Complex Numbers in Polar Form

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.

Re: Complex Numbers in Polar Form

Quote:

Originally Posted by

**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 $\displaystyle z^{4} = -16$

-----------------------

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.

DeMoivre's Theorem states that if $\displaystyle \displaystyle z = r\left(\cos{\theta} + i\sin{\theta}\right)$ then $\displaystyle \displaystyle z^n = r^n\left(\cos{n\theta} + i\sin{n\theta}\right)$ for positive integer values of $\displaystyle \displaystyle n$.

As for your other question, "Why start at $\displaystyle \displaystyle \pi$?", remember that you're finding the fourth root of $\displaystyle \displaystyle -1 + 0i$. This number makes an angle of $\displaystyle \displaystyle \pi$.

Re: Complex Numbers in Polar Form

Thanks! It makes sense now. DeMoivre's Theorem wasn't in my book, I suppose it was assumed pre-existing knowledge.