已知向量a=(√3sinπx/4,sinπx/4),向量b=(sinπx/4,cosπx/4),f（x）=ab-√3/2

利用公式cos2a=cos?a-sin?a=1-sin?a-sin?a=1-2sin?a
于是 sin?a=1/2-1/2cos2a
f(x)=ab-√3/2
=√3sin(πx/4)*sin(πx/4)+sin(πx/4)*cos(πx/4)-√3/2
=√3sin?(πx/4)+1/2[2*sin(πx/4)*cos(πx/4)]-√3/2
=√3[1/2-1/2cos2(πx/4）]+1/2sin2(πx/4)-√3/2
=√3/2-√3/2cos(π/2x)+1/2sin(π/2x)-√3/2
=1/2sin(π/2x)-√3/2cos(π/2x)
=sin(π/2x)*cosπ/3-sinπ/3*cos(π/2x)
=sin(π/2x-π/3)
(1)对称轴即 π/2x-π/3=π/2+kπ （k为整数）
所以x=5/3+2k （k为整数）
所以对称轴为 直线x=5/3+2k （k为整数）
(2）最小正周期T=2π÷(π/2)=4
而2012÷4=503
所以
f(1)+f(2)+f(3)+f(4)+.+f(2012)
=503[f(1)+f(2)+f(3)+f(4)]
=503×[sin(π/2-π/3)+sin(π-π/3)+sin(3/2π-π/3)+sin(2π-π/3)]
=503×[(sinπ/6π+sin(2/3π)+sin(7/6π)+sin(-π/3)]
=503×(sinπ/6π+sinπ/3-sinπ/6-sinπ/3)
=503×0
=0
所以 f(1)+f(2)+f(3)+f(4)+.+f(2012)=0