tailieunhanh - Handbook of mathematics for engineers and scienteists part 10

If two pairs of corresponding sides in a pair of triangles are in proportion and the included angles are congruent, then the triangles are similar. The areas of similar triangles are proportional to the squares of the corresponding linear parts (such as sides, altitudes, diagonals, etc.). | . Inverse Trigonometric Functions 31 Arccot x are multi-valued. The following relations define the multi-valued inverse trigonometric functions sin Arcsin x x cos Arccos x x tan Arctan x x coif Arccot x x. These functions admit the following verbal definitions Arcsin x is the angle whose sine is equal to x Arccos x is the angle whose cosine is equal to x Arctan x is the angle whose tangent is equal to x Arccot x is the angle whose cotangent is equal to x. The principal single-valued branches of the inverse trigonometric functions are denoted by arcsin x sin-1 x arcsine is the inverse of sine arccos x cos-1 x arccosine is the inverse of cosine arctan x tan-1 x arctangent is the inverse of tangent arccot x cot-1 x arccotangent is the inverse of cotangent and are determined by the inequalities -2 arcsin x 2 0 arccos x n -1 x 1 -2 arctan x y 0 arccot x n -to x to . The following equivalent relations can be taken as definitions of single-valued inverse trigonometric functions y arcsin x - 1 x 1 x sin y n n - 2 y 2 y arccos x - 1 x 1 x cos y 0 y n y arctan x - to x to x tan y n n - 2 y 2 y arccot x - to x to x cot y 0 y n. The multi-valued and the single-valued inverse trigonometric functions are related by the formulas Arcsin x -1 arcsin x nn Arccos x arccos x 2nn Arctan x arctan x nn Arccot x arccot x nn where n 0 1 2 . The graphs of inverse trigonometric functions are obtained from the graphs of the corresponding trigonometric functions by mirror reflection with respect to the straight line y x with the domain of each function being taken into account . . Arcsine y arcsin x. This function is defined for all x e -1 1 and its range is ye - y y . The arcsine is an odd nonperiodic bounded function that crosses the axes Ox and Oy at the origin x 0 y 0. This is an increasing function in its domain and it takes its smallest value y -y at the point x -1 it takes its largest value y y at the point x 1. The graph of the function y arcsin x is given in Fig. . 32 .