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.

Figure 1 |

If *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

Here are some more exercises in the use of the rules given in Equations 1,2, and 3.

**EXERCISE 1 **
Express the angle radians in (a) decimal form and (b) DMS
form.

**EXERCISE 2 **
Express the angle in radians.

**EXERCISE 3 **
Assume that City A and City B are located on the same
meridian in the Northern hemisphere and that the earth is a sphere of
radius 4000 mi.
The latitudes of City A and City B are
and , respectively.

**(a)**-
Express the latitudes of City A and City B
in decimal form.
**(b)**-
Express the latitudes of City A and City B
in radian form.
**(c)**- Find the distance between the two cities.

**
**

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Wed Dec 4 17:54:59 MST 1996

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