Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 50

the small circle at 42 degrees of latitude, is: 18578.6205/360 = 51.607
miles.
• This particular calculation scheme is a rich source of problems and
puzzles using geometry and trigonometry. Consider the following
question: At what latitude, angle θ , is the length of one degree of
longitude exactly half the value of one degree of longitude at the
Equator (69/2 = 34.5 miles)? These general steps are suggested as
guides for the reader: 34.5 = 2 π r /360 so that r = 1976.7, the radius
of the parallel at the desired latitude. Thus, r = R cos θ becomes
cos θ = 1976.7/3978.8769 = 0.496798, rounding to 0.5. Hence, θ = 60
degrees.
In the next section, you will have the opportunity to further explore the
connections between geography and mathematics through the exploration of
seasons, and specifically, calculating Sun angles for different latitudes and at
different times of the year.
2.9 Practice: Determine Sun angles at different seasons of the year
• The position of the Sun in the sky. On the date of the northern hemi-
sphere's summer solstice, usually on or close to June 21, the direct
ray of the Sun is overhead, or perpendicular to a plane tangent to
the Earth-sphere, at 23.5 degrees north latitude. The angle of the Sun
in the sky at noon with the ground at that latitude on that day is 90
degrees. What is the angle of the Sun in the sky, at noon on June 21,
at 42 degrees north latitude?
• Again, simple geometry and trigonometry solve the problem for
this value and for any other. Use the fact that 42 - 23.5 = 18.5
degrees; that there are 180 degrees in a triangle (look for a right
triangle with the right angle at 42 degrees north latitude); and,
that corresponding angles of parallel lines cut by a transversal
are equal. The answer works out to be 90 - 18.5, or 71.5 degrees.
Thus, on June 21 at local noon, in the northern hemisphere at 42
degrees north latitude the Sun will appear in the south at 71.5
degrees above the horizon. In the southern hemisphere at 42
degrees south on this day it will appear in the northern sky at
42 + 23.5 = 65.5, then 90 - 65.5 = 24.5 degrees above the horizon.
• Between the tropics, closer to the Equator, some interesting situa-
tions prevail (Arlinghaus, 1990). The use of this technique can be
important in calculating shadow and related matters in electronic
mapping including in, but not limited to, the formulation of virtual
reality models of tall buildings.
• Further Directions:
• The north and south poles are the Earth's geographic poles, located
at each end of its axis of rotation. All meridians meet at these poles.
The magnetized compass needle points to either of the Earth's two