One radian is the angle subtended by an arc that has a curved length equal to the radius of the arc.

Therefore, in a

unit circle:

Assumptions:
rad = radian
angle = angle subtended by a 1 unit arc
radius = 1
1 rad = {2π(radius)} * {(angle)/(360)}
1 rad = {2π1} * {(angle)/(360)}
1 rad = 2π(angle)/360
1 rad = π(angle)/180
So in a unit circle...
angle = 57.3°
π rad = 180°
2π rad = 360°
And the conversion ratios must be...
1° = 1(2π/360) radians = 1(π/180) radians
1rad = (360/2π)° = (180/π)°

Through trigonometry, it is simple to see that various functions have periods of π or 2π. For example:

Sine Function: f(x) = sin x
Domain: all real numbers
range: {-1, 1}
Period: 2π
Cosine Function: f(x) = cos x
Domain: all real numbers
range: {-1, 1}
Period: 2π
Tangent Function: f(x) = tan x
Domain: all real numbers
range: {-1, 1}
Period: π

Through these simple equations using radian measurement, we can easily create identities far more simply than with degree measurements. For example, through radian measurements, the following equation can exist:

_ _
lim | |
| (sin h) / h | = 1
h-->0 |_ _|

This equation is the basis for many trigonometric identities.