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
-----------------------
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.
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
, so is that four simply being brought into the sin and cos functions from the exponent of z? And if so, why is
instead of
?
After that step, they say this exactly:
"Consqeuently, |z|^4 = 16, so |z| = 2 while
and
."
What? I understand why |z| is 2, since |2|^4=16. But where did
and
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
. I don't understand this either. I can see what it's doing; taking
and adding n-number of 2pi to always return to the same location, but why start at pi?
Any help is appreciated.