Explanation:
Inverse trigonometric functions
In mathematics, the inverse trigonometric functions or cyclometric functions are the inverse functions of the trigonometric functions, though
they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions.
Since none of the six trigonometric functions are onetoone (by failing the horizontal line test), they
must be restricted in order to have inverse functions.
For example,
just as the square root function is defined such that y^{2} = x, the function y = arcsin(x) is defined
so that sin(y) = x. There are multiple numbers y such that sin(y) = x; for example, sin (0) = 0, but
also sin(π) = 0, sin(2π) = 0, etc. It follows that the arcsine
function is multivalued: arcsin (0) = 0, but also arcsin (0) = π, arcsin
(0) = 2π, etc. When only one value is desired, the function may be
restricted to its principal branch. With this restriction, for each x in the domain the expression
arcsin(x) will evaluate only to a single value, called its principal value. These
properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the
following table.
Name

Usual
notation

Definition

Domain
of x for real result

Range
of usual principal value
(radians)

Range
of usual principal value
(degrees)

arcsine

y =
arcsin x

x =
sin y

−1
≤ x ≤ 1

−π/2
≤ y ≤ π/2

−90°
≤ y ≤ 90°

arccosine

y =
arccos x

x =
cos y

−1
≤ x ≤ 1

0
≤ y ≤ π

0°
≤ y ≤ 180°

arctangent

y =
arctan x

x =
tan y

all
real numbers

−π/2
< y < π/2

−90°
< y < 90°

arccotangent

y =
arccot x

x =
cot y

all
real numbers

0
< y < π

0°
< y < 180°

arcsecant

y =
arcsec x

x =
sec y

x ≤
−1 or 1 ≤ x

0
≤ y < π/2 or π/2 < y ≤
π

0°
≤ y < 90° or 90° < y ≤
180°

arccosecant

y =
arccsc x

x =
csc y

x ≤
−1 or 1 ≤ x

−π/2
≤ y < 0 or 0 < y ≤
π/2

90°
≤ y < 0° or 0° < y ≤ 90°

(Our solved
example in mathguru.com uses this concept)
The notations
sin^{−1}, cos^{−1}, etc. are often used for arcsin,
arccos, etc., but this convention logically conflicts with the common semantics
for expressions like sin^{2}(x), which refer to numeric power
rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse.
http://en.wikipedia.org/wiki/Inverse_trigonometric_functions
The above explanation is copied from
Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons
Attribution ShareAlike 3.0 Unported License.