Question

在三角形ABC中,已知A＞B＞C,且A=2C.b=4,a+c=8,求a,c的长.用简单方法做,

Answer 1

A=2C

===> sinA=sin2C

===> sinA=2sinCcosC

===> cosC=(1/2)sinA/sinC

===> (a²+b²-c²)/(2ab)=(1/2)a/c

===> a²+16-c²=4a²/c

===> (a+c)(a-c)+16=4a²/(8-a)

===> 8(a-c)+16=4a²/(8-a)

===> 2[a+(a-8)]+4=a²/(8-a)

===> 2(2a-8)+4=a²/(8-a)

===> 4a-12=a²/(8-a)

===> (4a-12)(8-a)=a²

===> -4a²+44a-96=a²

===> 5a²-44a+96=0

===> (5a-24)(a-4)=0

===> a=24/5,或者a=4

已知A＞B＞C,所以：a＞b＞c

已知b=4

所以,a=24/5

则,c=8-a=16/5