The angular distribution functions of β-rays that are used to obtain the angular correlation functions are calculated taking into account the effect of the nuclear charge. The calculations are performed up to the second forbidden transitions for each β-decay interaction type of the Fermi theory separately but including the mixture of the nuclear matrix elements. To obtain the angular correlation functions easily in the case of the mixed nuclear matrix elements they are given in considerably simple forms in two interesting cases. Also in the appendix, transformation coefficients (j13m1m2|j13jm) are given.