hmmm let's start off by using the conjugate of the denominator to <u>rationalize the denominator</u>.
![\bf \cfrac{1+i}{1-i}\implies \cfrac{1+i}{1-i}\cdot \cfrac{1+i}{1+i}\implies \cfrac{(1+i)^2}{\underset{\textit{difference of squares}}{(1-i)(1+i)}}\implies \cfrac{\stackrel{FOIL}{1^2+2i+i^2}}{1^2-i^2} \\\\\\ \stackrel{\textit{recalling that }i^2=-1}{\cfrac{1+2i+(-1)}{1-(-1)}}\implies \cfrac{2i}{1+1}\implies \cfrac{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~i}{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies i\implies \stackrel{\textit{a+bi form}}{0 + 1i}](https://tex.z-dn.net/?f=%5Cbf%20%5Ccfrac%7B1%2Bi%7D%7B1-i%7D%5Cimplies%20%5Ccfrac%7B1%2Bi%7D%7B1-i%7D%5Ccdot%20%5Ccfrac%7B1%2Bi%7D%7B1%2Bi%7D%5Cimplies%20%5Ccfrac%7B%281%2Bi%29%5E2%7D%7B%5Cunderset%7B%5Ctextit%7Bdifference%20of%20squares%7D%7D%7B%281-i%29%281%2Bi%29%7D%7D%5Cimplies%20%5Ccfrac%7B%5Cstackrel%7BFOIL%7D%7B1%5E2%2B2i%2Bi%5E2%7D%7D%7B1%5E2-i%5E2%7D%20%5C%5C%5C%5C%5C%5C%20%5Cstackrel%7B%5Ctextit%7Brecalling%20that%20%7Di%5E2%3D-1%7D%7B%5Ccfrac%7B1%2B2i%2B%28-1%29%7D%7B1-%28-1%29%7D%7D%5Cimplies%20%5Ccfrac%7B2i%7D%7B1%2B1%7D%5Cimplies%20%5Ccfrac%7B~~%5Cbegin%7Bmatrix%7D%202%20%5C%5C%5B-0.7em%5D%5Ccline%7B1-1%7D%5C%5C%5B-5pt%5D%5Cend%7Bmatrix%7D~~i%7D%7B~~%5Cbegin%7Bmatrix%7D%202%20%5C%5C%5B-0.7em%5D%5Ccline%7B1-1%7D%5C%5C%5B-5pt%5D%5Cend%7Bmatrix%7D~~%7D%5Cimplies%20i%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Ba%2Bbi%20form%7D%7D%7B0%20%2B%201i%7D)
so the equation is really "i" in disguise, now, our coordinates are 0 and 1 for that complex point, we can't quite use tan⁻¹ to get the angle, since denominator is 0, meaning the point is lying on one of the axis, well, the rectangular (0,1) is right up above on the y-axis, namely at π/2.
![\bf \begin{cases} r = \sqrt{0^2+1^2}\\ \qquad 1\\ \theta =\frac{\pi }{2} \end{cases} \qquad \qquad 1\left[cos\left( \frac{\pi }{2} \right) +i~sin\left( \frac{\pi }{2} \right) \right]](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Bcases%7D%20r%20%3D%20%5Csqrt%7B0%5E2%2B1%5E2%7D%5C%5C%20%5Cqquad%201%5C%5C%20%5Ctheta%20%3D%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%5Cend%7Bcases%7D%20%5Cqquad%20%5Cqquad%201%5Cleft%5Bcos%5Cleft%28%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%5Cright%29%20%2Bi~sin%5Cleft%28%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%5Cright%29%20%5Cright%5D)