A Circular Coordinate System Worksheet
There are three basic circular coordinate systems commonly in use. In this class (Trigonometry) we use
what is commonly referred to as the polar coordinate or mathematicians polar system. The other two are
the azimuth system and the bearing system. The azimuth system is used primarily in navigation and the
bearing system is used primarily by surveyors but is becoming antiquated and is slowing fading from use.
In each case it is possible to report angles in radians, decimal degrees, or -degrees, minutes, seconds(DMS). Customarily radians are used in the polar system, decimal degrees in the azimuth system and
DMS in the bearing system.
In each circular coordinate system it is necessary to specify three things. An origin, a zero angle direction
and a “sense” for positive angles (i.e. CW or CCW). The bearing system is a bit odd as there are two zero
angle directions and the “sense” is based on a cardinal direction (i.e. E,W) rather than CW or CCW.
In circular coordinate systems we find a point in the plane by turning an angle from our zero angle
position and then travelling out a given distance from the origin. Like the Cartesian system, a point is
described by an ordered pair, but the first slot is the distance out from the origin along a ray/radius and the
second slot is the angle turned from the zero angle position in the proper “sense”.
Each of theses circular systems can have a Cartesian system superimposed on it. In which case every
point in the plane has a coordinate pair in both the circular/polar and Cartesian systems. They are
obviously not the same numbers in the pair, but they do indeed identify the same point in the plane. The
following drawing has the polar coordinate and Cartesian systems superimposed to show their
relationship.
Coordinate of Point
Cartesian: (x,y) = (4,3)
Polar: (r,)=(5,invtan(3/4))
Ok, Now that we understand how the various coordinate systems work let’s try a few exercises. First we
learn to transform angle measure from radians to decimal degrees to DMS
Convert to the indicated angle measure
 = degrees, DMS= degrees min sec, rad = radians,
Note: 1 minute=1/60 degree, 1 second = 1/60 minute = 1/3600 degree,  (rad) = 180
Radians(rad)
Decimal (  )
DMS (
 )

2
0
11/6
3/4
240.11
60 30 45

  
Now we need to be able to convert between the three systems of angle measure. Recall that we report
angles in radians for polar, decimal degrees for azimuth, and DMS for bearing. Also keep in mind that
the goal is to end up in the SAME location in the plane no matter how we do the “accounting”. Also note
that regardless of the circular system used the second slot in the coordinate pair is the length out from the
origin along the ray/radius which will be the same for all.
Convert the given DIRECTION to the indicated systems (either: polar, bearing, azimuth)
Polar
Azimuth
Bearing

0
0
11/6
3/4
240.11
S 60 30 45 W