Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 47

Use ArcGIS Online ( http://www.arcgis.com/home ), start a new map, use “Add”
and search for the “USA Topo” to find the USGS topographic maps. Add the USA
Topo map layer. Then search for the location “Hiawatha, KS.” Zoom in until you
can see the PLSS system with its sections, townships, and ranges identified.
Indicate the PLSS section for the location of the courthouse in Hiawatha. The map
shows the courthouse and most of the town as occupying Section 29. For
obtaining the township and range for Hiawatha, you may need to consult
additional map sources. Or, search and add the layer “Kansas Public Land Survey
System” to your map. Can you determine the township and range for Hiawatha?
Hiawatha is in Township 2 South Range 17 East.
2.7 Theory: Visual trigonometry review
Another part of mathematics that most of us learned in high school and used
in college in calculus courses and elsewhere involves trigonometric functions.
They were important in the time of Eratosthenes and permitted him to come
to his remarkable estimate of the circumference of the Earth. They are as
important today as they were then. They serve as a theoretical backbone of
much of the capability of Earth measurement that has enabled wonderful
technological advances such as handheld Global Positioning devices that let
us find our way around the complex world of today. Thus we offer a straight-
forward visual display of trigonometry to ensure that all are working from the
same conceptual base. A visual approach can make things clear that might
otherwise seem mysterious.
The geometric arrangement in Figure 2.3a shows the basic set of entities
(based on a unit circle in the Euclidean plane) that we shall use to illustrate
the geometric origins of trigonometric functions (Arlinghaus and Arlinghaus,
2005). Notice the importance of the idea of “axis” and “co-axis.” Here, the
word prefix “co” stands for “complementary,” as in “at a right angle.” The
significance of the naming will play out in the naming of trigonometric func-
tions, such as “sine” and “co-sine” (cosine). As with datums in Chapter 1 and
here, careful attention to axis arrangement is important.
Figure 2.3b shows the geometric derivations of the trigonometric functions
for the angle θ . The length of the green line, dropping from P to the axis,
measures the sine of θ : It is the opposite side of a right triangle over the
hypotenuse (here, a radius of the unit circle). Draw a tangent line to the circle
at (1,0) on the axis. The length of the red line, from the axis to P along the line
tangent to the circle, measures the tangent of θ . The length of the blue line,
from the origin to the red line (measured along the secant line), measures the
secant of θ . Using the Pythagorean Theorem, the reader should verify that it
follows easily from the geometry in Figure 2.3b that sec 2 θ - tan 2 θ = 1.
Figure 2.3c shows the geometric derivations of the trigonometric co-functions
(where “co” stands for “complementary”). The length of the green line, drop-
ping from P to the co-axis, measures the sine of co- θ ; hence, cosine of θ . Draw