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Math Help - algebra step?

  1. #1
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    algebra step?

    An online text explaining rotation of axes includes this step:

    x = X cos \theta - Y sin \theta
    y = X sin \theta + Y cos \theta

    By solving Equations (above) for X and Y we obtain

    X = x cos \theta +y sin \theta
    Y = -x sin \theta + y cos \theta

    How do they get from the upper equations to the lower ones? Thanks in advance.
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  2. #2
    Moo
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    Hello,

    If you know a bit about matrices :

    The first 2 equations can be written this way :

    \begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}\c  os\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}X\\Y\end{pma  trix}

    So \begin{pmatrix}X\\Y\end{pmatrix}=\begin{pmatrix}\c  os\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}^{-1}\begin{pmatrix}x\\y\end{pmatrix}

    Recall how to find the inverse of a 2x2 matrix, recall that \cos^2\theta+\sin^2\theta=1 and you're done.
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