Given a ring R, two classes $\mathcal{A}$ and $\mathcal{B}$ of R-modules are said to form a cotorsion pair $(\mathcal{A},\mathcal{B})$ in Mod R if $\mathcal{A}=\operatorname {Ker}\operatorname{Ext}^{1}_{R}(-,\mathcal{B})$ and $\mathcal{B}=\operatorname{Ker}\operatorname{Ext}^{1}_{R}(\mathcal{A},-)$ . We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of R. We show that Findim R<∞ if and only if the following dimensions are finite for some cotorsion pair $(\mathcal{A},\mathcal{B})$ in Mod R: the relative projective dimension of $\mathcal{A}$ with respect to itself, and the $\mathcal{A}$ -resolution dimension of the category $\mathcal{P}$ of all R-modules of finite projective dimension. Moreover, we obtain an analogous result for findim R, and we characterize when Findim R=findim R.