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So I have a rough idea on how to get the answer but I'm getting stuck on the angle or argument for the equation. The question is:

Find the 6th root of $-3+4i$.

I first find the $r$ value which is 5, now I need to find the angle that connects them and I'm told I need to plug $\tan(\frac ba)$ and that would give me $\tan(-\frac {4}{3}) $.

My teacher had mentioned using $\arctan$ to get the correct angle but I can't remember the process.

Chris Brooks
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Joshhw
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1 Answers1

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You won't be able to get a "nice" argument from this one.

If $z=-3+4\mathrm{i}$ then the modulus is nice, $(3,4,5)$ are a Pythagorean triple, and so $|z|=5$.

For the argument, shown in green, we look at the right-angled triangle with vertices $0+0\mathrm{i}$, $-3+0\mathrm{i}$ and $-3+4\mathrm{i}$. The argument can be found by finding the red angle and then subtracting that from $\pi$.

Using standard trigonometry, we have $\arg z = \pi - \tan^{-1}(4/3) \approx 2.214$

Note that $\arctan$ is an alternative notation for $\tan^{-1}$.

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Fly by Night
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