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Math Help - Trigonometric identities in matrix

  1. #1
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    Trigonometric identities in matrix

    Please can anyone shed some light on this?

    I would like to use trigonometric identities to simplify the 2x2 matrix in the attached word document.

    I have also included an example calculation from a paper but I don't understand how the simplified matrix was determined from the trig identities. Could anyone explain this to me?

    Any help would be much appreciated.
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  2. #2
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    Quote Originally Posted by stawbelly View Post
    Please can anyone shed some light on this?

    I would like to use trigonometric identities to simplify the 2x2 matrix in the attached word document.

    I have also included an example calculation from a paper but I don't understand how the simplified matrix was determined from the trig identities. Could anyone explain this to me?

    Any help would be much appreciated.
    Consider the 1,1 component of the example:
    sin(x)sin(z)cos(y) - cos(x)sin(y)

    We are told sin(z) = -1, so this is equal to
    -sin(x)cos(y) - cos(x)sin(y) = -[sin(x)cos(y) + cos(x)sin(y)]

    which is, according to identity 1 at the top of the page
    -sin(x + y)

    The rest of them are pretty much the same.

    For example in the question, look at the 1,2 component:
    -cos(x)sin(z)cos(y) + sin(x)sin(y)

    Again, sin(z) = -1 so this is equal to:
    cos(x)cos(y) + sin(x)sin(y) = cos(x - y) according to identity 4.

    -Dan
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  3. #3
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    Many thanks for your clear explanation.

    I have solved for the other components of the problem:

    1,1 component

    sin(x)sin(z)cos(y) + cos(x)sin(y)
    = -sin(x)cos(y) + cos(x)sin(y) = -sin(x-y) according to identity 3

    1,2 component (as shown in your post)

    -cos(x)sin(z)cos(y) + sin(x)sin(y)
    = cos(x)cos(y) + sin(x)sin(y) = cos(x - y) according to identity 4.

    2,1 component

    -sin(x)sin(z)sin(y) + cos(x)cos(y)
    = sin(x)sin(y) + cos(x)cos(y) = cos(x-y) according to identity 4

    2,2 component

    cos(x)sin(z)sin(y) + sin(x)cos(y)
    = -cos(x)sin(y) + sin(x)cos(y) = sin(x-y) according to identity 3

    I was a bit unsure about component 1,1 but if any of them are wrong, please let me know

    thanks!
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  4. #4
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    Quote Originally Posted by stawbelly View Post
    Many thanks for your clear explanation.

    I have solved for the other components of the problem:

    1,1 component

    sin(x)sin(z)cos(y) + cos(x)sin(y)
    = -sin(x)cos(y) + cos(x)sin(y) = -sin(x-y) according to identity 3

    1,2 component (as shown in your post)

    -cos(x)sin(z)cos(y) + sin(x)sin(y)
    = cos(x)cos(y) + sin(x)sin(y) = cos(x - y) according to identity 4.

    2,1 component

    -sin(x)sin(z)sin(y) + cos(x)cos(y)
    = sin(x)sin(y) + cos(x)cos(y) = cos(x-y) according to identity 4

    2,2 component

    cos(x)sin(z)sin(y) + sin(x)cos(y)
    = -cos(x)sin(y) + sin(x)cos(y) = sin(x-y) according to identity 3

    I was a bit unsure about component 1,1 but if any of them are wrong, please let me know

    thanks!
    They all look good to me.
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