Explanation:
Trigonometric functions
In mathematics, the trigonometric
functions (also called circular
functions) are functions of an angle. They
are used to relate the angles of a triangle to the lengths of the sides of a triangle.
Trigonometric functions are important in the study of triangles and modeling
periodic phenomena, among many other applications.
Rightangled
triangle definitions
The notion that there should be some standard correspondence between the
lengths of the sides of a triangle and the angles of the triangle comes as soon
as one recognizes that similar triangles maintain the
same ratios between their sides. That is, for any similar triangle the ratio of
the hypotenuse (for example) and another of the sides remains
the same. If the hypotenuse is twice as long, so are the sides. It is these
ratios that the trigonometric functions express.
To define the
trigonometric functions for the angle A, start
with any right triangle that contains the angle A. The three
sides of the triangle are named as follows:
§
The hypotenuse is the side opposite the right angle, in this case side h.
The hypotenuse is always the longest side of a rightangled triangle.
§
The opposite side is the side opposite to the angle we are interested in (angle A), in this
case side a.
§
The adjacent side is the side having both the angles of interest (angle A and
rightangle C), in this case side b.
In ordinary Euclidean geometry, according to the triangle postulate the inside angles of every triangle total 180° (π radians). Therefore, in a rightangled triangle, the two
nonright angles total 90° (π/2 radians), so each of these angles must be
in the range of (0˚,90°) as expressed in interval notation. The following
definitions apply to angles in this 0°  90° range. They can be
extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they
be periodic functions. For example, the figure shows
sin θ for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle (top) and as a graph (bottom).
The value of the sine repeats itself apart from sign in all four quadrants, and
if the range of θ is extended to additional rotations, this behavior repeats
periodically with a period 2π.
The trigonometric
functions are summarized in the following table and described in more detail
below. The angle θ is the angle between the hypotenuse and the adjacent
line  the angle at A in the accompanying diagram.
Function

Abbreviation

Description

Identities
(using radians)

Sine

sin

opposite /
hypotenuse


Cosine

cos

adjacent /
hypotenuse


Tangent

tan (or tg)

opposite /
adjacent


Cotangent

cot (or ctg or ctn)

adjacent /
opposite


Secant

sec

hypotenuse /
adjacent


Cosecant

csc (or cosec)

hypotenuse /
opposite


http://en.wikipedia.org/wiki/Trigonometric_functions
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°

General
solutions
Each of the trigonometric functions is periodic
in the real part of its argument, running through all its values twice in each
interval of 2π. Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2, and then reverse
themselves over 2πk +
π/2 to 2πk +
3π/2. Cosine and secant begin their period at 2πk, finish it
at 2πk + π, and
then reverse themselves over 2πk +
π to 2πk +
2π. Tangent begins its period at 2πk − π/2, finishes it at
2πk + π/2, and
then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2. Cotangent begins its
period at 2πk, finishes it at 2πk + π, and then repeats it
(forward) over 2πk +
π to 2πk +
2π.
This periodicity is reflected in the general
inverses where k is some integer:
(Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Inverse_trigonometric_functions#General_solutions
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.