已知锐角A,B满足sinB=2cos(A+B)sinA,1 .求证:tan(a+b)=3tana 2 .求tanb的最大值.
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已知锐角A,B满足sinB=2cos(A+B)sinA,1 .求证:tan(a+b)=3tana 2 .求tanb的最大值.
已知锐角A,B满足sinB=2cos(A+B)sinA,1 .求证:tan(a+b)=3tana 2 .求tanb的最大值.
已知锐角A,B满足sinB=2cos(A+B)sinA,1 .求证:tan(a+b)=3tana 2 .求tanb的最大值.
1
sinB=2cos(A+B)sinA
sinB=sin(180-B)=sin(A+C)=sinAcosC+sinCcosA
2cos(A+B)sinA=-2cosCsinA
sinAcosC+sinCcosA=-2cosCsinA
3sinAcosC=-sinCcosA
3sinA/cosA=-sinC/cosC
3tanA=-tanC=-tan(A+B)
2
tanB=tan[(A+B)-A]=[tan(A+B)-tanA]/[1+tan(A+B)tanA]=-4tanA/[1-3(tanA)^2]
tanB=-4tanA/[1-3(tanA)^2]=-4/[1/tanA-3tanA]
1/tanA-3tanA=cosA/sinA-3sinA/cosA=[(cosA)^2-3(sinA)^2]/(sinAcosA)=(2cos2A-1)/(sin2A/2)
sin2A=1时,1/tanA-3tanA最小
tanB最大=-4/[-1/(1/2)]=2