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
At the conclusion of this Concept, you'll be able to determine if the solutions of you and your friend are equivalent.
Watch This
Families of Polar Curves: Circles, Cardiods, and Limacons
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
Example A
Plot the following polar equations and compare the graphs.
a)
b)
Solution: By looking at the graphs, the result is the same. So, even though
b) These functions also result in the same graph. Here,
Example B
Graph the equations
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
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
Vocabulary
Equivalent Polar Curves: A set of equivalent polar curves are two equations that are different in appearance but that produce identical graphs.
Guided Practice
1. Write the rectangular equation
2. Determine if
3. Determine if
Solutions:
1.
Both equations produced a circle with center
2.
3.
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
And a graph of
As you can see from the plots, your friend is correct. Your graph and his are the same, therefore the equations are equivalent.
Practice
For each equation in rectangular form given below, write the equivalent equation in polar form.

x2+y2=4 
x2+y2=6y 
(x−1)2+y2=1 
(x−4)2+(y−1)2=17 
x2+y2=9
For each equation below in polar form, write another equation in polar form that will produce the same graph.

r=4+3sinθ 
r=2−sinθ 
r=2+2cosθ 
r=3−cosθ 
r=2+sinθ
Determine whether each of the following sets of equations produce equivalent graphs without graphing.

r=3−sinθ andr=3+sinθ 
r=1+2sinθ andr=−1+2sinθ 
r=3sinθ andr=3sin(−θ) 
r=2cosθ andr=2cos(−θ)  \begin{align*}r=1+3\cos \theta\end{align*} and \begin{align*}r=13\cos \theta\end{align*}
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Image Attributions
Here you'll learn to determine if two polar equations are equivalent by inspection of their respective graphs.