Is linear span of $\{\sin nt\}$ dense in $L^2((-\pi,\pi))$

Question

Is the linear span of the family $\{\sin nt\}^{\infty}_{n=1}$ dense in the space $L^2((-\pi,\pi))$?

I have absolutely no idea about how to proceed.

Answer 1

Notice that $\sin(nt)$ is a odd function, for $n \in \mathbb{Z}$. The linear combination of odd functions is an odd function. Since in $L^2((-\pi, \pi))$ there are function not odd, then the family cannot be dense in this space.

Answer 2

recall stuff about fourier series. we know the span of $\{ sin(nt)\}_{n=1}^{\infty}+\{ cos(nt)\}_{n=0}^{\infty}$ is a dense subspace of $L^2$. So, the question changes to "Can I approximate every element of $\{ cos(nt)\}_{n=0}^{\infty}$ with the sines?" Since Joseph Fourier include cosine in his formula, the answer should be clear.