For complex numbers, I get how we can solve for purely imaginary and/or purely real but letting sin or cos equal to zero. However, for some questions like the one below, the given set of k integers (for which a supposed n power makes the complex number zero or imaginary) is not always given by a set pattern. How would I approach and solve such questions?
View attachment 34497
Now prepare to witness the mind of a mathematician. The entire concept revolves around the fact that with a pattern where if we are finding something that is purely imaginary then for example we have
^{n}=0 )
which is simply
according to De Moivre's theorem.
The next step is just knowing the inverse of cosine when it's zero. This can be shown when
that is simply just
where
and many others that go in the cycle of
or
.
However, there is a variation of this type of question that asks us to find the smallest possible value of such an idea.
Consider the number
. Find the smallest value of
such that
is purely imaginary. Then what is happening there is just simply to write this is polar form which is just length of
which is
which is not too important here but might be important somewhere else such as finding mod-arg form, and then finding
so thus, as such we have
}+i\sin{\left(\frac{n\pi}{4}\right)})
. Depending on what you want which is either purely real or imaginary convert either cosine or sine of the trig function to be zero and then find vaues which can be
giving us
which then through simplification will give us
