Draw $r ≤ 3 + 2\sin (\theta)$

Question

Currently I'm stuck at this fairly easy task. All I have to do is sketch the region $r \le 3+2\sin \theta$. My guess would be that the circle has the origin $(0,3)$ with $r = 2$, as I use the formula $y = b + r \sin \theta$. But that's completely wrong.

EDIT: I am interesting in the right procedure for drawing a region like this, not the actual plot itself. This was a question for a previous test without the use of calculators.

Answer 1

This equation is describing a limacon.

You mention that you have to sketch a circle, but it's not a circle. You can find the drawing corresponding to the equation here.

$r=3+2\sin \theta$ is an equation that require to use polar coordinates to draw the graph. Be sure to be familiar with it.

Pay attention, in your title you are using "=" but "≤" in your question. "=" will be a line whereas "≤" will be the surface inside that line.

Answer 2

Sometimes a picture is worth 1000 words:

If you want to work "by hand," you can make a traditional (rectilinear) plots of $r$, $x$ and $y$ (actually, you only need $x$ and $y$), then sample pairs of coordinates at different $\theta$s and transfer them to your polar graph.