Density of $\sin(nx)$

Question

Let $\displaystyle \left(c_n\right)_{n \in \mathbb{N}}$ be the sequence defined by $$ c_n\left(x\right)=\cos\left(nx\right) $$ and the set $F$ as $$ F=\underset{n \in \mathbb{N}}{\text{Span}}\left(c_0,c_1, \dots , c_n\right) $$ I've proved that $F$ was dense to the set of $2 \pi $-periodic and continuous functions on $\left[0, \pi\right]$. Can I deduce from it that it is the same if i replace it by $s_n$ with $\displaystyle s_n(x)=\sin\left(nx\right)$ ? (since it is just $\cos(nx+\pi/2)$ )

Answer 1

How about: $s_n(0)=0$ for all $n$, any function $h$ in $$ H = \mathrm{span}(s_0,s_1,s_2,\cdots) $$ satisfies $h(0)=0$. But the constant $1$ is $2\pi$-periodic, and cannot be approximated uniformly (or even pointwise) by functions in $H$.