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Math Help - the rotation of a sinusoid curve

  1. #1
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    the rotation of a sinusoid curve

    Hello,
    I have a sinusoid curve in an euclidian orthonormal plane [OX,OY].
    the equation of the curve is y=a sin x.

    Now, I want to get the equation of the curve when I rotate it by an angle with respect to the (ox) axis like it is shown in the figure submitted.

    To do that,I'll define tehe vectorial euclidian rotation :in an oriented vectorial euclidian plan, a vectorial rotation is simply defined by its angle . Its matrix in an orthonormal direct basis is :


    In another way, a vector (x,y) has as image the vector (x',y') that can be calculated like that :



    then we have :

    and


    as we have

    , then we have



    the problem is that I must write in function of .

    Thank you for reading me. I'll be vey glad if you can help me
    Attached Thumbnails Attached Thumbnails the  rotation of a  sinusoid curve-sinusoidale.gif  
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  2. #2
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    Were you given restrictions on \alpha? Because for \alpha > 45^o, y' cannot be expressed as a function of x' (since it fails the vertical line test).
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  3. #3
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    Hello Snowtea,

    thank you for answering but I didn't understand why I must give restrictions to ?
    I tried to solve the problem like that but It seems to me complicated:

    As we have y=a sin x.

    and if we rotate the vector v(x',y') in an angle -, we have then u(x,y) and the matrix of rotation is:


    $\begin{pmatrix}<br />
x\\y\end{pmatrix}=\begin{pmatrix}-\cos\alpha&-\sin\alpha \\ \sin\alpha&-\cos\alpha\end{pmatrix}\begin{pmatrix}x'\\y'\end{p  matrix},$<br />

    then we have

    <br />
$x = -x' \cos \alpha - y' \sin \alpha\,$<br />
    and
    $y = x' \sin \alpha - y' \cos \alpha\,$

    then we replace x and y in the equation y=a sin x by their expressions in funtion of x' and y'.

    we then have:
    $$ x' \sin \alpha - y' \cos \alpha\ = a sin(-x' \cos \alpha - y' \sin \alpha)$$

    The problm now is how to write the equation in a simple form y'=f(x').


    Could you help me.
    Thank you
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  4. #4
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    You can only write y' = f(x') when for every x', you have a unique y' that satisfies the equation.

    If \alpha > \pi/4, just consider the simple case when \alpha = \pi/2.
    Then for x'=0, y'\in\{0, \pi, -\pi, ...\} are all solutions, so y' cannot be written as a function of x'.
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