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Math Help - polar forms

  1. #1
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    polar forms

    how can i express (1+i)/(sqrt(3)+i) in the x+iy form? Also, using those two smaller expressions, i must show that cos(pi/12) = (sqrt(3)+1)/(2sqrt(2)) and sin(pi/12) = (sqrt(3)-1)/(2sqrt(2))

    another one, i must show that if w is an nth root of unity, then w = 1/w. deduce that (1-w)^n = (w-1)^n. (showing that (1-w)^2n is real)
    the underscores go above the numbers, to show the conjugate
    Last edited by mistykz; September 10th 2007 at 11:39 AM. Reason: another question
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  2. #2
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    The quotient of two complex numbers can be obtained by dividing the moduli and subtrating the arguments.
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  3. #3
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    For any complex number we have \frac{1}{z} = \frac{{\overline z }}{{\left| z \right|^2 }}.
    So we have: \frac{{1 + i}}{{\sqrt 3  + i}} = \left( {1 + i} \right)\frac{{\sqrt 3  - i}}{2} = \left( {\frac{{\sqrt 3  + 1}}{2}} \right) + i\left( {\frac{{\sqrt 3  - 1}}{2}} \right).

    Now if w is an nth root of unity, from the above we see that \frac{1}{w} = \frac{{\overline w }}{{\left| w \right|^2 }} = \overline w.
    In general you need to show: \overline {\left( z \right)} ^n  = \left( {\overline z } \right)^n .
    Put together we get:
    \overline {\left( {1 - w} \right)} ^n  = \left( {1 - \overline w } \right)^n  = \left( {1 - \frac{1}{w}} \right)^n  = \left( {\frac{{w - 1}}{w}} \right)^n  = \frac{{\left( {w - 1} \right)^n }}{{w^n }} = \left( {w - 1} \right)^n
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