In polar coordinate system: The polar coordinates $r$ is called the radial coordinate and $\theta$ is called the angular coordinate, often called the polar angle.
I am confused when answering the question that: Is it true that in this case $r$ should be a function of $\theta$ (or Is it possible that $r$ may not be a function of $\theta$, still $(r,\theta)$ represent valid polar co-ordinate system)?
Could anyone please help me to answer this question.
Thanks in advance.
In general, a set of points in the plane may not be the graph of $r=f(\theta)$ ($r$ as a function of $\theta$) in polar coordinates. However, it is possible for the graph of $r=f(\theta)$ to be a self-intersecting curve, for some choices of the function $f$.
More generally, not only is it possible for the graph of $r=f(\theta)$ to have points at more than one distance $r$ along a single radial line, but there are several well-known functions that do exactly that. For example, if you plot $r = e^{\theta/20}$, you will get a spiral that wraps around the origin as many times as you want, depending on the values of $\theta$ that you plot. Each "wrap" is separated from the ones before and after it, so you can have a very large number of values of $r$ in the same direction. The secret is, one point is at $r=f(\theta),$ another is at $r=f(\theta+2\pi),$ another is at $r=f(\theta+4\pi),$ and so forth, because every time you add $2\pi$ to the polar angle you come back to the same radial line.
You can make a plot of this function by following this link.
That particular function does not create a self-intersecting curve, but other functions do. For example, you can plot $r = e^{-(\theta^2)/16}$ by following this link. The plot of this function intersects itself more than once.
This is unlike plots of $y=f(x)$ in Cartesian coordinates, where it is impossible to plot a self-intersecting curve in that way because each point has a unique pair of $(x,y)$ coordinates. Polar coordinates, on the other hand, allow you to plot the same point in many different ways.