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Thread: Help with Complex Number Question!

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    Help with Complex Number Question!

    Not sure how to do this question:

    If z = cos θ + isin θ, show that (1-z)/(1+z) = -itan(θ/2)

    Any help would be appreciated. Thanks!

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    Re: Help with Complex Number Question!

    1-z = 1 - cosx - isinx
    1+z = 1+cosx+isinx

    Times top and bottom by 1+cosx-isinx

    Numerator = (1-cosx-isinx)(1+cosx-isinx)
    = 1+cosx-isinx-cosx-cos^2x+isinxcosx-isinx-isinxcosx-sin^2x
    = 1-2isinx-(sin^2x+cos^2x)
    = -2isinx

    Denominator = (1+cosx+isinx)(1+cosx-isinx)
    = (1+cosx)^2 - (isinx)^2
    = 1+cos^2x+2cosx+sin^2x
    = 2(1+cosx)

    (1-z)/(1+z) = -isinx/(1+cosx)
    = -i(2sin(x/2)cos(x/2))/(1+1-2sin^2(x/2))
    = -i(sin(x/2)cos(x/2))/(cos^2(x/2))
    = -isin(x/2)/cos(x/2)
    = -itan(x/2)

    In summary: times top and bottom by conjugate of the denominator, i.e. realising it.
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    Re: Help with Complex Number Question!

    Quote Originally Posted by yleming View Post
    Not sure how to do this question:

    If z = cos θ + isin θ, show that (1-z)/(1+z) = -itan(θ/2)

    Any help would be appreciated. Thanks!






    pikachu975 likes this.

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    Re: Help with Complex Number Question!

    Thanks!

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    Junior Member HeroWise's Avatar
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    Re: Help with Complex Number Question!

    Quote Originally Posted by aa180 View Post








    How did you think of that? Like i would have done like pikachu but what made you go do the qtn in that way?

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    Re: Help with Complex Number Question!

    Quote Originally Posted by HeroWise View Post
    How did you think of that? Like i would have done like pikachu but what made you go do the qtn in that way?
    Just a trick I learnt at uni

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    Re: Help with Complex Number Question!

    If u can pls advice me senpai, teach me your ways
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    Re: Help with Complex Number Question!

    Quote Originally Posted by HeroWise View Post
    If u can pls advice me senpai, teach me your ways
    Lol, just maximize your exposure to various resources as well as students/teachers who are adept in this subject and you'll be fine

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