Dann Corbit wrote:

But in a very real sense it is a measure of rotation. We could call it a special measure, sort of like the way that e is a special base compared to all others.

That's not the only thing that radians are useful for, though.
Consider a weight bobbing up and down on a spring, or anything
else with periodic behaviour. It's common to represent its
position in the oscillation cycle in terms of an angle, even
though nothing is actually rotating.

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Greg
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