Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 49

a tangent line to the circle at (0,1) on the co-axis. The length of the red line,
from the co-axis to P along the line tangent to the circle, measures the tangent
of co- θ ; hence cotangent of θ . The length of the blue line, from the origin to the
red line (measured along the secant line), measures the secant of co- θ ; hence
cosecant of θ . Using the Pythagorean Theorem, the reader should verify that it
follows easily from the geometry in Figure 2.3c that csc 2 θ − cot 2 θ = 1.
Note that the green line in Figure 2.3c fits exactly into the space in Figure 2.3b
measured along the axis from the origin to the green line ( Figure 2.3d ) —that
is, that cos θ may be viewed as the adjacent side of a right triangle over the
hypotenuse (here, a radius of the unit circle). Now, the reader can verify in
a straightforward visual manner that sin 2 θ + cos 2 θ = 1 ( Figure 2.3d ). These
three trigonometric identities serve as a basic set from which others may be
verified by reducing them, through algebraic manipulation, to one of these
three forms.
In the next section, you will have the opportunity to determine the length of
one degree on the Earth.
2.8 Practice: Find the length of one degree on the Earth-sphere
• One degree of latitude, measured along a meridian or half of a great
circle, equals approximately 69 miles (111 km). Therefore, one minute
is 69/60, or just over a mile (1.15 miles), and one second is around 100
feet ((1.15 × 5280 = 6072 feet, divided by 60, or 101.2 feet), a pretty
precise location on a globe with a circumference of 25,000 miles).
Calculation: 25,000/360 = 69.444 miles in one degree of latitude along
a meridian. This value varies slightly because the Earth is not actually
a sphere; hence, the use of the generalized value of 25,000 for the
Earth's circumference.
• Because meridians converge at the poles, the length of a degree of
longitude varies, from about 69 miles at the Equator to zero at the
poles (longitude becomes a point at the poles). Calculation: At latitude
θ , find the radius, r , of the parallel, small circle, at that latitude. The
radius, R , of the Earth-sphere is R = 25,000/(2* π ) = 3978.8769 miles
(assuming the circumference of the Earth-sphere at 25,000 miles).
Thus, cos θ = r / R (using a theorem of Euclid that alternate interior
angles of parallel lines cut by a transversal are equal). Therefore,
r = R cos θ . Then, the circumference of the small circle is 2 r * π and the
length of one degree at θ degrees of latitude is 2 r * π /360.
Can you determine the length of one degree of longitude along any
latitude line?
• For example, at 42 degrees of latitude, r = R cos 42 = 3978.8769 * cos
42 = 2956.882. Thus, the circumference of the parallel at 42 degrees
north latitude is approximately 2 π *2956.882 = 18578.6205 miles.
Therefore, the length of one degree of longitude, measured along