radian

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

    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.

Ra"di*an (?), n. [From Radius.] Math.

An arc of a circle which is equal to the radius, or the angle measured by such an arc.

© Webster 1913.