若函数f(x)=sin(πx/4),f(1)+f(2)+...+f(2010)=_

f(1)=sinπ/4=√2/2
f(2)=sinπ/2=1
f(3)=sin3π/4=√2/2
f(4)=sinπ=0
f(5)=sin(π+π/4)=-√2/2
f(6)=sin(π+π/2)=-1
f(7)=sin(π+3π/4)=-√2/2
f(8)=sin(π+π)=0
f(9)=sin(2π+π/4)=√2/2
8是f(x)的周期
2010/8=251...2
又因为f(1)+f(2)+..+f(8)=0
∴
f(1)+f(2)+...+f(2010)=
f(2009)+f(2010)
=sin(2009π/4)+sin(2010π/4)
=sin(502π+π/4)+sin(502π+π/2)
=sinπ/4+sinπ/2
=1+√2/2