If we are decomposing into full F.S. $f(x)=x$ on interval $[-l, l]$ then it is the same as to decompose into sin-F.S. $f(x)=x$ on interval $[0, l]$.

But if we decompose into full F.S. $f(x)=x$ on interval $[0, l]$ then we have different picture: function is $l$-periodic and exactly this is on the picture (note that bolder line shows graph on the original interval)