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Math Help - graph in the xy plane is

  1. #1
    Super Member bigwave's Avatar
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    graph in the xy plane is

    The graph in the xy-plane represented by
    x=3+2\sin{\theta}
    y=2\cos{\theta}-1
    for -\pi\le\theta\le\pi

    the answer is a circle with the equation
    (x-3)^2+(y+1)^2=4

    but got stuck on how this was derived especially from the trig functions
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  2. #2
    MHF Contributor
    skeeter's Avatar
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    Quote Originally Posted by bigwave View Post
    The graph in the xy-plane represented by
    x=3+2\sin{\theta}
    y=2\cos{\theta}-1
    for -\pi\le\theta\le\pi

    the answer is a circle with the equation
    (x-3)^2+(y+1)^2=4

    but got stuck on how this was derived especially from the trig functions
    x^2 = 9 + 12\sin{t} + 4\sin^2{t}

    y^2 = 4\cos^2{t} - 4\cos{t} + 1<br />

    -----------------------------------

    x^2 + y^2 = 9 + 4 + 1 + 12\sin{t} - 4\cos{t}

    x^2 + y^2 = 14 + 12\sin{t} - 4\cos{t}<br />

    x^2 + y^2 = 18 + 12\sin{t} - 4 - 4\cos{t}

    x^2 + y^2 = 6(3 + 2\sin{t}) + 2 - 4\cos{t} - 6

    x^2 + y^2 = 6(3 + 2\sin{t}) - 2(2\cos{t} - 1) - 6

    x^2 + y^2 = 6x - 2y - 6<br />

    x^2 - 6x + y^2 + 2y = -6

    x^2 - 6x + 9 + y^2 + 2y + 1 = -6 + 9 + 1

    (x - 3)^2 + (y + 1)^2 = 4<br />
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