A set of meticulously designed procedures and ingenious methods for calculating the exact values of sine/cosine for 1° and any integer degree has been discovered and constructed. By calculating the sine/cosine values for 3°, and then using the expansion formulas for sine/cosine of triple angles, a univariate cubic equation satisfied by the sine/cosine values of 1° is derived. Subsequently, the exact expressions for sin/cos 1° are determined using Cardano's formula and its discriminant rules. After calculating the exact values of sine/cosine for all integer degrees up to 45° in the form of 3k°, the exact expressions for sine/cosine of all angles in the form of 3k±1° are calculated, and a table of exact sine/cosine values for any integer degree is compiled.

Hirota's bilinear derivative transformation is a direct method which is more convenient than the inverse scattering transform to deal with a nonlinear partial deferential equation. Based on Hirota's method, the soliton solutions of the modified nonlinear Schrdinger (MNLS) equation under standing wave boundary condition, are obtained; and by simple method of parameter vanishing, the corresponding soliton solutions of the derivative nonlinear Schrdinger (DNLS) equation under constant nonvanishing boundary condition, are gotten, which allows the existence of both bright and dark solitons. The evolution of bright/ dark-soliton solutions in time and space is demonstrated in figures. The results are consistent with what were obtained by the inverse scattering transform.