We can cut up circles in whatever size chunks we choose — we normally choose to cut them up so that the size of the angle of an entire circle is $2\pi$ or 360. Said differently, we choose units to be $\frac{1}{2\pi}$th or $\frac{1}{360}$th of a circle. I see no reason we can't define some unit, call it a Circunit, such that 1 circunit is the angle made by a full rotation, 1/2 Circunit = $\pi$ rad = 180º, 1/4 Circunit = $\pi/2$ rad = 90º, etc.
A part of me believes these might be nice units to work with:
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We can easily isolate rotations by looking at the whole and fractional parts of our value. $2\frac{1}{4}$ Circunits, is the exact same angle as 2 full circles plus 1/4 of a circle. Often when we have an angle like $4\frac{1}{2}\pi$ rad, we can treat that as similar to $\frac{\pi}{2}$ for trig, but the notation is much less suggestive about that than something like "2.25 Circunits is similar to .25 Circunits". This is a really nice property and I think it might make a lot of physics with waves, qm or dealing with $e^{i\pi\theta}$ a lot cleaner. I guess using $\tau$ has this same advantage.
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We can avoid keeping track of redundant information: we seem to almost always write radians in terms of $\pi$. In radians, we'll rarely write in the form sin($1\frac{3}{4}$) but often do write sin($1\frac{3}{4}\pi$). In this sense, the $\pi s$ that show up everywhere feel like redundant symbolism that doesn't add anything — can't the factor of pi just be absorbed into the unit? I can imagine this might lead to fewer $\pi s$ popping up in physics — sort of like what happens in GR when we use units that set the speed of light to equal 1. Using radians feels like working with nano-scale objects and rather than just using nm, ns, etc. writing everything in meters and secs, and appending $10^{-9}$ after everything.
So the question: Is such a unit commonly used in any branch of math or science? If so, what are its properties, advantages, and drawbacks? If it is not: are there compelling reasons to not use such a unit for measuring, writing down, and working with angles? If so, what are they (my suspicion is factors of $\pi$ might be forcefully introduced when we start differentiating/integrating trig fns)? Is there any other reason that such a unit has not been adopted, at least in certain use cases, and potentially for pedagogical reasons?
*Brownie points, although I presume answers to the main question will touch on this: why is $\pi$, the ratio of circumference to radius, a good choice for the (inverse) size of a unit of angle in the first place? It seems like fundamentally the radians unit is defined to have the property that sweeping x units of angle is the same as rolling x units of distance on the unit circle, but why is that a valuable feature for a unit of angle to have? I'm aware there are deep connections between sin/cos and the unit circle, but why is arc-length of that unit circle important at all here?
Best Answer
Your question/post can be reframed as
“Why is the angle measure ‘radian’ superior to degree/gradian/cycle/revolution—or, indeed—to ‘$2\pi$ rad’?”
Please click on that Answer to see how I mean. I shall not rehash it, except to point out that $$\sin'_\text{degree}(x)=\frac{\pi}{180}\cos_\text{degree}(x);$$ this parallels LittleO's comment:
OP: how can changing choice of units make it so that differentiating actually scales $\sin$ and $\cos?$
Changing the angle measure scales $x$-axis accordingly (without scaling the $y$-axis), which affects the gradient (derivative) accordingly.