On the unit circle, replace trigonometry by rational functions: $\sin\theta=(z-z^{-1})/(2i)$ and $d\theta=dz/(iz)$.Suggest a slogan
For \(\theta\in\RR\) and \(z=e^{i\theta}\) on the unit circle, \[ \begin{aligned} \sin(\theta) &= \frac{e^{i\theta} - e^{-i\theta}}{2i} = \frac{z - \inverseof{z}}{2i}, \\ d\theta &= \frac{dz}{iz}. \end{aligned} \]
How is $\sin(\theta)$ written in $z$ for a contour integral on the unit circle, and what is $d\theta$?