On the correctness of the determination of trigonometric functions by the system of functional equations

Abstract

The article shows that the system of the functional equations $S(x+y)=S(x)C(y)+S(y)C(x),\,C(x+y)=C(x)C(y)-S(y)S(x)$ defines a broader class of the functions than the trigonometric equations. The conditions, which must be added to this system to obtain the unique solution $S(x)=\sin{ax},\,C(x)=\cos{ax}$, have been developed. The method of the reduction of the functional equations to the differential equations is used.