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Inverse trigonometric functions

Inverse trigonometric functions. Multiple-valued functions.
Principal values of inverse trigonometric functions.

The relation  x = sin 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:

/ 2    arcsin x    + / 2 . 

A principal value of arccos x  is that its value, which is contained between  0  and     ( 0° and  +180° ), including the bounds:

  0    arccos x    .

A principal value of arctan x is that its value, which is contained between – / 2  and  + / 2   ( –90° and +90° ) without the bounds:

/ 2  <  arctan x  <  + / 2 .

A principal value of arccot x is that its value, which is contained between  0  and     ( 0° and  +180° ) without the bounds:

0  <  arccot x  <  .

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.

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