Any real number may be interpreted as the radian measure of an angle as follows: If , think of wrapping a length of string around the standard unit circle C in the plane, with initial point P(1,0), and proceeding counterclockwise around the circle; do the same if , but wrap the string clockwise around the circle. This process is described in Figure 1 below.
Observe that the length s of string gives the measure of the angle only when R=1.
As a matter of common practice and convenience, it is useful to measure angles in degrees, which are defined by partitioning one whole revolution into 360 equal parts, each of which is then called one degree. In this way, one whole revolution around the unit circle measures radians and also 360 degrees (or ), that is:
Each degree may be further subdivided into 60 parts, called minutes, and in turn each minute may be subdivided into another 60 parts, called seconds:
EXAMPLE 1 Express the angle in Degree-Minute-Second (DMS) notation.
Solution: We use Equation 3 to convert a fraction of a degree into minutes and a fraction of a minute into seconds:
EXAMPLE 2 Express the angle in radians.
Solution: From Equation 2 we see that
EXAMPLE 3 Find the length of an arc on a circle of radius 75 inches that spans a central angle of measure .
Solution: We use Equation 1, , with R=75 inches and
, to obtain