已知函数f(x)=2√3sinxcosx-3sin²x-cos²x+2若三角形ABC的内角A B C 的对边分别为a b c 且满足b/a=√3,[sin(2A+C)]/sinA=2+2cos(A+C),求f(B)的值
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![已知函数f(x)=2√3sinxcosx-3sin²x-cos²x+2若三角形ABC的内角A B C 的对边分别为a b c 且满足b/a=√3,[sin(2A+C)]/sinA=2+2cos(A+C),求f(B)的值](/uploads/image/z/1994474-2-4.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2%E2%88%9A3sinxcosx-3sin%26%23178%3Bx-cos%26%23178%3Bx%2B2%E8%8B%A5%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E7%9A%84%E5%86%85%E8%A7%92A+B+C+%E7%9A%84%E5%AF%B9%E8%BE%B9%E5%88%86%E5%88%AB%E4%B8%BAa+b+c+%E4%B8%94%E6%BB%A1%E8%B6%B3b%2Fa%3D%E2%88%9A3%2C%5Bsin%282A%2BC%29%5D%2FsinA%3D2%2B2cos%28A%2BC%29%2C%E6%B1%82f%EF%BC%88B%EF%BC%89%E7%9A%84%E5%80%BC)
已知函数f(x)=2√3sinxcosx-3sin²x-cos²x+2若三角形ABC的内角A B C 的对边分别为a b c 且满足b/a=√3,[sin(2A+C)]/sinA=2+2cos(A+C),求f(B)的值
已知函数f(x)=2√3sinxcosx-3sin²x-cos²x+2
若三角形ABC的内角A B C 的对边分别为a b c 且满足b/a=√3,[sin(2A+C)]/sinA=2+2cos(A+C),求f(B)的值
已知函数f(x)=2√3sinxcosx-3sin²x-cos²x+2若三角形ABC的内角A B C 的对边分别为a b c 且满足b/a=√3,[sin(2A+C)]/sinA=2+2cos(A+C),求f(B)的值
f(x)=2√3sinxcosx-3sin²x-cos²x+2
=√3(2sinxcosx)-(sin²x+cos²x)-(2sin²x+1) +3
=√3sin(2x)-1-cos(2x)+3
=√3sin(2x)-cos(2x) +2
=2[(√3/2)sin(2x)-(1/2)cos(2x)]+2
=2sin(2x-π/6)+2
b/a=√3,由正弦定理得sinB/sinA=√3
sinB=√3sinA
sin(A+C)=sinB=√3sinA
sin(2A+C)/sinA=2+2cos(A+C)
sin[(A+C)+A]/sinA=2-2cosB
[sin(A+C)cosA+cos(A+C)sinA]/sinA=2-2cosB
(√3sinAcosA-cosBsinA)/sinA=2-2cosB
√3cosA-cosB=2-2cosB
cosB=2-√3cosA
sin²B+cos²B=1
(√3sinA)²+(2-√3cosA)²=1
3sin²A+4-4√3cosA+3cos²A=1
2√3cosA=3
cosA=√3/2
cosB=2-√3cosA=2-√3(√3/2)=2-3/2=1/2
B为三角形内角,B=π/3
f(B)=2sin(2·π/3 -π/6)+2
=2sin(π/2)+2
=2·1+2
=4
f(B)=1 B=60, A=30 C=90