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I have $Z = {\frac{2+i}{2-i}}$. I need to write it in exponential form, i.e. $Z = r*e^{i{\phi}}$.

I simplified the given example to ${\frac{3 + 4i}{5}}$, i.e. $a={\frac{3}{5}}, b={\frac{4}{5}}$. Therefore, $r = 1$.

Then I calculate $cos({\phi}) = a / r = 3 / 5, sin({\phi})=b/r = 4/5$. How do I calculate ${\phi}$ from here?

The online Complex Number Exponential Form given ${\phi}=tan^{-1}({\frac{4}{3}})$.

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{Abs, Arg}@((2 + I)/(2 - I)) // Through
#1 Exp[I #2] & @@ %

{1, ArcTan[4/3]}

E^(I ArcTan[4/3])

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  • $\begingroup$ thanks for the answer, but as i said i get the right answer by using some tool, but i wanna understand how to calculate myself $\endgroup$ – user3132457 Nov 6 '18 at 18:50
  • $\begingroup$ If you're just interested in the math itself, and not how to do it in Mathematica, may I suggest asking it on math.se? $\endgroup$ – That Gravity Guy Nov 6 '18 at 18:58
  • $\begingroup$ yes, actually... $\endgroup$ – user3132457 Nov 6 '18 at 19:00

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