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Determining 4 intersection points of two polar graphs?

  1. Aug 26, 2007 #1
    1. The problem statement, all variables and given/known data
    r = 2
    r^2 = 9sin(2theta)
    Find the 4 points of intersection


    2. Relevant equations



    3. The attempt at a solution
    Since r = 2, 4 = 9sin(2theta)...
    4/9 = sin(2theta)

    Taking the inverse of 4/9 only gives me one answer on the calculator (obviously), and I do not know where to attain the 3 other points. (also, is there a way to do this without a calculator/calculus?) - IE 4/9 = 2sin(theta)cos(theta), although I do not see how this helps.
     
  2. jcsd
  3. Aug 27, 2007 #2
    [tex]\sin 2\theta=\frac{4}{9}\Rightarrow 2\theta =(-1)^k\arcsin\frac{4}{9}+k\pi\Rightarrow\theta =(-1)^k\frac{\arcsin\frac{4}{9}}{2}+\frac{k\pi}{2}[/tex]
    [tex]k=0\Rightarrow\theta=\frac{1}{2}\arcsin\frac{4}{9}[/tex]
    [tex]k=1\Rightarrow\theta=\frac{\pi}{2}-\frac{1}{2}\arcsin\frac{4}{9}[/tex]
    [tex]k=2\Rightarrow\theta=\pi+\frac{1}{2}\arcsin\frac{4}{9}[/tex]
    [tex]k=3\Rightarrow\theta=\frac{3\pi}{2}-\frac{1}{2}\arcsin\frac{4}{9}[/tex]
     
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