Here's an animated flux diagram, a cross section of the e-field pattern radiated by a short dipole emitter, from the MIT physics 8.02 course.
Here's a 3D flux pattern frozen in time. (Also lots more cool stuff from MIT 8.02)
Notice, no sine waves. Just expanding blobs (they'd be donut-shaped in 3D,) with the maximum wave-emission being broadside, and a zero node on the vertical axis. Near the EM source, the field pattern looks like expanding tori with the e-field wrapped around poloidially. Farther away from the dipole source we'd see it more as thin, expanding sphere waves with holes at the poles, with e-field and b-field flux lines "drawn upon the sphere," and at right angles to each other. The flux lines always close upon themselves to form squashed, outward-moving circles.
I suspect that these hand-waving sine-wave explanations (and even textbook explanations about "transverse waves") date to many decades ago, back when all light was "Transverse Waves In The Aether." Light is not a transverse wave, not like a shaking rope or shear-wave acoustic vibrations in solids. But it's very hard to make physics textbook authors change their language (search SJ Gould and the Fox Terrier clone problem.) The luminiferous aether was struck down, yet few authors stopped using the vibrating-string analogy for polarization, or stopped teaching us that EM radiation is "transverse wave."