I'm learning about polar equations and am trying to graph the curve r = 3 + 3 sin(theta), with theta between 0 and 2pi.
Based on what I've learned and the image below, since a = b = 3, it would seem as though the graph should be a cardioid with the indentation pointing to the side (a sideways heart, if you will). However, in Desmos, the graph seems to be like a one-loop limacon (an upside-down heart). What is causing this discrepancy? What rule can I use to determine the orientation of limacons and cardioids if the first image is not correct? Or, is Desmos incorrect in this case?
$\cos \theta$ and $\sin \theta$ are shifted from each other by an angle of $\frac \pi 2$. So you have to expect that $r = a \pm b\cos\theta$ and $r = a \pm b\sin\theta$ are going to be rotated by a right angle from each other. Also $r = a + b\sin\theta$ surely cannot look exactly like $r=a - b\sin\theta$. The first will have its maximum radius of $a + b$ at $\theta = \frac \pi2$ and its minimum radius of $a - b$ at $\theta = \frac {3\pi}2$. The latter will have them at the exact opposite points.
In other words, what that image is showing is only the shapes of the curves, not their orientations. The curves will always have the $x$ or $y$-axis as an axis of symmetry, but that still leaves four possible orientations, and all four are possible in each case, depending on whether you choose $\sin \theta$ or $\cos \theta$ and on whether the coefficient of the trig function is positive or negative. If you want to see curves whose symmetry axis is not the $x$ or $y$ axis, then you introduce a phase factor: $$r = a + b\sin(\theta - \theta_0)$$
I suggest you enter that equation into Desmos and set $a, b, \theta_0$ on sliders. See how the curve behaves as you change each of the values. It will help your intuition.