We study the minimal free resolution of the tangent cone of Gorenstein monomial curves in affine 4-space. We give the explicit minimal free resolution of the tangent cone of noncomplete intersection Gorenstein monomial curve whose tangentcone has fiveminimal generators and showthat the possible Betti sequences are (1,5,6,2) and (1,5,5,1). Moreover, we compute the Hilbert function of the tangent cone of these families as a result.