# What is the graph of r = 2a(1 + cosθ)?

Feb 23, 2016

Your polar plot should look something like this:

#### Explanation:

The question is asking us to create a polar plot of a function of angle, $\theta$, which gives us $r$, the distance from the origin. Before starting we should get an idea of the range of $r$ values we can expect. That will help us decide on a scale for our axes.

The function $\cos \left(\theta\right)$ has a range $\left[- 1 , + 1\right]$ so the quantity in parentheses $1 + \cos \left(\theta\right)$ has a range $\left[0 , 2\right]$. We then multiply that by $2 a$ giving:

$r = 2 a \left(1 + \cos \left(\theta\right)\right) \in \left[0 , 4 a\right]$

This is the ditance to the origin, which could be at any angle, so let's make our axes, $x$ and $y$ run from $- 4 a$ to $+ 4 a$ just in case:

Next, it's useful to make a table of the value of our function. We know that $\theta \in \left[0 , {360}^{o}\right]$ and let's break it up into 25 points (we use 25 because that makes 24 steps between points which are angles of ${15}^{o}$):

Where we have also included a calculation of the Cartesian coordinates of each point where $x = r \cdot \cos \theta$ and $y = r \cdot \sin \theta$. We now have a choice, we can plot the points using a protractor for the angle and a ruler for the radius, or just use the $\left(x , y\right)$ coordinates. When you are done, you should have something that looks like this: