向量m=(2cosC/2,-sinC) n=(cosC/2,2sinC) 向量m⊥n 角C=60° 若a²=2b²+c²求tanA的值

m⊥n
=>m.n=0
(2cos(C/2),-sinC).(cos(C/2),2sinC)=0
2(cosC/2)^2-2(sinC)^2=0
(2(cosC/2)^2-1)-2(sinC)^2+1=0
cosC-2(sinC)^2+1=0
cosC - (2-2(cosC)^2)+1=0
2(cosC)^2+cosC-1=0
cosC = (-1+3)/4 = -1/2
C=2π/3
sinC = √3/2
a^2 =2b^2+c^2
c^2 = a^2+b^2 - 3b^2
by cosine rule
-3b^2 =-2abcosC
cosC = (3b/(2a)) = -1/2
b= -a/3
a^2 =2b^2+c^2
= b^2 + c^2 +b^2
by cosine -rule
-2bccosA = b^2
cosA = -b/(2c)
= -b/ (2asinC/sinA)
tanA = -2asinC/b
= -2a(√3/2))/( -a/3)
= 3√3