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Math Help - Complex Number

  1. #1
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    Complex Number

    Simplify without the use of calculator

    [cos (pi/7) i sin (pi/7)]^3 / [ cos (pi/7) + i sin (pi/7)]^4

    How can I solve this?
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  2. #2
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    Hello geton,

    From your posts elsewhere I think you know the Re^{i\theta} form of a complex number.

    In this case you wont need the R.

    Once you have them in that form it should be easy to simplify.
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  3. #3
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    Quote Originally Posted by a tutor View Post
    Hello geton,

    From your posts elsewhere I think you know the Re^{i\theta} form of a complex number.

    In this case you wont need the R.

    Once you have them in that form it should be easy to simplify.
    I know z₁ = r(cosѲ + i sinѲ) & z₂ = r(cosѲ - i sinѲ). But how can I solve by this?
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  4. #4
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    Recall this basic principle:
    \left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3  = \left[ {\cos \left( {\frac{{ - \pi }}{7}} \right) + i\sin \left( {\frac{{ - \pi }}{7}} \right)} \right]^3  = \left[ {\cos \left( {\frac{{ - 3\pi }}{7}} \right) + i\sin \left( {\frac{{ - 3\pi }}{7}} \right)} \right].

    Thus:
    \frac{{\left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 }}{{\left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 }} = \left[ {\cos \left( { - \pi } \right) + i\sin \left( { - \pi } \right)} \right]
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  5. #5
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    Quote Originally Posted by Plato View Post
    Recall this basic principle:
    \left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3  = \left[ {\cos \left( {\frac{{ - \pi }}{7}} \right) + i\sin \left( {\frac{{ - \pi }}{7}} \right)} \right]^3  = \left[ {\cos \left( {\frac{{ - 3\pi }}{7}} \right) + i\sin \left( {\frac{{ - 3\pi }}{7}} \right)} \right].

    Thus:
    \frac{{\left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 }}{{\left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 }} = \left[ {\cos \left( { - \pi } \right) + i\sin \left( { - \pi } \right)} \right]

    But what about denominator. I didn't understand properly. Can you explain for me?
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  6. #6
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    Quote Originally Posted by geton View Post
    But what about denominator. I didn't understand properly. Can you explain for me?
    Using deMoivre's Theorem:


    \left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 = {\cos \left( {\frac{{4\pi }}{7}} \right) + i\sin \left( {\frac{{4\pi }}{7}} \right)}


    Then you just divide numerator by denominator using the usual rule for division of complex numbers expressed in polar form to get:


    \cos \left( -\frac{3 \pi}{7} - \frac{4 \pi}{7} \right) + i \sin \left( -\frac{3\pi}{7} - \frac{4\pi}{7} \right) = ...
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