Intersection Points of Polar Equations

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Discussion Overview

The discussion revolves around finding the intersection points of two polar equations: r=1+3sin(theta) and r=1-3cos(theta). Participants explore methods to identify these points, particularly those that may not be immediately apparent due to the nature of polar graphs and their angles of intersection.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in finding intersection points due to the graphs intersecting at different angles, suggesting a need for assistance in identifying these points for area calculations.
  • Another participant proposes a method of setting the equations equal to each other and solving for theta, indicating that this leads to specific angles where intersections occur.
  • A later reply emphasizes the need for additional intersection points, particularly those where the limacons pass through each other's inner loops, which the original poster has not been able to find.
  • One participant suggests adding and subtracting Pi to find multiple solutions for theta, indicating that there are infinitely many answers due to the periodic nature of the tangent function.
  • Another participant encourages examining the graphs in Cartesian coordinates to potentially reveal unexpected insights about the intersections.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method for finding all intersection points, and multiple approaches are presented without resolution. The discussion remains unresolved regarding the identification of all intersection points.

Contextual Notes

Participants express varying levels of understanding and familiarity with polar equations and their graphical representations, indicating potential limitations in their approaches. The discussion highlights the complexity of finding intersection points in polar coordinates, particularly when considering angles and graph behavior.

jnbfive
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I've been having a problem finding the intersection points of the following polar equations.

r=1+3sin(theta)

and

r=1-3cos(theta)

Now I've set the equations equal to each other to obtain those points. I've set each equation equal to zero. The problem I'm having is that when graphed, there are intersection points that can't be found due to each graph passing through the respective point at a different angle. I was wondering if anyone could tell me how I would go about finding those intersection points; I need these points in order to find certain areas of the graph.

It would be of MASSIVE help if anyone could provide me with information. Thank you.
 
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Maybe I am missing something but don't you just do
1+3sin(theta) = 1-3cos(theta)
Sin[x]/Cos[x]=-1

Tan[x]=-1
x= 3Pi/4 or -Pi/4
then plug this into r equation to find the corresponding r coordinate
 
I said in my first post, first line of the first paragraph.

"I've been having a problem finding the intersection points of the following polar equations.

r=1+3sin(theta)

and

r=1-3cos(theta)

Now I've set the equations equal to each other to obtain those points. I've set each equation equal to zero. The problem I'm having is that when graphed, there are intersection points that can't be found due to each graph passing through the respective point at a different angle. I was wondering if anyone could tell me how I would go about finding those intersection points; I need these points in order to find certain areas of the graph.

It would be of MASSIVE help if anyone could provide me with information. Thank you."


I have those points. I need the other points. Its easier to understand if you have a graphing calculator handy and plug them into it. The points are when each limacon passes through the inner loop of the other limacon. Those are what I can't find.

And I'm sorry if I came across as testy. Its just I've been working on this problem for the past 3 days. I've expended every possible resource that I know of; no one in my class knows how to mathematically obtain those points. It's just really bothersome that I can't figure it out.
 
Just add and subtract Pi
Tan[-Pi/4+Pi]=-1
Tan[3Pi/4+Pi]=-1
Tan[3Pi/4+Pi+Pi]=-1
Tan[-Pi/4+Pi+Pi]=-1
etc etc. there are an infinite amount of answers.
 
Well this seems to be revealing something about polar plots I never thought of before. Just do your graphs Cartesian-wise and see whether you don't see something unexpected! :wink:

Then you may be able to see what it is that is causing you this pain.
 

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