6.8: Equivalent Polar Curves
While working on a problem in math class, you get a solution with a certain equation. In this case, your solution is \begin{align*}3 + 2\cos (\theta)\end{align*}
At the conclusion of this Concept, you'll be able to determine if the solutions of you and your friend are equivalent.
Watch This
Learn more about families of polar curves by watching the video at this link.
Guidance
The expression “same only different” comes into play in this Concept. We will graph two distinct polar equations that will produce two equivalent graphs. Use your graphing calculator and create these curves as the equations are presented.
In some other Concepts, graphs were generated of a limaçon, a dimpled limaçon, a looped limaçon and a cardioid. All of these were of the form \begin{align*}r = a \pm b \sin \theta\end{align*}
Example A
Plot the following polar equations and compare the graphs.
a) \begin{align*}r & = 1 + 2 \sin \theta \\ r & = 1 + 2 \sin \theta\end{align*}
b) \begin{align*}r & = 4 \cos \theta \\ r & = 4 \cos (\theta)\end{align*}
Solution: By looking at the graphs, the result is the same. So, even though \begin{align*}a\end{align*}
b) These functions also result in the same graph. Here, \begin{align*}\theta\end{align*}
Example B
Graph the equations \begin{align*}x^2 + y^2 = 16\end{align*}
Solution:
Both equations, one in rectangular form and one in polar form, are circles with a radius of 4 and center at the origin.
Example C
Graph the equations \begin{align*}(x  2)^2 + (y + 2)^2 = 8\end{align*}
Solution: There is not a visual representation shown here, but on your calculator you should see that the graphs are circles centered at (2, 2) with a radius \begin{align*}2 \sqrt{2} \approx 2.8\end{align*}
Guided Practice
1. Write the rectangular equation \begin{align*}x^2 + y^2 = 6x\end{align*}
2. Determine if \begin{align*}r = 2 + \sin \theta\end{align*}
3. Determine if \begin{align*}r = 3 + 4 \cos ( \pi)\end{align*}
Solutions:
1.
\begin{align*}x^2 + y^2 & = 6x \\ r^2 & = 6(r \cos \theta) && r^2 = x^2 + y^2 \qquad \ \text{and} \qquad \ x = y \cos \theta \\ r & = 6 \cos \theta && \text{divide by}\ r \end{align*}
Both equations produced a circle with center \begin{align*}(3, 0)\end{align*}
2. \begin{align*}r =  2 + \sin \theta\end{align*}
3. \begin{align*}r = 3 + 4 \cos (  \pi)\end{align*}
Concept Problem Solution
As you learned in this Concept, we can compare graphs of equations to see if the equations are the same or not.
A graph of \begin{align*}3 + 2\cos (\theta)\end{align*}
And a graph of \begin{align*}3 + 2\cos (\theta)\end{align*}
As you can see from the plots, your friend is correct. Your graph and his are the same, therefore the equations are equivalent.
Explore More
For each equation in rectangular form given below, write the equivalent equation in polar form.

\begin{align*}x^2+y^2=4\end{align*}
x2+y2=4 
\begin{align*}x^2+y^2=6y\end{align*}
x2+y2=6y 
\begin{align*}(x1)^2+y^2=1\end{align*}
(x−1)2+y2=1 
\begin{align*}(x4)^2+(y1)^2=17\end{align*}
(x−4)2+(y−1)2=17 
\begin{align*}x^2+y^2=9\end{align*}
x2+y2=9
For each equation below in polar form, write another equation in polar form that will produce the same graph.
 \begin{align*}r=4+3\sin\theta\end{align*}
 \begin{align*}r=2\sin\theta\end{align*}
 \begin{align*}r=2+2\cos\theta\end{align*}
 \begin{align*}r=3\cos\theta\end{align*}
 \begin{align*}r=2+\sin\theta\end{align*}
Determine whether each of the following sets of equations produce equivalent graphs without graphing.
 \begin{align*}r=3\sin \theta\end{align*} and \begin{align*}r=3+\sin \theta\end{align*}
 \begin{align*}r=1+2\sin \theta\end{align*} and \begin{align*}r=1+2\sin \theta\end{align*}
 \begin{align*}r=3\sin \theta\end{align*} and \begin{align*}r=3\sin ( \theta )\end{align*}
 \begin{align*}r=2\cos \theta\end{align*} and \begin{align*}r=2\cos (\theta )\end{align*}
 \begin{align*}r=1+3\cos \theta\end{align*} and \begin{align*}r=13\cos \theta\end{align*}
Image Attributions
Description
Learning Objectives
Here you'll learn to determine if two polar equations are equivalent by inspection of their respective graphs.
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Date Created:
Sep 26, 2012Last Modified:
Feb 26, 2015Vocabulary
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