Inverse trigonometric functions
Principal values of inverse trigonometric functions.
The relation x = sin y permits to find both x by the given y , and also y by the given x ( at | x | 1 ). So, it is possible to consider not only a sine as a function of an angle, but an angle as a function of a sine. The last fact can be written as: y = arcsin x ( “arcsin” is read as “arcsine” ). For instance, instead of 1/2 = sin 30° it is possible to write: 30° = arcsin 1/2. At the second record form an angle is usually represented in a radian measure: / 6 = arcsin 1/2.
Definitions. arcsin x is an angle, a sine of which is equal to x . Analogously the functions arccos x , arctan x , arccot x , arcsec x , arccosec x are defined. These functions are inverse to the functions sin x , cos x , tan x , cot x , sec x , cosec x , therefore they are called inverse trigonometric functions. All inverse trigonometric functions are multiple-valued functions , that is to say for one value of argument an innumerable set of a function values is in accordance. So, for example, angles 30°, 150°, 390°, 510°, 750° have the same sine. A principal value of arcsin x is that its value, which is contained between – / 2 and + / 2 ( –90° and +90° ), including the bounds :
A principal value of arccos x is that its value, which is contained between 0 and ( 0° and +180° ), including the bounds :
A principal value of arctan x is that its value, which is contained between – / 2 and + / 2 ( –90° and +90° ) without the bounds :
A principal value of arccot x is that its value, which is contained between 0 and ( 0° and +180° ) without the bounds :
If to sign any of values of inverse trigonometric functions as Arcsin x , Arccos x , Arctan x , Arccot x and to save the designations: arcsin x , arcos x , arctan x , arccot x for their principal values, then there are the following relations between them:
where k – any integer. At k = 0 we have principal values.