Question

已知向量a=(cosA,sinA),b=(cosx,sinx),c=(sinx+2sinA，cosx+2cosA),其中0<A<180*

（1）若A=45*，求函数f(x)=向量a乘b的最小值及相应x的值
（2）若向量a与b的夹角为60*,且向量a垂直于b，求tan2A的值

Answer 1

1)
(a=π/4
c=(sinx+√2,cosx+√2)
f(x)=b·c=cosx(sinx+√2)+sinx(cosx+√2)
=2sinxcosx+√2(sinx+cosx)
设 sinx+cosx=t
∴t²=(sinx+cosx)²=1+2sinxcosx
∴ 2sinxcosx=t²-1
又t=sinx+cosx=√2sin(x+π/4)∈[-√2,√2]
∴f(x)=g(t)=t²-1+√2t=(t+√2/2)²-3/2
∴t=-√2/2时，g（t),即f(x) 取得最小值 -3/2
此时，√2sin(x+π/4）=-√2/2
∴sin(x+π/4）=-1/2
∵0<a<x<π ∴π/4<x<π ∴π/2<x+π/4<5π/4
∴x+π/4=7π/6 ∴x=11π/12
即x=11π/12时，f(x)取得π最小值 -3/2
(2)
∵a与b的夹角为π/3
∴a(b=|a||b|cosπ/3
∴ cosacosx+sinasinx=1*1*1/2=1/2
∴ cos(x-a)=1/2
∵0<a<x<π ∴0<x-a<π-a
∴x-a=π/3,x=a+π/3
∵a⊥c ∴a·c=0
∴cosa(sinx+2sina)+sina(cosx+2cosa)=0
∴sinxcosa+cosxsina+4sinacosa=0
∴sin(a+π/3)cosa+cos(a+π/3)sina+2sin2a=0
∴sin(2a+π/3)+2sin2a=0
∴sin2acosπ/3+cos2asinπ/3+2sin2a=0
∴5/2sin2a+√3/2cos2a=0
∴tan2a=-√3/5

Answer 2

(2)
f(x)
= a.b
=(cosA,sinA).(cosx,sinx)
= cos(x-A)
= cos(x-45°)
minf(x) = -1
at x= -45°
(2)
a与b的夹角为60*,且向量a垂直于b ?????????????
夹角 = 60°=90°???????????