Roy, W7EL wrote:
"When asked for the justification of the "cosine rule", he never offered
any, so its origin remains obscure."

It comes from the similarity of a standing wave antenna to a
transmission line. It isn`t a prominent feature of antenna texts because
energy isn`t confined to an antenna as it is to a transmission line.
Antenna behavior is more complicated. Terman refers the reader to his
transmission line section to explain standing wave antenna action
innstead of making a detailled explanation in his antenna section. Maybe
he was limited in number of pages. The explanation is available from a
collection of sources.

From "Transmission Lines, Antennas, and Wave Guides" by King, Mimno,
and Wing, page 93:
"---the distribution of current in antennas with h=WL/4 and with a wide
range of radii can be represented quite accurately by:

Iz = Io cos beta(z)

beta(z) is the distance from the antenna input in degrees.

It is obvious that the current must reverse directions at the open
circuit end of the antenna. This results in a total of forward and
reflected current of nearly zero while 1/4-wave back from the open
circuit, a current maximum results.

Kraus says that at the center of a 1/2-wavelength dipole, the effect of
wave cancellation is least and radiation is maximum perpendicular to the
wire at the center of the dipole, or words to that effect. The same is
true of the 1/4-wave vertical which along with its reflection in a
ground plane constitutes a 1/2-wavelength dipole.

Kraus also says that the tip cancellation accounts for the dipole
radiation pattern shaped as a doughnut impaled at the center of a
dipole. Again, these are my interpretations of Kraus` words as I
remember them.

Best regards, Richard Harrison, KB5WZI