# Graphing Basic Polar Equations

## Key Questions

• Limacons are polar functions of the type:
$r = a \pm b \cos \left(\theta\right)$
$r = a \pm b \sin \left(\theta\right)$
With $| \frac{a}{b} | < 1$ or $1 < | \frac{a}{b} | < 2$ or $| \frac{a}{b} | \ge 2$

Consider, for example: $r = 2 + 3 \cos \left(\theta\right)$
Graphically:

Cardioids are polar functions of the type:
$r = a \pm b \cos \left(\theta\right)$
$r = a \pm b \sin \left(\theta\right)$
But with $| \frac{a}{b} | = 1$

Consider, for example: $r = 2 + 2 \cos \left(\theta\right)$
Graphically:

in both cases:
$0 \le \theta \le 2 \pi$

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I used Excel to plot the graphs and in both cases to obtain the values in the $x$ and $y$ columns you must remember the relationship between polar (first two columns) and rectangular (second two columns) coordinates:

• You can find a lot of information and easy explained stuff in "K. A. Stroud - Engineering Mathematics. MacMillan, p. 539, 1970", such as:

If you want to plot them in Cartesian coordinates remember the transformation:
$x = r \cos \left(\theta\right)$
$y = r \sin \left(\theta\right)$

For example:
in the first one: $r = a \sin \left(\theta\right)$ choose various values of the angle $\theta$ evaluate the corresponding $r$ and plug them into the transformation equations for $x \mathmr{and} y$. Try it with a program such as Excel... it is fun!!!

• You consider a function of the type:
$r = f \left(\theta\right)$

So you give values of the angle $\theta$ and the function gives you values of $r$.

To graph polar functions you have to find points that lie at a distance $r$ from the origin and form (the segment $r$) an angle $\theta$ with the $x$ axis.

Take for example the polar function:
$r = 3$

This function describes points that for every angle $\theta$ lie at a distance of 3 from the origin!!!

Graphically:

The result is a circle of radius $r = 3$.

Now, the only complication is when $r$ becomes NEGATIVE ...how do I plot this?
We use a trick....we take the positive and flip it about the origin!!!!!!

Take for example the polar function:
$r = - 3$

This function describes points that for every angle $\theta$ lie at a distance of...-3 from the origin????
We use our trick!

Graphically:

Every point of the old graph flipped about the origin!!!!
It is a circle...again!!!!

Now try by yourself with:
$r = 2 \cos \left(\theta\right)$
Build a table of $\theta$ and $r$ and plot it...you should get another circle but with its center....on the $x$ axis (in $\left(1 , 0\right)$) and radius =1.

There are more complicated (and graphically beautiful) polar functions such as limacons, cardioids, roses, lemniscates, etc…try them!!!