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    Hello, I've been having a bit of trouble with finding some principle arguments using De Moivre's theorem. Here's one of my examples,

    Find the principle argument of (1+i)^5

    Let z=1+i

    |z|=\sqrt{2}

    So, z=\sqrt{2} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i)

    cos\theta =\frac{1}{\sqrt{2}} and sin\theta =\frac{1}{\sqrt{2}}

    \theta =\frac{\pi}{4}

    Using De Moivre's theorem, I multiply the argument by the power. So,

    \frac{\pi}{4}.5=\frac{5\pi}{4}.

    But on the answers, it says it's -\frac{3\pi}{4}. :cry2:


    Another example is the question where I have to find the principle argument of (\sqrt{3}-i)^7.

    So I let z=\sqrt{3}+i

    |z|=2

    z=2(\frac{\sqrt{3}}{2}+\frac{1}{  2}i)

    So cos\theta =\frac{\sqrt{3}}{2} and sin\theta =\frac{1}{2}

    So \theta =\frac{\pi}{6}

    Then, when I multiply it by the power, I get,

    \frac{\pi}{6}.7=\frac{7\pi}{6}, which isn't the right answer.

    The right answer, according to the answers, is in fact, -\frac{5\pi}{6}. :cry:

    So how do I do it? :puppyeyes:
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    (Original post by RamocitoMorales)
    Hello, I've been having a bit of trouble with finding some principle arguments using De Moivre's theorem. Here's one of my examples,

    Find the principle argument of (1+i)^5

    Let z=1+i

    |z|=\sqrt{2}

    So, z=\sqrt{2} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i)

    cos\feta =\frac{1}{\sqrt{2}} and sin\feta =\frac{1}{\sqrt{2}}

    \feta =\frac{\pi}{4}

    Using De Moivre's theorem, I multiply the argument by the power. So,

    \frac{\pi}{4}.5=\frac{5\pi}{4}.

    But on the answers, it says it's -\frac{3\pi}{4}. :cry2:
    I suspect your question asks for an answer between minus pi and pi?
    5pi/4 is 'equivalent' to -3pi/4 by adding 2 pi.
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    (Original post by assmaster)
    I suspect your question asks for an answer between minus pi and pi?
    5pi/4 is equivalent to -3pi/4 by adding 2 pi.
    Yes, it asks for an answer between -\pi amd \pi.

    But how would I know the answer was -\frac{3\pi}{4}, how would I work it out? What is it about the power that changes it like that? :cry2:
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    (Original post by RamocitoMorales)
    Yes, it asks for an answer between -\pi amd \pi.

    But how would I know the answer was -\frac{3\pi}{4}, how would I work it out? What is it about the power that changes it like that? :cry2:
    It's nothing to do with the powers. Your working is right to get to your final answer, you're just missing the final step.
    So obviously 5pi/4 > pi
    And because sine and cosine have a period of 2pi (think of the graphs repeating) it doesn't matter if you add or subtract multiples of 2pi. Which you need to do to get it into the range (minus pi, pi].
    So take 5pi/4 and subtract 2 pi, and you get -3pi/4, as required.
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    wth I'm learning this now in year 13
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    (Original post by RamocitoMorales)
    Yes, it asks for an answer between -\pi amd \pi.

    But how would I know the answer was -\frac{3\pi}{4}, how would I work it out? What is it about the power that changes it like that? :cry2:

    As assmaster said above, adding or subtracting 2\pi from the argument does not change the complex number simply because:

    \cos(\theta + 2\pi) = \cos(\theta) and \sin(\theta + 2\pi) = \sin(\theta)
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    You have done it right. In complex numbers we usually take the argument to be from -\pi to \pi and since your answer \frac{5\pi}{4} is bigger than \pi if you draw a "unit circle" you'll see that it will become -\frac{3\pi}{4}

 
 
 
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