Inverse trigonometric, or circular, functions

Inverse trigonometric, or circular, functions

These functions are obtained by inversion of thetrigonometric functions:

their symbols being respectively:

these symbols are simply abbreviated forms of description for the angle (or arc), of which the sine, cosine, tangent or cotangent is respectively equal to x.

We shall consider the function:

The graph of this function (Fig. 39) is obtained from the graph of y = sin x by the rule given in Inverse functions. This graph is wholly located in the vertical strip of width two, based on the interval - 1 < x < + 1 of the axis X, i.e. the function (22) is only defined in the interval -1 < x <+1. Furthermore, equation (22) is equivalent to the equation sin y = x; and, as is known from trigonometry, for a given x we obtain an infinite number of values of y. We see from the graph, in fact, that perpendiculars to the axis X from points in the interval -1 < x < +1 have an infinite number of points in common with the graph, i.e. function (22) is many-valued.

We see directly from Fig. 39 that function (22) becomes single valued if, instead of taking all the graph, we limit ourselves to the part shown in heavier type, which corresponds to stipulating that we shall consider only those values of the angle y, having a given sin y = x, which lie in the interval

Figures 40 and 41 illustrate the graphs of y= arc cos x and y= arctan x, the parts of the graphs in heavier type being those which must be kept in order to make the functions single-valued (we leave it to the reader to draw the figure for arc cot x). It may be noted here that the functions y = arc tan x and y = arc cot x are defined for all real values of x.

By noting from the figure, the interval of variation of y over the heavier part of the curve, we obtain a table of bounds, within which the function remains single-valued:

It can easily be shown that the functions thus defined, called the principal values of the inverse trigonometric functions, satisfy the relationships:

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek