Al Lorona wrote:
"How does one determine the correct antenna takeoff angle to communicate
over a certain sistance?"

Solve the trigonometry / geometry problem involving the target distance
and height of the reflecting layer.

King, Mimno and Wing in "Transmission Lines, Antennas, and Wave
Guides"say:
Elevation angles above 3-degrees are useful because they are not
immediately absorbed by earth loss. Low-angle radiation (above
3-degrees) makes a long distance trip to a distant receiver with the
fewest hops or bounces between the ionosphere and earth. At 3-degree
elevation, the distance per hop is about 3.500 km (2.100 miles). longer
distances are automatically broken up into units not exceeding 3.500 km.
Multipath transmission often exists and causes fading.

For distances less than 3,500 km, elevation angles of more than
3-degrees must be used. Often the antenna radiates a broad vertical
pattern. In any case, the useful ray becomes steeper as the receiver
becomes closer to the transmitter.

Kraus in edition No. 3 of "Antennas" shows how earth reflection affects
the pattern and impedance of a horizontal antenna. At 1/2-wavelength
elevation, the antenna has maximum radiation at 30-degrees over perfect
earth and the center-fed 1/2-wave horizontal dipole has a feedpoint of
73 ohms. The Zepp should have about the same vertical maximum radiation
angle but its impedance is something else.

Elevation angle is an inverse function of antenna height.

Best regards, Richard Harrison, KB5WZI