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De Moivre's Theorem help

So the question is:

Solve the equation z5=iz^{5}=i

giving your answers in the form of cosθ+isinθcos \theta + isin \theta

So far I have z=[cos(4k+12)π]+isin[(4k+12)π]z=[cos(\frac{4k+1}{2})\pi]+isin[(\frac{4k+1}{2})\pi]

To find the roots of this, what is the interval of the roots?

For example I assumed the interval is π<θπ-\pi<\theta\leq\pi so k=2,1,0,1,2k = -2, -1, 0, 1, 2 but the mark scheme states the roots are when k=0,1,2,3,4k = 0, 1, 2, 3, 4 so their interval lies within 0<θ2π0 < \theta \leq 2\pi

My question is how do I know for what interval do the nthnth roots of a complex number lie in?

The question is here, Question 5, http://pmt.physicsandmathstutor.com/download/Maths/A-level/FP2/Papers-Edexcel/January%202006%20QP%20-%20FP2%20Edexcel.pdf

Mark scheme is here (question 5),
http://pmt.physicsandmathstutor.com/download/Maths/A-level/FP2/Papers-Edexcel/January%202006%20MS%20-%20FP2%20Edexcel.pdf
What you've got is wrong. I think you need to post your working.
Hi,
If you use Euler and solve this in exponential form it is easier (same thing though).
z^5=e^i(4kpi+pi all over 2)
this becomes (after taking 5th root) z=e^i(4kpi+pi all over 10)
let k=0,plusminus1,plusminus2...
In your way of doing it, z becomes z=cos(4kpi+pi all over 10) plus isin(4kpi+pi all over 10). again, let k= all of those stated above.
Done :smile:
Reply 3
Original post by DFranklin
What you've got is wrong. I think you need to post your working.


32CC2EC3-D8B3-4DEA-8528-F69E65B4701F.jpg.jpeg

Sorry if it's not clear enough
Sorry, you#'re going to need to post that at a readable size.
Reply 5
Original post by DFranklin
Sorry, you#'re going to need to post that at a readable size.


de moivre.jpg

(I think the TSR app lowers the photo quality)
Original post by ManLike007
de moivre.jpg

(I think the TSR app lowers the photo quality)
OK; I'm not sure if you noticed, but this is rather different from what you posted in your original post.
Reply 7
Original post by DFranklin
OK; I'm not sure if you noticed, but this is rather different from what you posted in your original post.


Oh right yes it's 4k+110π\frac{4k+1}{10}\pi not (4k+12)π(\frac{4k+1}{2})\pi

Regardless, can you help with the values of the kk, apparently they're not right
OK, now I've seen what you actually did, I think your answer is fine; there's nothing in the question that says you need theta to be in a particular range, and you have all 5 roots. [Note that the question is asking for the roots, not the values of theta, so since k=-1 gives the same root as k = 4, it's the same answer even if it looks different).
Reply 9
Oh ok I see what you mean, thanks for your time

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