For some $n\in \mathbb N$, what is the set of points
$$\{\underline y=[\sin(x), \sin(2x),\dots \sin(nx)]^T,\ x\in [0,2\pi)\}?$$
The question naturally arises in the field of Fourier analysis, where the value of any ODD function is approximated by
$$f(x)\approx \sum_{i=1}^n a_i\sin(ix)=\langle \underline a, [\sin(x), \dots \sin(nx)]^T\rangle,$$
where $\underline a = [a_1, \dots a_n]^T$ are the Fourier coefficients.
For $n=2$, this set can be represented in the plane. I tried to plot it for 10k values of $x$ and this is what I got
