Question

在△ABC中,C=2AcosA=3/4,向量BA乘向量BC=27/2,则AC=

Answer 1

设|BA|=c,|BC|=a,AC=b

sinA=√[1-(3/4)²]=√7/4,

cosC=cos2A=2cos²A-1=1/8,sinC=√[1-(1/8)²]=3√7/8

cosB= cos(180-3A)=-cos3A = -cos(A+C)=sinAsinC-cosAcosC=9/16

向量BA乘于向量BC=c*a*cosB=27/2

∴ca=(27/2)/(9/16)=24

∴SΔABC=casinB/2=24(5√7/16)/2=15√7/4

∵b/sinB=c/sinC=a/sinA

∴b²=casin²B/sinCsinA=24(5√7/16)²/[(3√7/8)(√7/4)]=25

∴AC=b=5