We consider the problem of gathering bearing data to localize targets. We start with a commonly used notion of uncertainty based on geometric dilution of precision (GDOP) and study the following bicriteria problem. Given a set of potential target areas and an uncertainty level U, compute an ordered set of measurement locations for a single robot which 1) minimizes the total cost given by the travel time plus the time spent in taking measurements and 2) ensures that the uncertainty in estimating the target's location is at most U regardless of the targets' locations. We present an approximation algorithm and prove that its cost is at most 28.9 times the optimal cost while guaranteeing that the uncertainty is at most 5.5U. In addition to theoretical analysis, we validate the results in simulation and experiments performed with a directional antenna used for tracking invasive fish.