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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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1 Answer

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

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