# Inverse trigonometric functions

*Inverse trigonometric functions. Multiple-valued functions.*

Principal values of 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**are defined.

*x*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*:

*x*

+ / 2 .

A *principal value* of arccos *x* is that its value, which is contained between 0 and

( 0° and +180° ), *including the bounds*:

*x*

.

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.