The Neumann Laplacian acts on a dense subspace of square-integrable functions on a Riemannian manifold. If the manifold is compact then the spectrum is a countable discrete subset of the nonnegative reals. If the manifold is not compact then the spectrum can be mixture of eigenvalues and essential spectrum. In joint work with Luc Hillairet, we show that the generic triangle in the hyperbolic plane having a cusp has no nonconstant Neumann eigenfunctions, and hence there are no eigenvalues embedded in its essential spectrum. The proof involves a singular analytic perturbation, `asymptotic separation of variables', and an analysis of potential eigenvalue crossings.