Help understanding polar coordinates and conversion between polar and rectangular

Question

I'm not understand this. I understand that you can take normal functions with x's and y's and convert them into polar coordinates. I also understand that the polar form of that function will have the same exact graph because your getting the same points just a different way. What I do not get is this, if you take something like y=2x and just replace the x with θ and take the y and replace it with r, you will get the same exact outputs for he same inputs but the groans look completely different. Is this called something special? I don't get this because the way I see it, you can take functions like the one above and convert them into polar form(which just gives us the same graph because like I said it's just getting the inputs and outputs a different way) and then there's this thing that we can do by replacing the x with θ and the y with r. Are these two separate things? What's going on!?

Answer 1

In order to convert something from rectangular to polar, convert $x=r\cos\theta$ and $y=r\sin\theta$ Also, $x^2+y^2=r^2$.

The way the rectangular coordinate system works is that for each point, you define it's displacement vertically and horizontally from the origin. So suppose point (1,2) would be 1 unit to the right and 2 units up.

The way the polar coordinate system works is that for each point, you define an angle to point at, with $\theta=0$ at the positive $x$-axis, and the distance from the origin. So for example, ($\pi/4$,5) would be a point 5 units away from the origin pointing at the direction $\pi/4$.

The conversions from rectangular to polar coordinates are illustrated below:

EDIT: To answer more:
Converted $y=2x$ to $r=2\theta$ doesn't really mean anything. For the second equation, you are saying that you want the distance from the origin to be twice the radians that you are facing from the center. You graph these equations in different ways. The image below shows how to graph polar graphs:

In conclusion, $x$ and $y$ are just arbritary variables, as are $\theta$ and $r$. What is important is how you define them in terms of plotting them. Generally, we use $x,y$ for rectangular plots, and we use $r,\theta$ for polar graphs. They are completely different ways of graphhing things.