Two classes $\mathcal A$ and $\mathcal B$ of modules over a ring $R$ are said
to form a cotorsion pair $(\mathcal A, \mathcal B)$ if $\mathcal A={\rm Ker
Ext}^1_R(-,\mathcal B)$ and $\mathcal B={\rm Ker Ext}^1_R(\mathcal A,-)$. 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 $\Findim R<\infty$ if and only if the following dimensions are finite for
some cotorsion pair $(\mathcal A, \mathcal B)$ in $\mathrm{Mod} R$: the
relative projective dimension of $\A$ 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 $\Findim R=\findim R.$