Question

三角形ABC的内角A,B,C的对边分别为a,b,c,.已知3acosC=2ccosA,且b=2倍根

三角形ABC的内角A,B,C的对边分别为a,b,c,.已知3acosC=2ccosA,且b=2倍根号5,c=3
(1)求a的值
(2)求sin(B+4/pai)的值

Answer 1

3acosC=2ccosA
3sinAcosC=2sinCcosA

3sinAcosC+3sinCcosA=5sinCcosA
3sinB=5sinCcosA
3b=5ccosA
cosA=3b/5c=3×2✔5/(5×3)=2✔5/5
a²=b²+c²-2bccosA=20+9-12✔5×2✔5/5=5
a=✔5

追答

cosB=(a²+c²-b²)/(2ac)=(5+9-20)/(2✔5×3)=-✔5/5
sinB=✔(1-cos²B)=2✔5/5
sin(B+π/4)=✔2/2(sinB+cosB)=✔10/10