The paper deals with the derivation of the basic relations of the theory of finite rotations. A numerical study of the dependence of the guide cosine of the final rotation on the guide cosines of a fixed coordinate system and the coordinate system associated with a solid is carried out. The formulas for the cosines of the angle of final rotation are obtained. The developed program of calculation of the values of the guides of the cosines of the final turning of the quantities in the problem statement. A graph of the dependence of the guide cosine of the final turn on the sine of the angle of the final turn. This dependence is clearly displayed when the program is started.

Keywords: vector orientation in space, spherical trigonometry, finite rotation angle, guiding cosine, Euler angles, finite rotation