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Why is $2\pi$ radians not replaced by $1$ turn in formulas?

The majority of them would be simpler. If such a replacement was proposed earlier, why was it declined?

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I like "revolutions" as a measurement because I don't have to stoop to using degrees, but don't confuse people by using radians. (But, for mathematicians known to me, I will use radians, as they have desirable properties and come straight from the usual definitions of an angle) – Milo Brandt Mar 24 at 23:33

Because if you use $1$ for the turn instead of $2\pi$ (forgetting what a radian is), you get that $\frac{\mathrm d}{\mathrm dx} \sin x = 2\pi\cos x$, which is probably not what you want, and many other problems, such as $\sin x$ not being a solution to $y''+y=0$ etc.

You can discuss whether the full turn is $2\pi$ or $\tau$ (tau), but you can't change the fact that it's equal to $6.2831853071\cdots$.

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I think a better link for tau would be www.tauday.com which quite accurately describes tau's superiority to pi. – corsiKa Mar 24 at 19:59
    
Or if you want to link to Wikipedia, en.wikipedia.org/wiki/Tau_(mathematical_constant) would probably at least be a better target. – Ilmari Karonen Mar 24 at 20:53
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"You can discuss whether the full turn is 2π or τ (tau), but you can't change the fact that it's equal to 6.2831853071⋯." I strongly disagree. The radian is a derived unit. You can define angle to be measured with many arbitrary choices. We defined one full cycle of angle measurement to be $2\pi$. It is a natural choice only in that $\pi$ is an exceedingly pertinent parameter in the context of a circle. – zahbaz Mar 25 at 4:47
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@zahbaz Well, you can define the angle as you wish. My favourite definition of sine is that it's the solution to $y''+y=0$ for which $y(0)=0$ and $y'(0)=1$. Or that $\sin x = \frac{e^{ix}-e^{-ix}}{2i}$, where $e$ is such number that the derivative of $e^x$ at $0$ is $1$. Note that I as a mathematician cannot care less about SI and their units; sine does not take an angle as an argument, it takes a number. – yo' Mar 25 at 6:29
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So the question is: Do you want to use such a measure of angle that the mathematical sine is the same as the physical one? If yes, you can't but have 6.28 as the full turn. If no, you can do whatever you wish of course, but you're likely not simplifying things once you need any higher mathematics than pure Euclidean geometry involved. – yo' Mar 25 at 6:38

There is a movement to make $2\,\pi=\tau$ a fundamental constant instead of $\pi$ (read The Tau Manifesto.) But not as far as I know to use $1$ as the measure of the whole circunference. One possible reason is that $\pi$ encodes the relation between the radius an the length of a circumference: a circular sector of radius $r$ and angle $\alpha$ radians has a length equal to $\alpha\,r$; it would be $2\,\beta\,\pi\,r$, where $\beta$ is the measure of the angle in the units you propose. Moreover, you cannot avoid $\pi$ in formulas like $$ e^{\pi i}+1=0. $$

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And the final expression could then be $e^{\tau i} = 1$ – Henry Mar 24 at 12:10
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@yo': $e^{\tau i} = 1 + 0$ if you insist on having it. There's no reason to have the "$+ 1 = 0$" rather than "$= -1$" in the original. – Deusovi Mar 24 at 13:53
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Or $e^{i\frac{\tau}{2}}+1=0$, which includes 4 basic operations (now with division) and not 5 but 6 of the most important numbers in all mathematics. – Nathaniel Mar 24 at 14:15
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Aesthetic pickiness aside, the equation $e^{i\tau} = 1$ is strictly weaker than $e^{i\tau/2} = -1$. The former is consistent with $e^{i\tau/2} = 1$, too. – Ryan Reich Mar 24 at 23:40
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@RyanReich And you can say the same about $e^{i\pi}=-1$ as opposed to $e^{i\pi/2}=i$, since it is also consistent with $e^{i\pi/2}=-i$. And so it goes all the way down; you won't be able to pin down the $e^{ix}$ function using single point equations. – Mario Carneiro Mar 25 at 2:43

In fact, "radians" are not a unit like meters or second, so you can't rescale them to make $2\pi$ become $1$ (as sometimes you do in physics, rescaling e.g. meters to make $c=1$).

As an aside, degrees are an absolutely artificial concept, and you should try to never use them.

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"From what I've heard from my physicist colleagues, they frequently assume that all constants are equal to 1" is just purely wrong. What physicists do is to re-scale measurement processes so that some measured units end up having a different numerical value with respect to the original units used for the measurement itself. Whoever upvoted that comment should be truly prosecuted. – Gennaro Tedesco Mar 24 at 13:53
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As a matter of fact, physicists do occasionally set pi equal to one (as a very coarse approximation rather than as a definition) - see Fermi estimation. – Nathaniel Mar 24 at 14:18
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"you should try to never use them" unless you want to communicate with the other 99% of the population! – Oddthinking Mar 24 at 15:21
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@Nathaniel Even as a very coarse approximation, it's probably not wise: $\pi^2\approx 10$, so if you have multiple factors of $\pi$ you're at risk to start losing orders of magnitude. – Semiclassical Mar 24 at 16:31
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@Semiclassical This would be in a context where a few orders of magnitude lost here or there are not a problem. Like estimating how many stars are in the universe, or the field strength of a graviton (where the real answer is "way too weak to ever detect" and a few orders of magnitude won't change that). – David Z Mar 24 at 17:54

Radians are the natural, dimensionless choice of unit to measure angle. We could certainly define the full turn as $1$. That's a nice, wholesome unit. It's nicer than pushing our historical comfort with base $60$ with $360^\circ$ or dealing with cumbersome $400 \text{ gon}$. But does $1$ really reflect the system we're trying to describe?

In choosing an appropriate unit, we should ask "What is an angle measure?" One appropriate definition is the measure $\theta$ of the angle subtended by an arc length $s$ of a circle of radius $r$. This definition, though, needs some work to become a practical tool. Given the obvious relationship between an angle and a circle, we can call upon Euclidean geometry to find ourselves a nice parameter. Behold the ratio of circumference to diameter: $C/d=\pi$! Now we observe that the relationship between arc length and circumference goes as

$$s = (\text{some fraction})\times C = \dfrac{\theta \times C}{\theta_{turn}}$$

We're getting somewhere. Let's adopt a convenient parametrization of angle measure, something we can throw numbers at and interpret easily. Perhaps a convenient choice would necessitate a convenient circle. Well, the unit circle has a nice radius and area. Its circumference is $2\pi$. If the distance around the unit circle is $2\pi$, then why don't I adopt this for my unit of angle measure? I'll define the angle measure $\theta_{turn}$ that takes me around the circle to be $2\pi$.

Arc length is now cleanly given by

$$s = r\theta $$

That's the definition of radian in that last line. Why is it a superior choice? Any other choice of units would have left over some other constants. With the choice of $2\pi$, we have a clean, relevant, and dimensionless unit.

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Just a complement to zahbaz answer.

With angles in radian, you get the nice formulas:

sin' = cos
cos' = -sin

sin(x) = 1/2 * (exp(ix) - exp(-ix))
cos(x) = 1/2 * (exp(ix) + exp(-ix))

Of course those are mathematical considerations, and that's the reason why in common life we use degrees.

But there are other units. In artillery, the thousandth (millième in french) is used. You have 6400 of then in one turn (why not...) but as an approximation, it gives an elevation of 1m at 1km of distance. If you have this in binoculars and can guess the height of something, you can easily compute a distance - even a military can...

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