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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:

  1. 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.

  2. 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?

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  • $\begingroup$ Considered whether this is better for Mathematics Educators, but I'm less interested in the pedagogy and history of using radians, and more on concrete and practical mathematical reasons that is/isn't a valuable choice of unit $\endgroup$
    – David Lalo
    Nov 3, 2021 at 6:32
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    $\begingroup$ If we define $\text{Sin}(x)$ and $\text{Cos}(x)$ to be the sine and cosine of $x$ circunits, then we would have $\text{Sin}(x) = \sin(2\pi x)$. So $\text{Sin}'(x) = 2\pi \cos(2\pi x) = 2 \pi \text{Cos}(x)$. That's a less beautiful formula than $\frac{d}{dx} \sin(x) = \cos(x)$. $\endgroup$
    – littleO
    Nov 3, 2021 at 6:39
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    $\begingroup$ see wiki entry of turn $\endgroup$ Nov 3, 2021 at 6:45
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    $\begingroup$ In engineering a term “revolution” is also widespread (as in rpm = revolutions per minute) $\endgroup$ Nov 3, 2021 at 11:04
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    $\begingroup$ @DavidLalo With radians the fourth derivative of $\sin$ is itself. That is nice and makes the Taylor series much simpler. $\endgroup$
    – badjohn
    Nov 24, 2021 at 11:01

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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:

If we define $\text{Sin}(x)$ and $\text{Cos}(x)$ to be the sine and cosine of $x$ Circunits, then $$\text{Sin}'(x) =2 \pi \text{Cos}(x).$$ That's a less beautiful formula than $\sin'(x) = \cos(x)$.

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.

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  • $\begingroup$ Appreciate the answer. I think the claim You make in the last sentence is really what I need to understand my confusion, but I'm not really sure I understand the claim other than just restating that "changing choice of units makes diff'ing scale sin/cos". $\endgroup$
    – David Lalo
    Nov 28, 2021 at 2:53
  • $\begingroup$ If the question wasn't clear, my confusion is: if You have an equation to represent some physical real thing modeled by the equation (in radians) x = cos(t), and therefore x' = -sin(t). If I have the same system and I want to represent it in circunits/turns instead now x' = -2πsin(t), which is a physically different system, which has a higher maximum speed. Changing how I measure angles should have no impact on how I measure speeds (or should it?) $\endgroup$
    – David Lalo
    Nov 28, 2021 at 2:57
  • $\begingroup$ This definitely helps but it's still not completely clear to me. x and x' here need not correspond to angles and angular velocities but rather might represent the literal x-axis of something in space (normally angular position/velocity are parameters for the trig fn, and in this case I was hoping t would evoke time). In my example I was thinking about waves, or an ideal spring, which are well described with trig fns, but there are no physical angles being measured that we are ever going to actually scale to match our measuring system. $\endgroup$
    – David Lalo
    Nov 29, 2021 at 1:17
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    $\begingroup$ @DavidLalo Switching from radians to Circunits involves scaling <position> graph's horizontal axis accordingly, which affects the the gradient; this effect is equivalent to the <velocity> graph's value having that 2π adjustment factor. $\endgroup$
    – ryang
    Nov 29, 2021 at 3:12
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    $\begingroup$ @DavidLalo Here's a concrete (position, velocity) example: $(\sin_r 2t, 2\cos_r 2t)$ is equivalent to $\displaystyle(\sin_c \frac t\pi, 2\cos_c \frac t\pi).$ Observe that to preserve the input-output correspondence, changing the trigo variant goes hand in hand with altering the function specification. $\endgroup$
    – ryang
    Nov 29, 2021 at 4:19

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