in looking up some specific info on polar coordinates, I came across a plot of the polar conversion of $y=sin(6x)+2$ on the wikipedia page. This equation converts to $r=sin(6\theta)+2$. I've been trying to prove this to myself, but can't figure out how to do it.
I begin by writing $r\cdot sin\theta =sin(6\cdot r \cdot cos\theta)$ and get stuck rather quickly. I've tried looking for trig identities that may help me simplify this, but I haven't been able to figure it out. I also tried converting the sines and cosines to complex exponetials using Euler's formula but I still was unable to get to the equation in polar form.
I'm looking for any guidance on how to make this conversion. Thank you.
Polar and Cartesian coordinates are two coordinate systems for describing the same points - so, for example, the point one unit above the origin is $(0,1)$ in Cartesian coordinates and $(\frac{\pi}{2},1)$ in polar (using the order $(\theta, r)$). Think of this as a different way of talking about the same point.
So when you "convert" an equation from Cartesian into polar, you shouldn't be changing the graph - you're just changing how you talk about the graph. For example, the equation $y = \sqrt{1 - x^2}$ in Cartesian coordinates can be rewritten as $r = 1$, $0 \leq \theta \leq \pi$ in polar. Notice that these equations look nothing alike - they produce the same graph, but the coordinate systems are so different that the equations have to be written very differently.
Likewise, $y = 6\sin{x} + 2$ can't be "converted" into polar by writing $r = 6\sin{\theta} + 2$ - a quick test with any graphing software you like should convince you that $r = 6\sin{\theta} + 2$ makes a big round shape, while $y = 6\sin{x} + 2$ makes a wiggly curve extending infinitely in each direction. Since those are different graph, this isn't a conversion.
But you can think of it as a transformation - the idea is that the act of "replacing" coordinates in one system with coordinates in the other system can be thought of as a function that turns points into other points. For example, the point $(0,1)$ in Cartesian coordinates could be transformed into the point $(0,1)$ in polar coordinates - which would mean $r = 1$ and $\theta = 0$, which would be the point with Cartesian coordinates $(1,0)$. Turning $y = 6\sin{x} + 2$ into $r = 6\sin{\theta} + 2$ is doing exactly this: you're transforming every point on the plane this way, and watching where the ones along your curve wound up.