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Polar to Cartesian conversion - how related?

  1. Jan 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the cartesian equation for the curve r=csctheta

    3. The attempt at a solution
    I understand how to get the answer, by changing it to r=1/sin, and then rsin=1, and then since y=rsin, then y=1.

    What I'm not understanding is the relationship between y=1 an r=csc. I thought that when you convert to a cartesian equation, it's supposed to either look the same when graphed or be able to be used to plot the r and theta coordinates onto a polar plane. But y=1 translates to r=1 on a polar plane, and r=csc is nothing like that?? What am I not understanding here?
     
  2. jcsd
  3. Jan 25, 2010 #2

    LCKurtz

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    Draw a cartesian coordinate system and draw the horizontal line y = 1. Then draw a line from the origin to the point (3,1) on your horizontal line and make that a radius vector r. Label the polar coordinate angle [itex]\theta[/itex] in the proper position. Drop a line from your point (3,1) to the x axis at (3,0) forming a little triangle with r and [itex]\theta[/itex] as part of it. The sides of that triangle are r, 3, and 1 units long. Notice that for that triangle that

    [tex]\csc\theta = \frac r 1[/tex]

    Then notice if you do the same thing using (x,1) instead of (3,1) you will still get

    [tex]\csc\theta = \frac r 1[/tex]

    So that equation [itex]r =\csc\theta[/itex] literally gives the same graph as y = 1.

    [edit] fixed two typos
     
    Last edited: Jan 26, 2010
  4. Jan 26, 2010 #3
    Excellent! That helps a lot. But the last bit about using (x,1) - not sure how you're getting csc=r/x. Wouldn't it still be r/1?
     
  5. Jan 26, 2010 #4

    HallsofIvy

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    Yes, that was probably a typo. It is precisely because [itex]csc(\theta)= r/1[/itex] that [/itex]r= (1)csc(\theta)[/itex].

    "r= 1" is NOT "y= 1". It's graph is the circle with center at the origin and radius 1.
     
  6. Jan 26, 2010 #5

    LCKurtz

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    Yes. That was a typo as Halls surmised. There was another one too. I'll fix both.
     
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