A radian is a measurement of angle, based upon a fundamental property of circles. Its unit is that angle where the amount of circumference subtended is equal to the radius of that circle. This turns out to be 57.296 degrees. There are 2*pi radians in a circle (360 degrees).

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:
    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.

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