1. Understand the molecule

   $1,1,1$-Trichloroethane has the structure: $\mathrm{CCl_3-CH_3}$

   We consider the Newman projection along the $\mathrm{C-C}$ bond.

2. Meaning of dihedral angle

   The dihedral angle is the angle between a bond on the front carbon and the nearest bond on the back carbon in the Newman projection.

3. Staggered conformation

   In any staggered Newman projection of an ethane-type molecule, the bonds on the back carbon are offset by: $60^\circ$ from the bonds on the front carbon.

   This is true for $\mathrm{CCl_3-CH_3}$ as well. The substituents are different, but the definition of the staggered arrangement remains the same.

4. Therefore

   The dihedral angle in the staggered form is: $60^\circ$

5. Rounded to nearest integer

   $60$

Final integer answer: $60$