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Math Help - help with cosine and sine difference formulas and axes of rotation.

  1. #1
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    help with cosine and sine difference formulas and axes of rotation.

    I am trying to derive the clockwise rotation linear combination...

    So, the counter-clockwise rotation matrix is

    \cos\alpha (-\sin\alpha)
    \sin\alpha \cos\alpha

    ``derived" from \cos(\alpha + \beta), s.t.

    x = r[\cos(\alpha + \beta)] = r[\cos\alpha \cos\beta - \sin\alpha \sin\beta]

    = \cos\alpha (r \cos\beta) - \sin\alpha (r \sin\beta)

    = X \cos\alpha - Y \sin\alpha.

    Similarly,

    y = r[\sin(\alpha + \beta) = r[\sin\alpha \cos\alpha + \cos\alpha \sin\beta]

    = \sin\alpha (r \cos\beta) + \cos\alpha (r \sin\beta)

    = X \sin\alpha + Y \cos\alpha.

    Now, I know you make a linear combination out of these using matrix multiplication as I listed the counter-clockwise matrix above...and that you can just use the fact that a negative \beta will ``switch" the matrix to a clockwise rotation matrix, i.e., the same thing as multiplying the matrix by negative 1.

    However, I am not able to solve correctly for y in the sine difference formula, if I start with the negative \beta in the first place, and this is where I need some help...

    For starters, let's solve for x...as follows:

    x = r[\cos(\alpha + (-\beta))] = r[\cos\alpha \cos(-\beta) - \sin\alpha \sin(-\beta)]

    = r[\cos\alpha \cos\beta + \sin\alpha \sin\beta]

    = \cos\alpha (r \cos\beta) + \sin\alpha (r \sin\beta)

    = X \cos\alpha + Y \sin\alpha.

    Now here's the rub, watch this...

    y = r[\sin(\alpha + (-\beta))] = r[\sin\alpha \cos(-\beta) + \cos\alpha \sin(-\beta)]

    = r[\sin\alpha \cos\beta - \cos\alpha \sin\beta]

    = \sin\alpha (r \cos\beta) - \cos\alpha (r \sin\beta)

    = X \sin\alpha - Y \cos\alpha.

    As you see, this gives an incorrect clockwise rotation matrix, as follows:

    \cos\alpha \sin\alpha
    \sin\alpha (-\cos\alpha).

    Whereas it should be...

    \cos\alpha \sin\alpha
    (-\sin\alpha) \cos\alpha.

    Please advise, thanks,

    carmamer
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  2. #2
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    Re: help with cosine and sine difference formulas and axes of rotation.

    I solved the relevant issues pertaining to this question - so this thread/question can be retired.
    Last edited by carmamer; June 12th 2012 at 03:03 PM.
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