Dense set in the unit circle- reference needed For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit circle $B(0,1)$ of $\mathbb R^2$. Could someone please give me a proof or reference for a proof?
From $x\notin \pi{\mathbb Q}$ it follows that $e^{ikx}\ne1$ for all $k\in{\mathbb Z}\setminus\{0\}$, and this implies that the numbers $e^{inx}\in S^1$ $(n\geq0)$ are all different. Assume that a point $\zeta\in S^1$ and an $\epsilon&gt;0$ is given. Since $S^1$ has finite length we then can find two numbers $n_1&lt;n_2$ with $|e^{in_2x}-e^{in_1 x}|&lt; \epsilon$. Put $n':=n_2-n_1$. Then
$$\bigl|e^{in'x} -1\bigr|=|e^{in_2x}-e^{in_1 x}|&lt;\epsilon\ .$$
Put $n_k:=k\&gt;n'$ $(k\geq0)$. Then the successive points $e^{i n_k x}\in S^1$ $(k\geq0)$ have a distance $&lt;\epsilon$. It follows that there is a $k\geq0$ with $|e^{in_k x}-\zeta|&lt;\epsilon$.
You can proove that every rational number other than 1/2 comes from an irrational angle.  


a = cos(pi d/z) + i sin(pi d/z) solves some equation of the form a^z + 1 = 0
Prove that a can never be a rational, by supposing that where

$$a^n = \sum^{z-1}_{e=0} x_i a^e$$


that if a were a fraction p/q, multiply through by p^n, then the LHS is not a multiple of q, and the RHS is, unless q=1.  

Therefore the equation a can never be rational unless it is half-integer, which means that the rational numbers arise from irrational angles.

Since the rational numbers are dense on the circle, the irrational angles are too.