Since is a real number (1.54308...)
Does this mean that I can substitute "i" as follows?
If not, why not?
Also, since does this mean that
The first one follows. If , then
The second equation follows as well. Provided that , then by taking the power of both sides of the equation, we have:
I'm not sure what you are going to use these facts to solve, but each of these steps are correct. Are you certain that ? Are we speaking in terms of hypothetical? Imaginary numbers are tricky, it would help if I knew where you were coming from.
I suspect that each of these steps are correct, but that the premise may be in error.
According to wikipedia's page on the imaginary unit:
The page doesn't show how this is derived.
I don't have a particular use in mind for this info. I was just cruising wikipedia and this jumped out at me.
Even if cos(i) is a real number, I guess this does not give us a real number value for "i" itself. Acos(cos(i)) does not exist, or is imaginary.
Same with .