# What is the name of the shape graphed by the function r =1+3 sin theta?

May 13, 2017

It is called a Limacon, French for snail.

#### Explanation:

Limacons were studied by Pascal.

May 13, 2017

Limaçon

#### Explanation:

Consider the following shapes for polar graphs:

Cardiod:
$r = a \pm b \cos \theta$ and $r = a \pm b \sin \theta$
where $a = b$

Limaçon:
$r = a \pm b \cos \theta$ and $r = a \pm b \sin \theta$
where $a \ne b$

Rose Curve:
$r = a \cos \left(n \theta\right)$ and $r = a \sin \left(n \theta\right)$
if $n$ is odd, then $n$ petals
if $n$ is event, then $2 n$ petals

Lemniscate
${r}^{2} = {a}^{2} \cos \left(2 \theta\right)$ and ${r}^{2} = {a}^{2} \sin \left(2 \theta\right)$

Based on this list, we can determine that the given polar equation is a Limaçon