在三角形ABC中,角A,B,C的对边分别是a,b,c,已知M=(SinC,CosC),N=(1,-根号3).C=2,MN=01,求Sin2〔(A+B)/2〕+Cos2C 2,若SinC+Sin(B-A)=2Sin2A,求三角形ABC的面积
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![在三角形ABC中,角A,B,C的对边分别是a,b,c,已知M=(SinC,CosC),N=(1,-根号3).C=2,MN=01,求Sin2〔(A+B)/2〕+Cos2C 2,若SinC+Sin(B-A)=2Sin2A,求三角形ABC的面积](/uploads/image/z/673515-27-5.jpg?t=%E5%9C%A8%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E4%B8%AD%2C%E8%A7%92A%2CB%2CC%E7%9A%84%E5%AF%B9%E8%BE%B9%E5%88%86%E5%88%AB%E6%98%AFa%2Cb%2Cc%2C%E5%B7%B2%E7%9F%A5M%3D%28SinC%2CCosC%29%2CN%3D%281%2C-%E6%A0%B9%E5%8F%B73%29.C%3D2%2CMN%3D01%2C%E6%B1%82Sin2%E3%80%94%28A%2BB%29%2F2%E3%80%95%2BCos2C+2%2C%E8%8B%A5SinC%2BSin%28B-A%29%3D2Sin2A%2C%E6%B1%82%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E7%9A%84%E9%9D%A2%E7%A7%AF)
在三角形ABC中,角A,B,C的对边分别是a,b,c,已知M=(SinC,CosC),N=(1,-根号3).C=2,MN=01,求Sin2〔(A+B)/2〕+Cos2C 2,若SinC+Sin(B-A)=2Sin2A,求三角形ABC的面积
在三角形ABC中,角A,B,C的对边分别是a,b,c,已知M=(SinC,CosC),N=(1,-根号3).C=2,MN=0
1,求Sin2〔(A+B)/2〕+Cos2C 2,若SinC+Sin(B-A)=2Sin2A,求三角形ABC的面积
在三角形ABC中,角A,B,C的对边分别是a,b,c,已知M=(SinC,CosC),N=(1,-根号3).C=2,MN=01,求Sin2〔(A+B)/2〕+Cos2C 2,若SinC+Sin(B-A)=2Sin2A,求三角形ABC的面积
MN=0
则M垂直N,
(sinC*1+cosC*(-√3)=0,
sinC*1/2-√3/2*cosC=0,
sin(C-60)=0,
C=60度.
A+B=180-C,(A+B)/2=90-(C/2).
sin(A+B)/2=cos(C/2)=cos45.
Sin2〔(A+B)/2〕+Cos2C
=(cos45)^2+cos120
=1/2-1/2
=0.
2.若SinC+Sin(B-A)=2Sin2A
2sin(C+B-A)/2*cos(C+A-B)/2=2Sin2A,
而,2sin(C+B-A)/2=cosA,cos(C+A-B)/2=sinB,
即有,cosA*sinB=sin2A=2sinA*cosA,
cosA(sinB-2sinA)=0,
cosA=0,sinB-2sinA=0,
A=90度,
sinB=2sinA,
b=2a,而,A=90度,a^2=b^2+c^2,c=2.
a^2=4a^2+4,(不合,舍去).
A≠90度,则有
b=2a,
而,C=60度,
cos60=(a^2+b^2-c^2)/2ab,c=2.
1/2=(a^2+4a^2-4)/4a^2,
a^2=4/3,
a=2√3/3.
b=2*2√3/3=4√3/3.
三角形ABC的面积=1/2*sinC*ab=1/2*√3/2*(2√3/3)*(4√3/3)
=2√3/3.