  Generalized sine and cosine functions, $\sin_{n}$ and $\cos_{n}$, that
parametrize the generalized unit circle $x^n+y^n=1$ are, much like their
classical circular counterparts, extendable as complex analytic functions. In
this article, we identify the natural domain on which $\sin_{n}$ is a conformal
equivalence from a polygon to the complex plane with $n$ slits. We also give
some geometric and analytic applications.
