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Math Help - matrix transformation for trig identities

  1. #1
    Member
    Joined
    May 2009
    Posts
    91

    Wink matrix transformation for trig identities

    Hi folks,

    Given that the following matrix creates an anticlockwise rotation of the x-y plane about the origin through an angle \theta :

    \left( \begin{array}{cc} <br />
\cos\theta & -\sin\theta \\<br />
\sin\theta & \cos\theta <br />
\end{array} \right)<br />

    use this fact to obtain the standard trig identities:

    \sin(\theta + \alpha) = \sin\theta\cos\alpha + cos\theta\sin\alpha<br />
and

    \cos(\theta + \alpha) = \cos\theta\cos\alpha - \sin\theta\sin\alpha<br />

    I can obtain the identities using geometry and I can see how the matrix transformation creates an anticlockwise rotation, but I can't see how to use the matrix trransformation to generate the identities. Can anyone help?
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  2. #2
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    Calculate

    <br />
\left( \begin{array}{cc} <br />
\cos\theta & -\sin\theta \\<br />
\sin\theta & \cos\theta <br />
\end{array} \right)<br />
\left( \begin{array}{cc} <br />
\cos\alpha & -\sin\alpha \\<br />
\sin\alpha & \cos\alpha <br />
\end{array} \right)<br />

    and note that this is equivalent to rotating through by \theta + \alpha, which is the matrix

    <br />
\left( \begin{array}{cc} <br />
\cos(\theta + \alpha) & -\sin(\theta + \alpha) \\<br />
\sin(\theta + \alpha) & \cos(\theta + \alpha) <br />
\end{array} \right)
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  3. #3
    Member
    Joined
    May 2009
    Posts
    91
    Ahhhhh! Now I see. Thanks very much Icemanfan.
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