Summary: Let \(R\) be an associative ring with identity, and \((\mathcal{A}, \mathcal{B})\) a hereditary cotorsion pair generated by a set in \(R\)-Mod. Then \((\text{dw}\tilde{\mathcal{A}}, (\mathrm{dw} \tilde{\mathcal{A}}^\bot)\) is a complete and hereditary cotorsion pair (we call it the degreewise cotorsion pair) in the category of \(R\)-complexes, where \(\text{dw}\tilde{\mathcal{A}}\) denotes the class of all complexes \(X\) with components \(X_n\in\mathcal{A}\) for all \(n \in\mathbb{Z}\). For any complexes \(X\) and \(Y\) and any \(n \in\mathbb{Z}\), we define the relative cohomology groups \(\text{Ext}^n_{\text{dw} \tilde{\mathcal{A}}}(X,Y )\) based on the degreewise cotorsion pair and investigate the vanishing of the relative cohomology groups. Specifically, we introduce the relative cohomology dimension of \(X\) related to \(\text{dw} \tilde{\mathcal{A}}\)-precovers, and then show that such a dimension of \(X\) is equal to the least integer \(n\) for which \(\text{Ext}^n_{\text{dw} \tilde{\mathcal{A}}}(X,Y )=0\) for all \(i>n\) and all \(R\)-modules \(Y \in\mathcal{B}\), which recovers the result on relative cohomology dimensions (defined by Z. Liu [J. Algebra 502, 79–97 (2018; Zbl 1442.16009)]) of complexes related to Gorenstein projective precovers.