Earth-skimming neutrinos are those which travel through the Earth’s crust at a shallow angle. For Ultra-High-Energy (E$_{\nu}$ > 1 PeV; UHE) earth-skimming tau neutrinos, there is a high-probability that the tau lepton created by a neutrino-Earth interaction will emerge from the ground before it decays. When this happens, the decaying tau particle initiates an air shower of relativistic sub-atomic particles which emit Cherenkov radiation. To observe this Cherenkov radiation, we propose the Trinity Observatory. Using a novel optical structure design, pointing at the horizon, Trinity will observe the Cherenkov radiation from upward-going neutrino-induced air showers. Being sensitive to neutrinos in the $1 - 10^4$ PeV energy range, Trinity's expected sensitivity will have a unique role to play filling the gap between the observed astrophysical neutrinos observed by IceCube and the expected sensitivity of radio UHE neutrino detectors.