What equation represents a straight line at $n$ degrees/radians in polar coordinate system?
I know in Cartesian coordinates: $y = \sin\frac{2\pi}{3}x +0$
Polar conversion: $x = r \cos\rho$ and $y = r \sin\rho$
So I substitue: $0 = \sin \frac{2\pi}{3}x - y$
$0 = \sin \frac{2\pi}{3} r \cos \theta - r \sin \theta$
$0 = r(\sin \frac{2\pi}{3} \cos \theta - \sin \theta)$
So $r$ is zero???
A polar equation is given using $r$ (the radius) and $\theta$ (the angle), just like a cartesian equation is given using $x$ and $y$.
The straight line through the origin making an angle $\alpha$ with the positive $x$-axis would then simply be given by the equation $$\theta=\alpha.$$ This equation leaves $r$ free to take any (positive, negative, or zero) value, so it corresponds to all points on that line. For $\alpha=\frac{2\pi}{3}$ radians (120 degrees), the equation would be $$\theta=\frac{2\pi}{3}.$$
You can also see this from the equations of the cartesian line. The line would be $$\begin{align*} y &= \frac{\sin \alpha}{\cos\alpha}x\\ (\cos\alpha)y &= (\sin\alpha)x\\ (\cos\alpha)r\sin\theta &= (\sin\alpha)r\cos\theta. \end{align*}$$ The value of $r$ is irrelevant, so we can cancel (when $r=0$ we get the same point regardless of the value of $\theta$). This leaves $$\tan\theta = \tan\alpha,$$ which gives $\theta=\alpha+n\pi$ for an arbitrary integer $n$. Now simply note that $\theta=\alpha+n\pi$ for any integer $n$ describes the same line.