How am I solving for the graphing of a polar equation incorrectly?

Question

I am asked to graph the polar equation: $r = 1 + 2\cos(\theta)$

I create a table, I've read that this table should start at $0$ and increment by $\pi/2$, unless there is some 'n' in the theta of the polar equation, in that case it's $n\theta = \pi/2$ and we increment by that result.

So what I did was set up the table like so:

$\theta........r$

$======$

$0........3$$

$\pi/4......1$

$\pi/2.....-1$

$3\pi/4.....1$

$\pi........3$

$5\pi/4.....1$

$3\pi/2....-1$

I got the $r$ values from evaluating the function with those thetas. I graph it by first point in the direction of theta and going the distance of $r$ always wrapping around counter clockwise.

My graph looked like this:

Is there something off with my approach? My table? the increments?

Please help

Thank you

Answer 1

This is what you should get (I used python/matplotlib):

Answer 2

Your values for $r$ are incorrect.

$$ \begin{align*} r(0) &= 1+2(1) = 3 \\ r(\pi/4) &= 1 + 2(\sqrt{2}/2) = 1 + \sqrt{2} \approx 2.414 \\ r(\pi/2) &= 1 + 2(0) = 1 \\ r(3\pi/4) &= 1 + 2(-\sqrt{2}/2) = 1 - \sqrt{2} \approx -0.414 \\ r(\pi) &= 1 + 2(-1) = -1 \\ r(5\pi/4) &= 1 + 2(-\sqrt{2}/2) = 1 - \sqrt{2} \approx -0.414 \\ r(3\pi/2) &= 1 + 2(0) = 1 \end{align*} $$

Now plot these points and you should get something that looks like the plot in Andrei's answer.