The Hamiltonian for nanocones with curvature-induced spin-orbit coupling have been derived. The effect of curvature-induced spin-orbit coupling on the electronic properties of graphitic nanocones is considered. Energy spectra for different numbers of the pentagonal defects in the tip of the nanocones are calculated. It was shown that the spin-orbit interaction considerably affects the local density of states of the graphitic nanocone. This influence depends on the number of defects present at the tip of the nanocone. This property could be applied in atomic force microscopy for the construction of the probing tip.