Question

Find z : cos z = 3.
I need the logic for the last part.
Please, help. I will post my work

Answer 1

\(cos z =\dfrac{e^{iz}+e^{-iz}}{2}=3\implies e^{iz}+e^{-iz}=6\)

Answer 2

multiple both sides by e^(iz) , I got \(e^{2iz} -6e^{iz}+1=0\)
Let \(w = e^{iz}\) and solve for w

Answer 3

I have \(w = 3\pm 2\sqrt 2\)
Do I have to solve one by one or I combine them on 1 argument?
I meant: \(e^{iz}= 3 + 2\sqrt 2\) and then \(e^{iz}= 3-2\sqrt 2\)separately, right?

Answer 4

From \(e^{iz} = 3 + 2\sqrt 2\), \(iz = log (3+2\sqrt2 ) + i 2\pi n, n\in \mathbb Z\), hence
\(z = 2\pi n - ilog (3+2\sqrt 2) , n\in \mathbb Z\)

Answer 5

But if \(e^{iz} = 3-2\sqrt 2\), with the same argument, I have \(z = 2\pi n -i log(3-2\sqrt2)\)
I do not know how to do. Should I combine them (how?) or let them as they are?