在三角形ABC中,向量m=(2cos(c/2),-sinc),n=(cos(c/2),2sinc).且m垂直n.若a^2=2b^2+c^2,求tanA的值

在三角形ABC中,向量m=(2cos(c/2),-sinc),n=(cos(c/2),2sinc).且m垂直n.若a^2=2b^2+c^2,求tanA的值
在三角形ABC中,向量m=(2cos(c/2),-sinc),n=(cos(c/2),2sinc).且m垂直n.若a^2=2b^2+c^2,求tanA的值

在三角形ABC中,向量m=(2cos(c/2),-sinc),n=(cos(c/2),2sinc).且m垂直n.若a^2=2b^2+c^2,求tanA的值
m垂直n
=>m.n=0
(2cos(C/2),-sinC).(cos(C/2),2sinC)=0
2(cos(C/2))^2-2(sinC)^2=0
cosC +1 - 2(sinC)^2=0
cosC +1-2(1-(cosC)^2) =0
2(cosC)^2+cosC -1=0
cosC = (-1+3)/4 = -1/2
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