The addition formulae

Transcription

1 The addition formulae mc-ty-addnformulae There are six so-called addition formulae often needed in the solution of trigonometric problems. Inthisunitwestartwithoneandderiveasecondfromthat.Thenwetakeanotheroneasgiven, andderiveasecondonefromthat. Finallyweusethesefourtohelpusderivethefinaltwo. This exercise will improve your familiarity and confidence in working with the addition formulae. The proofs of the formulae are left as structured exercises for you to complete. In order to master the techniques explained here it is vital that you undertake the practice exercises provided. Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: work with the six addition formulae Contents 1. Introduction. Thefirsttwoadditionformulae: sin(a ± B) 3. Thesecondtwoadditionformulae: cos(a ± B) 3 4. Derivingthetwoformulaefor tan(a ± B) 5 5. Examplesoftheuseoftheformulae c mathcentre 009

2 1. Introduction There are six so-called addition formulae often needed in the solution of trigonometric problems. Inthisunitwestartwithoneandderiveasecondfromthat.Thenwetakeanotheroneasgiven, andderiveasecondonefromthat.andthenwearegoingtousethesefourtohelpusderivethe final two. This exercise will improve your familiarity and confidence in working with the addition formulae.. The first two addition formulae: sin(a ± B) Theformulawearegoingtostartwithis Thisiscalledanadditionformulabecauseofthesum A + Bappearingtheformula. Notethat itenablesustoexpressthesineofthesumoftwoanglesintermsofthesinesandcosinesof the individual angles. Wenowwanttolookat sin(a B). Wecanobtainaformulafor sin(a B)byreplacingthe Bintheformulafor sin(a + B)by B.Then sin(a B) sin A cos( B) + cosasin( B) Wenowusethefollowingimportantfacts: cos( B) cosb,but sin( B) sin B.Then This is the second of our addition formulae. sin(a B) sin A cosb cos A sin B sin(a B) sin A cosb cosasin B c mathcentre 009

3 Exercise 1 S 1 T R B O A P Q 1.ByusingrightangledtriangleOSR,inwhichthelengthofOSequals1,determinethe lengthoforintermsofangleb..byusingtheanswerofpart1andrightangledtriangleorqdeterminethelengthofqr intermsofanglesaandb. 3.ByusingtheanswerofpartdeterminethelengthofPT. 4.Whatis TRO? 5.Whatis TRS? 6.Whatis RST? 7.ByusingrightangledtriangleOSRdeterminethelengthofRS. 8.Byusingtheanswerofpart7andrightangledtriangleRSTdeterminethelengthofTS. 9.Byusingtheanswersofparts3and8determinethelengthofPS. 10. Byusingtheanswerofpart9andrightangledtriangleOSPdetermine sin(a + B). 3. The second two addition formulae: cos(a ± B) Thistime,theadditionformulawearegoingtostartwithis cos(a + B) cosacosb sin A sin B cos(a + B) cosacosb sin A sin B 3 c mathcentre 009

4 Wewanttousethistoderiveanotherformulafor cos(a B).Todothis,asbefore,wereplace Bwith B.Thisgives cos(a B) cosacos( B) sin A sin( B) But cos( B) cosband sin( B) sin B,andso cos(a B) cos A cosb + sin A sin B cos(a B) cosacosb + sin A sin B So we ve now got four addition formulae. We will summarise them all here: sin(a B) sin A cosb cosasin B cos(a + B) cosacosb sin A sin B cos(a B) cosacosb + sin A sin B Exercise ReferbacktothefigureinExercise1.Useasimilarstrategytothatofexercise1todetermine lengthspq(tr),oqandhenceop.fromthisdetermine cos(a + B). 4 c mathcentre 009

5 4. Deriving the two formulae for tan(a ± B) Fromthefourformulaewehaveseenalready,itispossibletoderivetwomoreformulae.Wecan deriveaformulafor tan(a + B)fromtheearlierformulaebynotingthat Then, sin(a + B) cos(a + B) sin(a + B) cos(a + B) sin A cosb + cosasin B cosacosb sin A sin B Thisresultgives tan(a + B)intermsofsinesandcosines.Wenowlookathowwecanwriteit directlyintermsof tan Aand tan B. Wedothisbydividingeveryterm,bothtopandbottom, ontheright-handsideby cosacosb.thisproduces sin A cos B cos A cosb cosacosb cosacosb Cancelling common factors where possible produces so that + cosa sin B cos A cos B sin A sin B cosacosb sin A cos B cos A cosb + cosa sin B cos A cos B cosa cosb sin A sin B cos A cosb cosacosb tana + tanb 1 tanatan B Wecandothesamewith tan(a B)whichwouldproduce tan(a B) tana tan B 1 + tanatan B tan A + tan B 1 tan A tanb tan A tanb tan(a B) 1 + tan A tanb 5 c mathcentre 009

6 5. Examples of the use of the formulae Let shavealookatsomefairlytypicalexamplesofwhenweneedtousetheadditionformulae. Example Supposeweknowthat sin A 3 5 andthat cosb where Aand Bareacuteangles.Suppose 5 13 wewanttousethisinformationtofind sin(a+b)and cos(a B).Beforewecanusetheaddition formulaeweneedtoknowexpressionsfor cosaand sin B.Wecanfindthesebyreferringtothe right-angled triangle in Figure A Figure1.Aright-angledtriangleconstructedfromthegiveninformation: sin A 3 5 UsingPythagoras theoremwecandeducethatthelengthofthethirdsideis4asshownin Figure.Hence cosa A Figure.Fromtheright-angledtriangle, cos A 4 5 Similarly, giventhat cosb 5,thenbyreferencetothetriangleinFigure3andbyusing 13 Pythagoras theoremwecandeducethat sin B B Figure3.Fromthetriangle sin B Wearenowinapositiontousetheadditionformulae: cos(a B) cosacosb + sin A sin B Thisisonewayinwhichtheformulaecanbeused. 6 c mathcentre 009

7 Example Supposeweareaskedtofindanexpressionfor sin 75,notbyusingacalculatorbutbyusinga combinationofotherknownquantities. Notethatwecanrewrite sin 75 as sin( )and thenuseanadditionformula.wehavespecificallychosenthevalues 45 and 30 becauseofthe standardresultsthat sin 45 cos 45 1, sin 30 1 and cos 30 3.Then sin( ) sin 45 cos 30 + cos 45 sin Example Supposewewishtofindanexpressionfor tan15 usingknownresults.notethat andalsothat tan 60 3and tan45 1. tan15 tan(60 45 ) tan60 tan tan60 tan Itwouldbemoreusualtotidythisresultuptoavoidleavingarootinthedenominator. This canbedonebymultiplyingtopandbottombythesamequantity,asfollows: 3 1 ( 3 1) ( 3 1) c mathcentre 009

8 Example In this Example we use an addition formula to simplify an expression. Supposewehave sin(90 + A)andwewanttowriteitinadifferentform. Wecanusethefirstadditionformulaasfollows: sin(90 + A) sin 90 cos A + cos 90 sin A cosa since sin 90 1and cos 90 0.So sin(90 + A)canbewritteninthesimplerform cosa. Example Supposewewishtosimplify cos(180 A). since cos 180 1and sin cos(180 A) cos 180 cosa + sin 180 sin A cos A So we can see that these addition formulae help us to simplify quite complicated expressions. Exercise 3 1.Verifyeachofthethreeadditionformulae(i.e. for sin(a + B), cos(a + B), tan(a + B)) for the cases: a) A 60, B 30 and b) A 45, B 45..Verifyeachofthethreesubtractionformulae(i.e.for sin(a B), cos(a B), tan(a B)) for the cases: a) A 90, B 60 and b) A 90, B Angles A, Band C areacuteanglessuchthat sin A 0.1, cosb 0.4, sin C 0.7. Withoutfindingangles A, Band C,usetheadditionformulaetocalculate,todecimal places, a) sin(a + B) b) cos(b C) c) sin(c A) d) cos(a + C) e) tan(b A) f) tan(c + B) [Hint:Workto4decimalplaceswhenfinding cosa, tan A,etc.] 4.Byfindingtheangles A, Band Cinquestion3verifyyouranswers. Answers Exercise 1 1. cosb. sin A cosb 3. sin A cosb 4. A 5.90 o A 6. A 7. sin B 8. cosasin B 9. sin A cosb + cos A sin B 10. sin A cosb + cosasin B Exercise PQTRRS sin A sin A sin B OQOR cosa cosacosb OPOQ-PQcosAcosB sin A sin B cos(a + B) OP cosacosb sin A sin B Exercise 3 3. a) 0.95 b) 0.93 c) 0.63 d) 0.64 e) 1.78 f) c mathcentre 009