已知函数f(x)=二分之根号三Sin2x-cosx的平方-1/2,x属于R设△ABC的内角A,B,C的对边分别为a,b,c,且c=√3,f(C)=0,若sinB=2sinA,求a,b的值
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![已知函数f(x)=二分之根号三Sin2x-cosx的平方-1/2,x属于R设△ABC的内角A,B,C的对边分别为a,b,c,且c=√3,f(C)=0,若sinB=2sinA,求a,b的值](/uploads/image/z/1660979-11-9.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D%E4%BA%8C%E5%88%86%E4%B9%8B%E6%A0%B9%E5%8F%B7%E4%B8%89Sin2x-cosx%E7%9A%84%E5%B9%B3%E6%96%B9-1%2F2%2Cx%E5%B1%9E%E4%BA%8ER%E8%AE%BE%E2%96%B3ABC%E7%9A%84%E5%86%85%E8%A7%92A%2CB%2CC%E7%9A%84%E5%AF%B9%E8%BE%B9%E5%88%86%E5%88%AB%E4%B8%BAa%2Cb%2Cc%2C%E4%B8%94c%3D%E2%88%9A3%2Cf%EF%BC%88C%EF%BC%89%3D0%2C%E8%8B%A5sinB%3D2sinA%2C%E6%B1%82a%2Cb%E7%9A%84%E5%80%BC)
已知函数f(x)=二分之根号三Sin2x-cosx的平方-1/2,x属于R设△ABC的内角A,B,C的对边分别为a,b,c,且c=√3,f(C)=0,若sinB=2sinA,求a,b的值
已知函数f(x)=二分之根号三Sin2x-cosx的平方-1/2,x属于R
设△ABC的内角A,B,C的对边分别为a,b,c,且c=√3,f(C)=0,若sinB=2sinA,求a,b的值
已知函数f(x)=二分之根号三Sin2x-cosx的平方-1/2,x属于R设△ABC的内角A,B,C的对边分别为a,b,c,且c=√3,f(C)=0,若sinB=2sinA,求a,b的值
先化简:f(x)=√3/2sin2x-cos²x-1/2=√3/2sin2x-(1+cos2x)/2-1/2=√3/2sin2x-1/2cos2x-1=sin(2x-π/6)-1,∵f(C)=0,由于C∈(0,π)解得C=π/3,根据正弦定理得sinB=2sinA,可得b=2a,根据余弦定理得cosC=a²+b²-c²/2ab,即cosπ/3=1/2=a²+(2a)²-(√3)²/2×2a×a,解得a=1,b=2a=2
【参考答案】
f(C)=(√3 /2)sin2C-cosC^2-(1/2)
=(√3/2)sin2C-[(cos2C+1)/2]-(1/2)
=sin2Ccos(π/6)-sin(π/6)cos2C-1
=sin(2C- π/6)-1
=0
则 sin(2C- π/6)=1, 即C=π/3
由sinB=2sinA得
b/s...
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【参考答案】
f(C)=(√3 /2)sin2C-cosC^2-(1/2)
=(√3/2)sin2C-[(cos2C+1)/2]-(1/2)
=sin2Ccos(π/6)-sin(π/6)cos2C-1
=sin(2C- π/6)-1
=0
则 sin(2C- π/6)=1, 即C=π/3
由sinB=2sinA得
b/sinB=a/sinA
b/(2sinA)=a/sinA
b/2=a
b=2a
cosC=-1/2=[a^2 +(4a^2)-3]/(2a×2a)
1/2=(5a^2 -3)/(4a^2)
5a^2 -3=2a^2
3a^2=3
a=1,于是 b=2
不理解之处欢迎追问
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