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**tangibleLime** Not totally sure if this belongs in abstract algebra, but it's in my abstract algebra book and it seems more complicated that regular algebra and requires trigonometry, etc...

Problem:

Find all solutions of $\displaystyle z^{4} = -16$

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I am following an example from the book. The first step they take is to put this in polar form, part of which I do not understand.

$\displaystyle |z|^{4}(cos4\theta + isin4\theta) = 16(-1+0i)$

I'm confused about where the 4 comes from in the cos and sin functions. I know the polar coordinate version of a complex number is $\displaystyle z = |z|(cos\theta + isin\theta)$, so is that four simply being brought into the sin and cos functions from the exponent of z? And if so, why is $\displaystyle cos(4\theta)$ instead of $\displaystyle cos^4(\theta)$?

After that step, they say this exactly:

"Consqeuently, |z|^4 = 16, so |z| = 2 while $\displaystyle cos4\theta = -1$ and $\displaystyle sin(4\theta) = 0$."

What? I understand why |z| is 2, since |2|^4=16. But where did $\displaystyle cos4\theta = -1$ and $\displaystyle sin(4\theta) = 0$ come from? This book (which annoyingly says "Clearly, <statement>" and "Obviously, <statement>" as if we already know the entire book, is giving me no obvious information about why these two trigonometric functions need to equal -1 and 0 respectively.

They then go on to say "We find that $\displaystyle 4\theta = \pi + n(2\pi)$. I don't understand this either. I can see what it's doing; taking $\displaystyle \pi$ and adding n-number of 2pi to always return to the same location, but why start at pi?

Any help is appreciated.