sinB bsinC csinB 5sin120 7sin120sinB06185895741317Barcsin06185895741317382FinallyA180120382218 Example252SASSolvethetriangleifa3b7andC37SolutionWearegiventwosidesa ID: 486176
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sinB b=sinC csinB 5=sin120 77sinB=5sin120sinB=5 7sin120sinB=0:6185895741317B=arcsin(0:6185895741317)38:2Finally,A=180�(120+38:2)=21:8: Example25.2(SAS)Solvethetriangleifa=3;b=7andC=37:Solution.Wearegiventwosidesandtheincludedangle.Wemustndthethirdside.Themissingsideisc:BytheLawofCosinesc2=a2+b2�2abcosCc2=9+49�2(3)(7)cos37c2=58�42cos37c=p 58�42cos374:9:NowusetheLawofSinesandndthesmallestangle.Thesmallestangleisdenitelyanacuteangle.TheLawofSinescannotdistinquishbetweenacuteandobtusebecausebothanglesgiveapositiveanswer.Thesmallestangleisoppositesidea;theshortestside.sinA 3=sin37 4:94:9sinA=3sin37sinA=3 4:9sin37sinA=:36845817A21:63 ThentheareaofthetriangleisK=1 2heightbase=1 2hb:ButsinA=h corh=csinA:Thus,K=1 2bcsinA:Usingsimilararguments,onecanestablishtheareaformulasK=1 2acsinBandK=1 2absinC:Example25.4GivenA=62;b=12meters;andc=5:0meters,ndtheareaofthetriangleABC:Solution.Usingtheformulaforarea,wehaveK=1 2bcsinA=1 2(12)(5:0)sin6226m2: Example25.5Afarmerhasatriangulareldwithsides120yards,170yards,and220yards.Findtheareaoftheeldinsquareyards.Thenndthenumberofacresif1acre=4840squareyards.Solution.WeneedtondananglesowecanusetheareaformulaSo,leta=120;b=170;c=220:WestartbyndingC.c2=a2+b2�2abcosC48400=14400+28900�40800cosC5100=�40800cosC�5100 40800=cosCarccos(�5100 40800)=C97:2CNowndthearea5 aregiven.Inwhatfollows,welets=a+b+c 2(i.e.sishalftheperimeterofthetriangle).Solet'slookatFigure25.4. Figure25.4UsingthePythagoreantheoremwecanwritea2�p2=c2�(b�p)2ora2�p2=c2�b2�p2+2bp:Solvingforpwendp=a2+b2�c2 2b:Asimplearithmeticshowsthefollowingh2=a2�p2=(a�p)(a+p)=a�a2+b2�c2 2ba+a2+b2�c2 2b=2ab�a2�b2+c2 2b2ab+a2+b2�c2 2b=c2�(a�b)2 2b(a+b)2�c2 2b=(c�a+b)(c+a�b)(a+b�c)(a+b+c) 4b2=(a+b+c)(�a+b+c)(a�b+c)(a+b�c) 4b2=(2s)2(s�a)2(s�b)2(s�c) 4b2=4s(s�a)(s�b)(s�c) b2Thus,h=2p s(s�a)(s�b)(s�c) bButtheareaofthetriangleisK=1 2bh=1 2b 2p s(s�a)(s�b)(s�c) b!=p s(s�a)(s�b)(s�c):ThislastformulaisknownasHeron'sformula.7