Plane Trigonometry and Numerical Computation

56. Radian Measure. In certain kinds of work it is more convenient in measuring angles to use, instead of the degree, a unit called the radian. A radian is defined as the angle at the center of a circle whose subtended are is equal in length to the radius of the circle (Fig. 45). Therefore, if an angle θ at the center of a circle of radius r units subtends an arc of s units, the measure of θ in radians is

          s
(1)  θ = ———.
          r

 2πr
————— = 2π radians = 360°,
  r

(2)  π radians = 180°.

1 radian = 180/π = 57° 17' 45" (approximately).

Fig. 45

It is important to note that the radian^[1] as defined is a constant angle, i.e. it is the same for all circles, and can therefore be used as a unit of measure.

From relation (2) it follows that to convert radians into degrees it is only necessary to multiply the number of radians by 180/π, while to convert degrees into radians we multiply the number of degrees by π/180. Thus 45° is π/4 radians; π/2 radians is 90°.

[1] the symbol ^r is often used to denote radians. Thus 2^r stands for 2 radians, π^r for π radians, etc. When the angle is expressed in terms of π (the radian being the unit), it is customary to omit r, Thus, when we refer to angle π, we mean an angle of π radians. When the word radian is omitted, it should be mentally supplied in order to avoid the error of supposing π means 180. Here, as in geometry, π = 3.14159....