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

If

Observe that the length

As a matter of common practice and convenience, it is useful to measure angles in

Each degree may be further subdivided into 60 parts, called

Therefore, .

, to obtain

MORE EXAMPLES TO UNDERSTAND

Express the angle radians in (a) degree form and (b) DMS form.

(b) To express in DMS form, proceed as follows:

*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.Figure 1 |

*Q*(*x*,*y*) is the point on the circle where the string ends, we may think of as being an angle by associating to it the central angle with vertex*O*(0,0) and sides passing through the points*P*and*Q*. If instead of wrapping a length*s*of string around the unit circle, we decide to wrap it around a circle of radius*R*, the angle (in radians) generated in the process will satisfy the following relation: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:Therefore, .

**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

MORE EXAMPLES TO UNDERSTAND

Express the angle radians in (a) degree form and (b) DMS form.

*Solution:*(a) To express radians in degree form, use the fact that to obtain(b) To express in DMS form, proceed as follows: