(1)已知sinA+sinB+sinC=0,cosA+cosB+cosC=0.求cos(B-C)的值.(2)在三角行ABC中,已知a^4+b^4+c^4=2c^2(a^2+b^2),则∠C=?(3)在三角行ABC中,已知∠A>∠B>∠C且∠A=2∠C,b=4,a+c=8,求a,c的长.
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/05 12:07:09
![(1)已知sinA+sinB+sinC=0,cosA+cosB+cosC=0.求cos(B-C)的值.(2)在三角行ABC中,已知a^4+b^4+c^4=2c^2(a^2+b^2),则∠C=?(3)在三角行ABC中,已知∠A>∠B>∠C且∠A=2∠C,b=4,a+c=8,求a,c的长.](/uploads/image/z/10127323-19-3.jpg?t=%281%29%E5%B7%B2%E7%9F%A5sinA%2BsinB%2BsinC%3D0%2CcosA%2BcosB%2BcosC%3D0.%E6%B1%82cos%28B-C%29%E7%9A%84%E5%80%BC.%282%29%E5%9C%A8%E4%B8%89%E8%A7%92%E8%A1%8CABC%E4%B8%AD%2C%E5%B7%B2%E7%9F%A5a%5E4%2Bb%5E4%2Bc%5E4%3D2c%5E2%28a%5E2%2Bb%5E2%29%2C%E5%88%99%E2%88%A0C%3D%3F%283%29%E5%9C%A8%E4%B8%89%E8%A7%92%E8%A1%8CABC%E4%B8%AD%2C%E5%B7%B2%E7%9F%A5%E2%88%A0A%3E%E2%88%A0B%3E%E2%88%A0C%E4%B8%94%E2%88%A0A%3D2%E2%88%A0C%2Cb%3D4%2Ca%2Bc%3D8%2C%E6%B1%82a%2Cc%E7%9A%84%E9%95%BF.)
(1)已知sinA+sinB+sinC=0,cosA+cosB+cosC=0.求cos(B-C)的值.(2)在三角行ABC中,已知a^4+b^4+c^4=2c^2(a^2+b^2),则∠C=?(3)在三角行ABC中,已知∠A>∠B>∠C且∠A=2∠C,b=4,a+c=8,求a,c的长.
(1)已知sinA+sinB+sinC=0,cosA+cosB+cosC=0.求cos(B-C)的值.
(2)在三角行ABC中,已知a^4+b^4+c^4=2c^2(a^2+b^2),则∠C=?
(3)在三角行ABC中,已知∠A>∠B>∠C且∠A=2∠C,b=4,a+c=8,求a,c的长.
(1)已知sinA+sinB+sinC=0,cosA+cosB+cosC=0.求cos(B-C)的值.(2)在三角行ABC中,已知a^4+b^4+c^4=2c^2(a^2+b^2),则∠C=?(3)在三角行ABC中,已知∠A>∠B>∠C且∠A=2∠C,b=4,a+c=8,求a,c的长.
(1)已知sinA+sinB+sinC=0,cosA+cosB+cosC=0.求cos(B-C)的值.
sinA=-(sinB+sinC)
cosA=-(cosB+cosC)
sinA^2+cosA^2=1
带入整理可得cos(B-C)=-0.5
(2)在三角行ABC中,已知a^4+b^4+c^4=2c^2(a^2+b^2),则∠C=?
a^4+b^4+c^4=2c^2(a^2+b^2)
a^4+b^4+c^4-2c^2a^2-2c^2b^2=0
(a^2+b^2-c^2)^2=2a^2b^2
a^2+b^2-c^2=正负(根号2)ab
cosC=(a^2+b^2-c^2)/(2ab)=正负(根号2)/2
C=45度,或135度
(3)在三角行ABC中,已知∠A>∠B>∠C且∠A=2∠C,b=4,a+c=8,求a,c的长.
A=2C
sinB=sin(180-B)=sin(A+C)=sin3C
sinA=sin2C
由正弦定理得
b/sinB=(a+c)/(sinA+sinC)
4/sin3C=8/(sinA+sinC)
2sin3C=sin2C+sinC
之后用3倍2倍角及恒等式可得出8cos^2C-2cosC-3=0
所以cosC=3/4 (cosC=-1/2舍去)
sinC=√7/4
sinA=sin2C=3√7/8
a/c=sinA/sinc=3:2
所以a=24/5 c=16/5