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To clarify my objection is with this statement

> no one ever writes down numbers in the unit of radians, they already convert to half rotations or full rotations by scaling with pi

That statement is true only if you are talking about geometry, If you are working in any other domain the trigonometric functions operate on real numbers and have nothing to with rotations or angles and if I call sin(1) I expect to get back 0.7847 not 0

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Still, the discussion is only about convenience. For example, e^ix = cos x + i sin x (with sin/cos taking an argument in radians) would become e^ix = cos 2pi x + i sin 2pi x (with sin/cos taking an argument in turns). It's more cumbersome than the radian-based definitions, but it's not strictly different.

As mentioned elsewhere in this thread, with turn-based rotation units, this relationship can be simplified to avoid transcendental numbers entirely:

> (-1)^(2x) = cos(x) + i sin(x)

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