在△abc中 角a b c所对的边分别为a b c 若sinA sinB sinC=根号3/2(sin^2A+sin^2B-sin^2C)(1)求角C的大小(2)求y=sinA+sinB(A≥π/4)的值域
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![在△abc中 角a b c所对的边分别为a b c 若sinA sinB sinC=根号3/2(sin^2A+sin^2B-sin^2C)(1)求角C的大小(2)求y=sinA+sinB(A≥π/4)的值域](/uploads/image/z/613028-20-8.jpg?t=%E5%9C%A8%E2%96%B3abc%E4%B8%AD+%E8%A7%92a+b+c%E6%89%80%E5%AF%B9%E7%9A%84%E8%BE%B9%E5%88%86%E5%88%AB%E4%B8%BAa+b+c+%E8%8B%A5sinA+sinB+sinC%3D%E6%A0%B9%E5%8F%B73%2F2%28sin%5E2A%2Bsin%5E2B-sin%5E2C%29%281%29%E6%B1%82%E8%A7%92C%E7%9A%84%E5%A4%A7%E5%B0%8F%EF%BC%882%EF%BC%89%E6%B1%82y%3DsinA%2BsinB%28A%E2%89%A5%CF%80%2F4%EF%BC%89%E7%9A%84%E5%80%BC%E5%9F%9F)
在△abc中 角a b c所对的边分别为a b c 若sinA sinB sinC=根号3/2(sin^2A+sin^2B-sin^2C)(1)求角C的大小(2)求y=sinA+sinB(A≥π/4)的值域
在△abc中 角a b c所对的边分别为a b c 若sinA sinB sinC=根号3/2(sin^2A+sin^2B-sin^2C)
(1)求角C的大小
(2)求y=sinA+sinB(A≥π/4)的值域
在△abc中 角a b c所对的边分别为a b c 若sinA sinB sinC=根号3/2(sin^2A+sin^2B-sin^2C)(1)求角C的大小(2)求y=sinA+sinB(A≥π/4)的值域
(1) sinA sinB sinC=根号3/2(sin^2A+sin^2B-sin^2C)
sinC=(√3/2)(sin^2A+sin^2B-sin^2C)/sinAsinB
由正弦定理,右边转为边的形式
sinC=√3(a^2+b^2-c^2)/(2ab)
由余弦定理知,sinC=√3cosC
tanC=√3
C=π/3
(2) B=π-A-C=2π/3-A
y=sinA+sinB=sinA+sin(2π/3-A)
=2sin(π/3)cos(π/3-A)
=√3cos(π/3-A)
=√3sin(A+π/6)
因π/4≤A
(sinA)^2+(sinB)^2+(sinC)^2+2sinAsinB+2sinBsinA+2sinCsinA
=3(sinA)^2+3(sinB)^2+3(sinC)^2
2(sinA)^2+2(sinB)^2+2(sinC)^2-2sinAsinB-2sinBsinA-2sinCsinA=0
[(sinA)^2+(sinB)^2-2sinAsinB]+[(sinB)^...
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(sinA)^2+(sinB)^2+(sinC)^2+2sinAsinB+2sinBsinA+2sinCsinA
=3(sinA)^2+3(sinB)^2+3(sinC)^2
2(sinA)^2+2(sinB)^2+2(sinC)^2-2sinAsinB-2sinBsinA-2sinCsinA=0
[(sinA)^2+(sinB)^2-2sinAsinB]+[(sinB)^2+(sinC)^2-2sinBsinC]+[(sinC)^2+(sinA)^2-2sinCsinA]=0
(sinA-sinB)^2+(sinB-sinC)^2+(sinC-sinA)^2=0
∴sinA=sinB=sinC
∴∠A=∠B=∠C
等边三角形
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