Question

已知在三角形ABC中，如果sinA:sinB:sinC=2:3:4，那么cosC等于？

Answer 1

sinA/2=sinB/3=sinC/4 sinA=sinC/2 sinB=3sinC/4
sinC=sin(180-A-B)=sin(A+B)=sinAcosB+conAsinB=sinCcosB/2+3cosAsinC/4
cosB/2+3cosA/4=1 [√1-(sinB)^2]/2+3[√1-(sinA)^2]/4=1
4-2√1-(3sinC/4)^2=3√1-(sinC/2)^2
(sinC)^2=15/16 cosC=1/4

Answer 2

sinA/2=sinB/3=sinC/4 sinA=sinC/2 sinB=3sinC/4
sinC=sin(180-A-B)=sin(A B)=sinAcosB conAsinB=sinCcosB/2 3cosAsinC/4
cosB/2 3cosA/4=1 [√1-(sinB)^2]/2 3[√1-(sinA)^2]/4=1
4-2√1-(3sinC/4)^2=3√1-(sinC/2)^2
(sinC)^2=15/16 cosC=1/4

Answer 3

正弦定理，sinA/sinB=a/b,sinB/sinC=b/c,
sinA:sinB:sinC=a:b:c=2:3:4,
设三边长分别为2m,3m,4m,
cosC=(a^2+b^2-c^2)/(2ab)=([(2m)^2+(3m)^2-(4m)^2]/(2*2m*3m)=-1/4,