Tringle formule m-ty-tringleformule-009-1 ommonmthemtilprolemistofindthenglesorlengthsofthesidesoftringlewhen some,utnotllofthesequntitiesreknown.itislsousefultoeletolultethere of tringle from some of this informtion. In this unit we will illustrte severl formule for doing this. In order to mster the tehniques explined here it is vitl tht you undertke the prtie exerises provided. fterredingthistext,nd/orviewingthevideotutorilonthistopi,youshouldeleto: solve tringles using the osine formule solve tringles using the sine formule findresoftringles ontents 1. Introdution. The osine formule 3 3. The sine formule 5 4. Someexmplesoftheuseoftheosinendsineformule 6 5. Thereoftringle 9 6. Summry 1 www.mthentre..uk 1 mthentre 009
1. Introdution onsidertringlesuhsthtshowninfigure1. Figure1.tringlewithsixpieesofinformtion:nglest,,nd ;sides, nd. Thereresixpieesofinformtionville: nglest, nd,ndthesides, nd. Thenglet isusullywritten,ndsoon.notiethtwelelthesidesordingtothe following onvention: side isoppositethengle side isoppositethengle side isoppositethengle Nowifwetkethreeofthesesixpieesofinformtionwewill(exeptintwospeilses)e le to drw unique tringle. Let s del first with the speil ses. The first speil se Thefirstspeilseiswhenweknowjustthethreengles. Thenhvingdrwnonetringle withthesengles,wendrwsmnymoretringlesswewish,llwiththesmeshpes theoriginl,utlrgerorsmller. llwillhvethesmenglesutthesizesofthetringles willedifferent. Wennotdefineuniquetringlewhenweknowjustthethreengles. This ehviour is illustrted in Figure where the orresponding ngles in the two tringles re the sme, ut lerly the tringles re of different sizes. Figure. Given just the three ngles we nnot onstrut unique tringle. www.mthentre..uk mthentre 009
The seond speil se Thereisseondspeilsewhereyifweregiventhreepieesofinformtionitisimpossile toonstrutuniquetringle. Supposeweregivenonengle, sy,ndthelengthsoftwo ofthesides.thissitutionisillustrtedinfigure3().thefirstgivensideismrked//.the seondgivensideismrked/;thisnepledintwodifferentlotionssshowninfigures 3) nd 3). onsequently it is impossile to onstrut unique tringle. () () () Figure3.Itisimpossiletodrwuniquetringlegivenonenglendtwosidelengths. prt from these two speil ses, if we re given three piees of informtion out the tringle wewilleletodrwituniquely. Therereformulefordoingthiswhihwedesrieinthe following setions.. The osine formule Wenusetheosineformulewhenthreesidesofthetringleregiven. osine formule Key Point When given three sides, we n find ngles from the following formule: os = + os = + os = + www.mthentre..uk 3 mthentre 009
The osine formule given ove n e rerrnged into the following forms: Key Point = + os = + os = + os Ifweonsidertheformul = + os,ndrefertofigure4wenotethtwen useittofindside whenweregiventwosides(nd )ndtheinludedngle. Figure4.Usingtheosineformuletofind ifweknowsides nd ndtheinludedngle. Similr oservtions n e mde of the other two formule. Sotherereinftsixosineformule,oneforehofthengles-tht sthreeltogether,nd oneforehofthesides,tht snotherthree.weonlyneedtolerntwoofthem,oneforthe ngle,oneforthesidendthenjustylethelettersthroughtofindtheothers. Exerise 1 Throughout ll exerises the stndrd tringle nottion(nmely side opposite ngle, et.) is used. 1.Findthelengthofthethirdside,to3deimlples, ndtheothertwongles, to1 deiml ple, in the following tringles () = 1, =, = 30 () = 3, = 4, = 50 () = 5, = 10, = 30 www.mthentre..uk 4 mthentre 009
.Findthengles(to1deimlple)inthefollowingtringles () =, = 3, = 4 () = 1, = 1, = 1.5 () =, =, = 3 3. The sine formule Wenusethesineformuletofindside,giventwosidesndnnglewhihisNOTinluded etween the two given sides. sin = where R is the rdius of the irumirle. sin = Key Point sin = R R Figure 5. The irumirle is the irle drwn through the three points of the tringle. Ristherdiusoftheirumirle-theirumirleistheirlethtwendrwthtwillgo throughllthepointsofthetringlesshowninfigure5. Tkingjustthefirstthreetermsintheformulewenrerrngethemtogive sin = sin ndwenusetheformuleinthisformswell. = sin www.mthentre..uk 5 mthentre 009
4. Some exmples of the use of the osine nd sine formule Exmple Supposeweregivenllthreesidesoftringle: = 5, = 7, = 10 We will use this informtion to determine ngle using the osine formul: os = + = 7 + 10 5 7 10 = = 14 140 49 + 100 5 140 1 14 = os 140 = 7.7 (1 d.p.) The remining ngles n e found y pplying the other osine formule. Exmple Supposeweregiventwosidesoftringlendnngle,sfollows = 10, = 5, = 10 It s not immeditely ovious wht informtion we hve een given. In the lst Exmple it ws veryler. Sowemkeskethtomrkouttheinformtionwehveeengivensshownin Figure 6. = 5 10 o = 10 Figure 6. The informtion given in the Exmple. FromtheFigurewendeduethtwehveeengivensidesndtheinludedngle.Wen usetheosineformultodeduethelengthofside. = + os = 10 + 5 10 5 os10 = 100 + 5 100 os 10 ( = 15 100 1 ) = 175 = 175 = 13. (3 s.f.) Nowthtwehveworkedoutthelengthofside,wehvethreesides.Weouldusetheosine formule to find out either one of the remining ngles. www.mthentre..uk 6 mthentre 009
Exmple Suppose we re given the following informtion: = 8, = 1, = 30 Notethtweregiventwosidelengthsndnnglewhihisnottheinludedngle.Referring k to the speil ses desried in the Introdution you will see tht with this informtion there is the possiility tht we n otin two distint tringles with this informtion. seforeweneedskethinordertounderstndtheinformtion(figure7.) 30 o = 1 = 8 Figure7.Weregiventwosidesndnon-inludedngle. eusewehveeengiventwosidesndnon-inludedngleweusethesineformule. or sin = sin = sin sin = sin = sin euseweregiven, nd weusethefollowingprtoftheformulinordertofindngle. sin sin 1 sin = = sin sin 30 = 8 1 sin 30 8 = 1 1 8 = 6 8 = 3 4 = 0.75 = sin 1 0.75 = 48.6 (1 d.p.) Now there is potentil omplition here euse there is nother ngle with sine equl to 0.75.Speifilly, ouldequl 180 48.6 = 131.4. www.mthentre..uk 7 mthentre 009
Inthefirstsethenglesofthetringlerethen: = 30, = 48.6, = 180 78.6 = 101.4 Intheseondsewehve: = 30, = 131.4, = 180 161.4 = 18.6. ThesitutionisdepitedinFigure8.Inordertosolvethetringleompletelywemustdelwith thetwosesseprtelyinordertofindthereminingunknown. 8 30 o 1 8 Figure 8. There re two possile tringles. se1.here = 30, = 48.6, = 101.4.Weusethesineruleintheform from whih sin = sin 1 sin 101.4 = sin 48.6 = 15.7 (1 d.p.) se.here = 30, = 131.4, = 18.6.ginwenusethesineruleintheform from whih Exerise sin = sin 1 sin 18.6 = sin 131.4 = 5.1 (1 d.p.) 1.Findthelengthsoftheothertwosides(to3deimlples)ofthetringleswith () =, = 30, = 40 () = 5, = 45, = 60 () = 3, = 37, = 54 www.mthentre..uk 8 mthentre 009
.Findllpossiletringles(givethesidesto3deimlplesndthenglesto1deiml ple) with () = 3, = 5, = 3 () =, = 4, = 63 () =, = 1, = 108 5. The re of tringle Wenowlooktsetofformulewhihwillgiveusthereoftringle.stndrdformulis re = 1 se height height se Figure9.Thereofthetringleis 1 se height Letusssumeweknowthelengths, nd,ndthenglet. onsidertheright-ngled tringleontheleft-hndsideoffigure9.inthistringle nd so, y rerrnging, sin = height height = sin Thenfromtheformulforthereofthelrgetringle,, re = 1 se height = 1 sin Now onsider the right-ngled tringle on the right-hnd side in Figure 9. nd so, y rerrnging, sin = height height = sin So,thereofthelrgetringle,,islsogiveny Itislsopossiletoshowthttheformul willlsogivethereofthelrgetringle. re = 1 sin re = 1 sin www.mthentre..uk 9 mthentre 009
Key Point Whenweregiventwosidesndtheinludedngle,thereofthetringlenefoundfrom one of the three formule: Figure 10. re = 1 sin = 1 sin = 1 sin Theseformuledonotworkifwerenotgivennngle.nnientGreekythenmeofHero (orheron)derivedformulforlultingthereoftringlewhenweknowllthreesides. Hero s formul: Key Point where re = s(s )(s )(s ) s = + + = semi-perimeter The semi-perimeter, s the nme implies, is hlf of the perimeter of the tringle. www.mthentre..uk 10 mthentre 009
Exmple Supposeweregiventhelengthsofthreesidesoftringle: = 5 = 7 = 10 We n use Hero s formul: re = s(s )(s )(s ) where s = + + = 5 + 7 + 10 = 11 Then re = 11(11 5)(11 7)(11 10) = 11 6 4 1 = 64 = 16. (3 s.f.) Sothereis16.squreunits. Exmple Supposewewishtofindthereoftringlegiventhefollowinginformtion: = 10 = 5 = 10 sketh illustrtes this informtion. = 5 10 o = 10 Figure11.Weregiventwosidesndtheinludedngle. Weregiventwosidesndtheinludedngle. re = 1 sin = 1 10 5 sin 10 = 5 sin 10 = 1.7 (3 s.f.) Sothereis1.7squreunits. www.mthentre..uk 11 mthentre 009
Exerise 3 1.Findtheresofehofthetringles(to3deimlples)inExerise1,Question1..Findtheresofehofthetringles(to3deimlples)inExerise1,Question. 6. Summry osine Formule forfindingnngleusingthethreesides: os = + os = + os = + forfindingsideusingtwosidesndtheinludedngle = + os Sine Formule = + os = + os Usewhenyouregiventwosidesndthenon-inludedngle,ortwonglesndside: sin = sin Formuleforthereoftringle sin = = sin sin = R = sin re = 1 sin = 1 sin = 1 sin where re = s(s )(s )(s ) s = + + = semi-perimeter www.mthentre..uk 1 mthentre 009
Exerise 4 1.Determinethelengthsofllthesides(to3deimlples),thesizesofllthengles(to 1deimlple)ndthere(to3deimlples)ofehofthefollowingtringles. nswers () = 5, = 3, = 6 () = 5, =, = 4 () = 3, = 40, = 60 (d) =, = 73, = 41 (e) = 5, = 3, = 4 (f) =, = 5, = 78 Exerise 1 1.() = 1.39, = 3.8, = 16. () = 3.094, = 48.0, = 8.0 () = 6.197, = 3.8, = 16...() = 9.0, = 46.6, = 104.5 () = 41.4, = 41.4, = 97. () = 41.4, = 41.4, = 97. Exerise 1.() =.571, = 3.759 () = 7.044, = 6.14 () = 1.806, =.47.() Two possile tringles: = 6.0, = 86.0, = 5.647nd = 118.0, = 30.0, =.833 () = 7.0, = 88.0, = 4.487 () =.457, = 3., = 51.9 Exerise 3 1.() 0.5 () 0.587 () 1.5.().905 () 0.496 () 1.984 Exerise 4 1.() = 56.3, = 9.9, = 93.8,re=7.483 () = 3.760, = 117., = 0.9,re=3.346 () = 1.958, =.638, = 80,re=.544 (d) =.915, =.785, = 66,re=.663 (e) = 90, = 36.9, = 53.1,re=6 (f) = 5.017, = 79.0, = 3.0,re=4.908 www.mthentre..uk 13 mthentre 009