I want to write the difference between $n$-th roots of unity in the form $re^{i \theta}.$
It is enough to find the polar form of $1 - \zeta^k$. By thinking geometrically, I can guess the formula $$1 - \zeta^k = 2 \sin\Big( \frac{k \pi}{n} \Big) e^{i (-\frac{\pi}{2} + \frac{k \pi}{n} )},$$ but I'm not sure how to derive this algebraically.
OK, I see it. Following the comment, $$1 - e^{-2i \varphi} = \frac{e^{i \varphi} - e^{-i \varphi}}{e^{i \varphi}} = 2i \sin(\varphi) e^{-i\varphi},$$ so when we let $\varphi = -\frac{\pi k}{n}$, $$1 - \zeta^k = 1 - e^{\frac{2\pi i k}{n}} = 2i \sin(\frac{-\pi k}{n}) e^{\frac{\pi i k}{n}} = -2i \sin(\frac{\pi k}{n}) e^{\frac{\pi i k}{n}}$$$$= 2 \sin(\frac{\pi k}{n}) e^{-\frac{\pi i}{2} + \frac{\pi i k}{n}}.$$