How to convert this complex number to exponential form?

Question

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 ${\phi}=tan^{-1}({\frac{4}{3}})$.

Answer 1

$$\cos\phi=\frac{3}{5}\\ \sin\phi=\frac45$$ Divide the second by the first $$\tan\phi=\frac{\sin\phi}{\cos\phi}=\frac{4/5}{3/5}=\frac43$$Therefore $\phi=\arctan\frac43$