Question

在△ABC中，已知A＞B＞C，且A＝2C，b=4，a+c=8，求a，c

Answer 1

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

Answer 2

超简便方法，求采纳
a=2c
a+c=8
2c+c=8
3c = 8
c = 8/3
a = 2c
a = 2(8/3)
a = 16/3
a = 16/3 , b = 4 , c = 8/3
符合 a > b >c

Answer 3

a=16/3 c=8/3