)60 37.IfxcosA=1andtanA=y,thenx2 y2 isequal to:[2010(T-I)](a)tanA(b)1(c)0(d)tanA 38.[cos4 Asin4 A]isequalto:[2010(T-I)](a)2cos2 A+1(b)2cos2 A1 (c)2sin2 A1(d)2sin2 A+1 39.Thevalueoftheexpression[(sec21) (1cosec2 )]is:[2010(T-I)] (a)1(b)1(c)0(d) 127 40.If ( ) A B 13 and sin , A 12 thenthevalue ofBis: [2010(T-I)](a)45(b)60(c)0(d)15 41.InABCrightangledatB,tanA=1,thevalue of2sinAcosAis:[2010(T-I)](a)1(b)2(c)3(d)1 42.If2 60 1 sin ( ) , thenthevalueofis:[2010(T-I)](a)45(b)15(c)60(d)30 43.sin(60+)cos(30)isequalto:[2010(T-I)](a)2cos(b)2sin(c)0(d)1 44.Giventhatcos 12, thevalueof 212sectan + is: [2010(T-I)] (a)1(b)2(c) 12 (d)0 45.Inthefigure,AD=3cm,BD=4cmand CB=12cm,thentanequals:[2010(T-I)]BCAD (a) 34 (b) 512 (c) 43 (d) 125 46.Ifcot 78, then the value of ( cos )( sin )( cos )( sin )1111++ is: [2010(T-I)] (a) 4964 (b) 87 (c) 6449 (d) 78 47.Thevalueofsincos(90)+cos sin(90)is:[2010(T-I)](a)1(b)0(c)2(d)1 48.Iftan=cot,thenthevalueofsecis:[2010(T-I)] (a)2(b)1(c) 23 (d) 2 49.IfcosA+cos2A=1,thensin2A+sin4Ais:[2010(T-I)](a)1(b)0(c)1(d)2 50.Fromthefgure,thevalueofcosecA+cotA is:[2010(T-I)] CA Babc