[Math] How to know that a limaçon is going to inner loop

Question

I was setting up the graph for this polar function:
r = 1 + 2sin(Θ)

I set up a table incrementing by π/2

Θ__|__r

0____1

π/2____3

π____1

3π/2____-1

2π____1

5π/2____3

3π____1

7π/2____-1

4π____1

My problem Is I set up the plot points on the graph and begin connecting the dots. But there comes a point, where I have it in red that I don't know where it should go, how do I know it's suppose to loop inwards like this? How do I know it's suppose to go through the origin and loop?

This is what I originally thought (connecting the dots):

And the part in red where I got confused:

Please help

Thank you

Answer 1

You need to plot the points at finer intervals. $\sin \theta$ has period $2 \pi$ so your table is twice as long as it needs to be. Here it is with intervals of $\pi/16$ in $\theta$. I had to convert the points to rectangular to get Excel to plot it.