The sun's light can cast the shadow of another object, but does it ever cast its own shadow?
Sun and Shadows – How the Sun Can Cast Shadows
electromagnetic-radiationshadowsunvisible-light
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The color of the object casting the shadow is irrelevant. What this demonstrates is that our eyes perceive color partially in relative terms.
When your eyes look at the general area illuminated by the blue light, they start to cancel out the blue color. This is a lot like a camera sensing ambient light to determine the white ballance. You still see blue because not all the blue is cancelled out. However, the small shadow will also have the presumed ambient blue subtracted from it, which makes it look orange.
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Seen from their summits almost all mountain shadows look triangular regardless of the mountain's shape. This is a perspective effect. You are standing at the top edge of a long tunnel of shadowed air and looking along its length. The tunnel's cross section is the shape of the mountain but its "end" is so far away that it looks insignificant. The finite size of the sun also causes the umbral (fully shaded) parts of the shadow to converge and eventually taper away. The tapering sets limits to the umbral length of shadows. That of the Earth is over a million miles. That of a high mountain can be two to three hundred miles. Triangular shadows are not seen from objects much smaller than mountains because their shadows are not long enough.
On another page, it says:
Mountain shadows at sunrise and sunset are immensely long tunnels of unlit air, crepuscular rays in fact.
From the summit, perspective effects nearly always make the shadow triangular regardless of the mountain's profile. You are standing at the top edge of the shadowed tunnel and looking out along its length which can be more than a hundred miles. Only the shadow's end carries much information about the mountain shape and it is so far away and in any event blurred by the 0.5ยบ angular spread of the sun's rays that it is hardly visible
To elaborate on "unlit air", as an answer to your "There don't appear to be any clouds or anything," air itself reflects light, as is evident at night when it is dark. It is also seen in the shadow of the earth as the sun rises on a clear sky, a deep blue separated often from the pink/orange of sunrise and sunset. That was the beginning and end of a day in the variable hour ancient calendars.
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Yes. For example, on October 8th 1970 Earth was in the Sun's radiofrequency shadow with respect to quasar 3C 279. In other words, quasar 3C 279 was occluded by the sun.
Observation from just before and after the occulation permitted measurement of the bending of radiowaves as a test of general relativity.
The sun would also block other frequencies of electromagnetic radiation including visible light.