Let . What are all possible solutions of:
So, after converting to polar form I have:
I have the answer which includes 7 solutions and I'm a little confused on how to get to them. I guess this is more a trig question?
Hmm I still seem to be missing a step in my head. I don't understand how the k multiples of pi got inside the sin and cos functions. I can see that
can be written as
but from there I thought I just applied the power to the and then multiplied by which gives me:
What am I missing in between those steps?
Ok, I think I get the process. My understanding is very superficial though. I only learned how to "apply the formula" and didn't cover what I am actually doing by solving one of these which is what I think my problem is.
Take a similar example:
. Find all the solutions for:
I convert to polar form:
I notice that so I figure there will be 4 solutions (not sure why).
I then apply the exponent 1/4 to get:
Then to get my solutions I just generate 4 answers by multiplying the exponent 1/4 by and adding it to cycling k from 0 - 3 to get:
I can see that it makes one revolution and the answers are recorded periodically and evenly depending on the denominator of the increments, and I seem to be able to generate the answers for these types of problems, I just feel awkward not knowing why I'm doing each step if you understand?
Will these problems always have a coefficient that can be expressed as an integer raised to a power so you can determine how many answers there should be?