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I think you are missing a bit:
if I face the Sun AT SUNRISE
In theory, the sun rises exactly due East on the day of the equinox.
in practice, the refraction (an effect due to the atmosphere) causes the sun to rise a few degrees left of due East (for observers in the Northern hemisphere). That is because the refraction lifts the image of the Sun, making it appear a bit higher than it really is.
You have to wait for the Sun to be a bit more than its own diameter above the visible horizon (if you do have a flat horizon at the same level as where you are) before you can say the sun is now due East.
If you live near the equator, this is irrelevant (the azimuth of the Sun does not change while it rises). As you move north, the difference gets bigger and bigger.
You could be almost 5 degrees off if you simply take the first flash of sunlight as sunrise.
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The best kind of horizon for this is the sea horizon, while the observer is not far above the water surface, on a ship.
If you are very high relative to the horizon (e.g., standing on a cliff by the sea), then another factor, called dip makes it worst.
If you live West of a mountain range, you will have the opposite effect: by the time you see the Sun appear above the mountain range, you are well passed the theoretical sunrise.
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