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Math Help - Sketching x(t) and y(t) graphs for solution

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    Sketching x(t) and y(t) graphs for solution

    Hi, given the ellipse in the xy plane I have to sketch the x(t) & y(t) graphs for the solution. I have included a photo of the problem from the textbook (its problem # 24, the ellipse) as well as my attempt at a solution.
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    Re: Sketching x(t) and y(t) graphs for solution

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    Re: Sketching x(t) and y(t) graphs for solution

    Let $a$ be the radius of the ellipse along the x-axis and $b$ be the radius along the y-axis. Let $(h,k)$ be the center of the ellipse. Then the parametrization of the ellipse is:

    $x = h + a\cos t$
    $y = k + b\sin t$

    Your ellipse has $a=2, b=1, h=2, k=1$

    So, $x(t) = 2(1+\cos t)$, $y(t) = 1+\sin t$

    Hence, your graph for $x(t)$ looks correct, but your graph of $y(t)$ is not. Reflect $y(t)$ over the line $y=1$ and it will be what you drew.
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    Re: Sketching x(t) and y(t) graphs for solution

    Thank you very much, I really appreciate it. I was wondering, do I need to label the t axis with values, or are they unknown? At first I put that each complete wavelength took 2pi but for some reason I wasn't sure if there was enough info to determine the time it takes to complete each wavelength
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    Re: Sketching x(t) and y(t) graphs for solution

    Quote Originally Posted by grandmarquis84 View Post
    Thank you very much, I really appreciate it. I was wondering, do I need to label the t axis with values, or are they unknown? At first I put that each complete wavelength took 2pi but for some reason I wasn't sure if there was enough info to determine the time it takes to complete each wavelength
    Well, I gave you the standard parametrization of an ellipse. But, you can really parametrize it to have any period you want. But, the period for each must be the same. If you want to go with the standard, you are correct that it is $2\pi$.
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