1. Unique characteristics of circles:
 All points on their perimeters are equidistant
from their centers
 Circles are formed by rotations of line
segments in fixed planes about fixed points
 The “same” circles have the same radii, and
thus the same diameters, circumferences, and
areas
 “Different” circles have different radii, and
thus different diameters, circumferences, and
areas.
The unit circle is a special circle centered at the origin of the x-y plane with a radius
of 1 and a circumference of 2π. Π is a special number referred to as transcendental
because it cannot be expressed algebraically and only as various limits as n
increases indefinitely, but more on π and limits later. 
The unit circle gives a visual representation of sine and cosine.
The equation that gives the unit circle in Cartesian x- and y-
coordinates is x2 + y2 = 1. Notice that x and y values are related
via the Pythagorean theorem. The equation, interpreted
correctly, states that no matter what, all points on the
graph of the unit circle are a distance of 1 from the origin and
that this distance is obviously dependent on each of the points’

2. location in the x-y plane, represented by two perpendicular displacements from
each axis, or x and y values. Because the equation that gives the unit circle in the x-y
plane is x2 + y2 = 1 and because sin2θ + cos2θ also equals 1, x and y represent sinθ
and cosθ. But now the question is, which represents which?
First off and most importantly, the equation for the unit circle is not a function, not
even if you value x and y arbitrarily, call them dummy variables, and switch them.
The circle formed by graphing the equation fails both the vertical and horizontal line
tests. If you take any value of x in between -1 and 1 and plug it into the equation for
the unit circle, out pop two possibilities for y. The same goes for y. Rather, x and y
are functions of the angle formed between the radius extending from the origin to
the point to which the x and y values belong and the original x-axis. This becomes
obvious after once again realizing that x and y are actually just values of sinθ and
cosθ, which themselves are functions of θ, except we still haven’t answered the
original question highlighted in green. To answer the question in green, x values
represent cosθ and y values represent sinθ, because:
As the diagram shows, the radius connecting each
point on the circle and the center of the circle (the
origin) is the hypotenuse of the right triangle
unique to each point, formed by perpendicular x
and y displacements (legs) whose squares add up
to 1. In mathematics, we refer to the angle that
increases naturally with circumference swept as
the independent variable of the sine and cosine
functions, labeled θ in the diagram.
The other angle in the triangle (not
the right angle in the diagram),
approaches being a right angle as θ
approaches 0, x approaches 1, and
y approaches 0, and behaves
strangely relative to the
counterclockwise ‘sweeping’ of
circumference. So since cosine is
defined as length of side adjacent
to reference angle, θ, divided by length of hypotenuse (always 1), and the x-
displacement is adjacent to θ, the pink x-displacement in the diagram represents
cosθ and the red y-displacement represents sinθ.
Because amount of circumference, or arc length, swept is proportional to the angle
in degrees between the radius reaching a point on the circle from the origin and the
original x-axis, and because the unit circle is a very natural representation of all the
possible combinations of sine and cosine values, we often use arc length swept to
represent corresponding angles in units of radians, simply because they are a more
natural unit of domain for the sine and cosine functions than degrees, as further
knowledge of the trigonometric functions and their graphs will help explain. The
θ

3. circumference of the unit circle is 2π, so 2π radians represents 360 degrees of angle
swept.
*Pictures taken from http://circlemaths.wordpress.com/page/4/,
http://en.wikipedia.org/wiki/Unit_circle, and
http://www.regentsprep.org/Regents/math/algtrig/ATT5/unitcircle.htm