How does the radiation pattern of a horizontally polarized antenna vary with increasing height above ground?
Why A horizontal antenna's elevation pattern is formed by interference between the direct wave and the wave reflected from the ground, which acts like an image antenna the same distance below the surface. The peak of the lowest lobe occurs at roughly sin(theta) = lambda/(4h), so as the height h in wavelengths grows, that angle gets smaller. A dipole at 1/2 wavelength peaks near 30 degrees, at 1 wavelength near 15 degrees, which is why DX operators want their antennas high.
Watch out The beamwidth choices confuse planes: height above ground reshapes the vertical (elevation) pattern, while the azimuth pattern of a horizontal wire is set by the wire itself and changes little with height. The choice that says the takeoff angle increases has it backwards; that is what happens as you lower the antenna.
Higher antenna, lower angle. Half wave up gives about 30 degrees, one wavelength up gives about 15 degrees.