Is this function a surjection? Consider the set $S_n$ defined as:
$S_n=\lbrace \mathbf{x} \in \mathbb{R}^n: x_i \neq0,\text{} \forall \text{ }i=1,2,..,n\rbrace$,
i.e. $S_n$ is the set of real vectors of dimension $n$ that have all entries different than zero.
Is the function $f_n(r,\theta) = \left(r\cos(\theta),r\cos(2\theta),...,r\cos(n\theta)\right)^T$ a surjection from $\mathbb{R}^2$ to $S_n$ for any natural $n$?
How to prove it?
Thank you in advance!
 A: $n=2$: I'll use $t$ for $\theta.$ Let's first take $r=1, t\in [-\pi/4, \pi/4].$ Recall that $\cos 2t = 2\cos^2t - 1.$ So the set of points traced out by $f$ from this subdomain has the form $(\cos t, 2\cos^2t-1), t\in [-\pi/4, \pi/4].$ This is exactly the set $(x,2x^2 -1): x\in [-1/\sqrt 2, 1/\sqrt 2],$ i.e., it's the graph of $y=2x^2-1$ over this interval.
If we look at $r=-1, t\in [-\pi/4, \pi/4],$ we will get the graph of $y=1-2x^2, x\in [-1/\sqrt 2, 1/\sqrt 2].$ In other words, we get the reflection of the first graph with respect to the $x$-axis.
The union of the above two graphs, let's call it $G,$ has the property that for every $a\in [0,2\pi],$ there exists $s(a) >0$ such that $s(a)e^{ia} \in G.$  Now $G\subset f(\mathbb R^2).$ Since $uf(\mathbb R^2)\subset f(\mathbb R^2)$ for any real $u,$ the line thru the origin determined by $e^{ia}$ is contained in $f(\mathbb R^2)$ for every $a\in [0,2\pi].$ Thus $f(\mathbb R^2)=\mathbb R^2.$
$n>2:$ Here $f$ is nonsurjective to say the least. Suppose $n=3.$ Let $$\gamma = \{(\cos t, \cos 2t, \cos 3t):t\in [0,2\pi]\}.$$ Then $f(\mathbb R^2) = \mathbb R\cdot \gamma.$ But $\gamma$ only contains one point in the $y-z$ plane, namely, $(0,-1,0).$ Thus the intersection of $\mathbb R\cdot \gamma $ with the $y-z$ plane is just the line $(0,y,0): y \in \mathbb R.$ The result for $n>3$ follows.
Another way to proceed for $n>2$ is through measure theory: If $f:\mathbb R^j\to \mathbb R^k$ is a smooth map and $j<k,$ then $f(\mathbb R^j)$ has measure $0$ in $\mathbb R^k.$ Hence such an $f$ misses being surjective by miles.
A: No. For example $(1,0, \dots, 0)$ is not in the image.
