A reconfigurable intelligent surface (RIS) consists of a large number of low-cost elements that can control the propagation environment seen from a transmitter by intelligently applying phase shifts to impinging signals before reflection. This paper studies an RIS-assisted communication system where a transmitter wants to transmit a common signal to many users residing in a wide angular area. To cover this sector uniformly, the RIS needs to radiate a broad beam with a spatially flat array factor, instead of a narrow beam as normally considered. To achieve this, we propose to use a dual-polarized RIS consisting of elements with orthogonal polarizations and show that the RIS can produce a broad beam if the phase shift configuration vectors in the two polarizations form a so-called Golay complementary sequence pair. By utilizing their properties, we also present a method for constructing configuration for large RISs from smaller ones, while preserving the broad radiation pattern of the smaller RIS. The numerical results corroborate the mathematical analyses and highlight the greatly improved coverage properties.