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Math Help - Difficult complex number question

  1. #1
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    Difficult complex number question

    The question:
    Let z = (1 + \sqrt{3} i) and w = (1 + i). Find Arg(z) and Arg(w) and hence Arg(zw). Evaluate zw and hence show that cos\frac{7\pi}{12} = \frac{1 - \sqrt{3}}{2\sqrt{2}}. Find a similar expression for sin \frac{7\pi}{12}

    I found:
    Arg(z) = \frac{\pi}{3}
    Arg(w) = \frac{\pi}{4}
    Arg(zw) = \frac{\pi}{12}
    zw = 2\sqrt{2}e^{i\frac{\pi}{12}}

    The last bit of the question is giving me trouble. I tried using de Moivre's theorem as follows:

    (2\sqrt{2}e^{i\frac{\pi}{12}})^7
    (2\sqrt{2})^7e^{i\frac{7\pi}{12}}

    This gives me the argument I want. But I'm unsure of how to solve this. Changing it to sin/cos form (not sure of the proper name for it) seems to be the way to go, but I'm still stumped:

    (2\sqrt{2})^7(cos(\frac{7\pi}{12}) + isin(\frac{7\pi}{12}))

    Any assistance would be great!
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  2. #2
    A Plied Mathematician
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    I agree with arg(z) and arg(w). However, I don't agree with your arg(zw). How do arguments behave under multiplication?
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  3. #3
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    Oh whoops, that should be 7Pi/12.

    I guess that makes zw:

    2\sqrt{2}e^{i\frac{7\pi}{12}}

    I'm still not sure how to solve it though.
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  4. #4
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    Try multiplying zw in both the polar form and the rectangular form, and then setting them equal. What does that give you?
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  5. #5
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    Ahh! Got it! Thank you very much.

    This is what I did, for anyone that's looking on:

    zw = (1-\sqrt{3}) + i(1 + \sqrt{3}) (rectangular form)
    zw = 2\sqrt{2}(cos(\frac{7\pi}{12}) + isin(\frac{7\pi}{12}))

    Equating real parts:
    2\sqrt{2}(cos(\frac{7\pi}{12})) = 1-\sqrt{3}
    Therefore,
    cos\frac{7\pi}{12} = \frac{1 - \sqrt{3}}{2\sqrt{2}}
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  6. #6
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    Looks good to me. You're very welcome.
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