ln z = ln (mod of z) + i Arg z +- (2n pi) i
ln (e^{3i })
You are overcomplicating things. The argument is the angle that is made. You already read off that this angle is 3 (radians).
Of course if you were going to do it your way, you'd note that sin(x)/cos(x) = tan(x) and so your arctangent cancels off anyway.
You should have noticed that the ln and e^ functions are inverses of each other, so in at least some branch, ln(e^(z)) = z. Then think about what the angle/magnitude of this would be, whether this result makes sense and if not, what you need to do to make it make sense.ln (e3i )