Pre-Calc

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Convert r=1+3sin2x from a polar equation to a rectangular equation.

  • Pre-Calc -

    I assume you mean

    r = 1 + 3 sin 2t
    where t = theta

    x = r cos t
    y = r sin t

    sin 2 t = 2 sin t cos t
    so
    r = 1 + 6 sin t cos t
    r = 1 + 6 (y/r)(x/r)
    r^3 = 1 + 6 x y
    but r^2 = x^2+y^2
    r= (x^2+y^2)^.5
    so
    (x^2+y^2)^(3/2) = 1 + 6 x y
    (x^2+y^2)^3 = 1 + 12 x y + 36 x^2 y^2
    you can multiply the left out

  • Pre-Calc -

    wouldnt the "1"in the 1+.... in the 4th step become r^2 when you multiply both sides by r^2.

  • Pre-Calc -

    You are right
    r^3 = r^2 + 6 x y
    (x^2+y^2)^3 = (x^2+y^2)^2 + 6 x y

  • Pre-Calc -

    x^6 + 3 x^4y^2 + 3 x^2 y^4 + y^6
    = x^4 + 2 x^2y^2 + y^4 + 6 x y

  • Pre-Calc -

    Thanks a lot for your help

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