Question

在△ABC中，角A，B，C的对边为a，b，c，已知b=4，c=6，C=2B，①cosB的值为。②△

在△ABC中，角A，B，C的对边为a，b，c，已知b=4，c=6，C=2B，①cosB的值为。②△ABC的面积为？

Answer 1

∵b=4，c=6
根据正弦定理：
b/sinB=c/sinC
∵C=2B
∴sinC=sin2B=2sinBcosB
∴b/sinB=c/(2sinBcosB)
∴cosB=c/(2b)=6/(2*4)=3/4

sinB = √(1-cos²B) = √(1=9/16) = √7/4
sinC = 2sinBcosB = 2*√7/4*3/4 = 3√7/8
cosC = cos2B = 2cos²B-1 = 2×9/16-1 = 1/8
sinA = sin(B+C) = sinBcosC+cosBsinC = √7/4*1/8+3/4*3√7/8 = 5√7/16
S = 1/2bcsinA = 1/2*4*6*5√7/16 = 15√7/4