Movable antenna (MA) has attracted increasing attention in wireless communications recently. As compared to conventional fixed-position antennas (FPAs), the geometry of MAs can be dynamically reconfigured, such that more flexible beamforming can be achieved for different purposes. In this paper, we investigate the application of MAs to wide-beam coverage, aiming to jointly optimize the MAs' beamforming weights and positions within a line segment to maximize the minimum beam gain among all possible directions in a target region. However, the resulting optimization problem is non-convex and difficult to be optimally solved. To tackle this difficulty, we first derive a closed-form optimal solution to this problem in the special case with two MAs. While for the case with more than two MAs, an alternating optimization (AO) algorithm is proposed to obtain a high-quality suboptimal solution, where the MAs' beamforming weights and positions are alternately optimized by applying the successive convex approximation (SCA) technique. To reduce computational complexity, we further propose a more efficient MA position optimization method by leveraging the frequency modulation continuous wave (FMCW) design. Specifically, we construct a spatial FMCW-based continuous phase profile for the entire line segment and then select an optimal set of MA positions to optimize the wide-beam coverage performance with their FMCW-based phase profiles, thus greatly simplifying the wide-beam design. Furthermore, we extend the proposed AO and FMCW-based algorithms for the linear MA array to the planar MA array. Numerical results show that both our proposed algorithms can significantly outperform conventional FPAs even with optimized beamforming weights.