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In this report based on the development of the method to determine the roots (zeros) of the Hermite polynomials, research has been initiated for different searches and some deductions have been reached with the conclusions produced from them. We have also focused on the answer to the question of the well-known cosine function asymptotically for even Hermite polynomials and sine function asymptotically for odd Hermite polynomials in a more detailed form than existing in the related scientific literature. In the construction of the trigonometric function coefficient in the Maclaurin expansion, we have proposed two power series in the scaled independent variable x such that first and second power series are individually multiplied by cos(x) and sin(x) respectively and are of even and odd functions. Appropriate recursions have been constructed to evaluate the coefficients these power series. So the power series become to be determined uniquely as long as necessary initial values are given these recursions. After having the capability of evaluating these power series, their certain degree polynomial truncations have been used for numerical approximations. After all these, the roots of the truncated approximants have been evaluated individually for the desired roots. What we have observed in the evaluations has been the fact that the roots far from the origin need more numerical effort for a specified numerical precision. Although it seems possible to combine this proposed method with the perturbation expansion.