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Math Help - totally stuck

  1. #1
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    totally stuck

    I have to find all complex number solutions and I have to leave my answer in trig form:

    x^5-i=0

    i have no idea how to start this problem. How would I go about getting the 1st degree to plug into the trig form?
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  2. #2
    Super Member

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    Hello, DroMan!

    Solve: . x^5-i\:=\:0
    Leave answers in trig form.

    We have: . x^5 \:=\:i \quad\Rightarrow\quad x \:=\:i^{\frac{1}{5}}

    Convert to polar form: . i \;=\;\cos\left(\frac{\pi}{2} + 2\pi n\right) + i\sin\left(\frac{\pi}{2} + 2\pi n\right)


    Use Demoivre's Theorem:

    . . i^{\frac{1}{5}} \;=\;\bigg[\cos\left(\frac{\pi}{2} + 2\pi n\right) + i\sin\left(\frac{\pi}{2} + 2\pi n\right)\bigg]^{\frac{1}{5}}

    . . . . = \;\cos\left(\frac{\pi}{10} + \frac{2\pi}{5}n\right) + i\sin\left(\frac{\pi}{10} + \frac{2\pi}{5}n\right)\quad\hdots\quad \text{for }n \,=\,0,1,2,3,4


    Therefore: . x \;=\;\begin{Bmatrix}\cos\frac{\pi}{10} + i\sin\frac{\pi}{10} \\ \\[-4mm] \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} &=& i \\ \\[-4mm] \cos\frac{9\pi}{10} + i\sin\frac{9\pi}{10} \\ \\[-4mm]<br />
\cos\frac{13\pi}{10} + \sin\frac{13\pi}{10} \\ \\[-4mm] \cos\frac{17\pi}{10} + i\sin\frac{17\pi}{10}\end{Bmatrix}

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  3. #3
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    thnaks so much
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