(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
帶入整理可得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),則∠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=正負(fù)(根號(hào)2)ab
cosC=(a^2+b^2-c^2)/(2ab)=正負(fù)(根號(hào)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的長.
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