(2/2)乘向量n等于－1.若a＝2倍根3,c＝2,求三角形ABC面积

题目

(2/2)乘向量n等于－1.若a＝2倍根3,c＝2,求三角形ABC面积

答案

m=(2cosA,√3sinA),n=(cosA,-2cosA)
m.n=-1
(2cosA,√3sinA).(cosA,-2cosA) =-1
2(cosA)^2-2√3sinAcosA=-1
cos2A-√3sin2A = -2
(cos2A)^2+4cos2A+4= 3(sin2A)^2
4(cos2A)^2+4cos2A+1=0
(2cos2A+1)=0
cos2A = -1/2
2A =2π/3
A=π/3
a/sinA = c/sinC
2√3/(√3/2) = 2/sinC
4sinC =2
C =π/6
A+B+C= π
B= π-π/3-π/6
=π/2
a=2√3,c=2
S△ABC= (1/2)acsinB
=(1/2)(2√3)2
=2√3