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We consider an inverse problem for the scattering of an obliquely incident electromagnetic wave by an impedance cylinder. We show that the cross-section of the cylinder can be uniquely determined from the far-field patterns of only the electric or magnetic field, through the use of special polarized incident plane waves. The proof is based on the generalized mixed reciprocity principle and the singularity analysis of the reflected solutions for point sources. Our result would be helpful in practical situations where only one type of measurement is available. Moreover, our argument is reconstructive, and can be used directly to set up a numerical scheme. In fact, the singular behavior of the reflected solutions for point sources enables us to reconstruct the shape of the cross-section numerically. To recover the impedance, we use a more singular source to derive a pointwise calculation formula.