Resolution principle for sheaf cohomology

If $0\to \mathcal{F}\to {\mathcal{A}}^0\to {\mathcal{A}}^1\to \cdots$ is an exact sequence of sheaves of abelian groups on $X$ and $H^q(X,{\mathcal{A}}^j)=0$ for every $q>0$ and $j\ge 0$, then sheaf cohomology is computed by the cochain complex of global sections:In particular a flasque resolution computes sheaf cohomology. Acyclicity is required on the space on which cohomology is being computed.