> But math never decreed that sine and cosine have to take radian arguments!
That is not entirely true. It comes from the relationship between those functions and the complex numbers via the Euler formula.
ix
e = cos x + i sin x
There may be arithmetic/numerical inconveniences, but that's not all there is to "math".
Let's define ncos and nsin ("nice cos, nice sin") as follows:
nsin x = sin 2πx
ncos x = cos 2πx
So then what do we make of:
ncos x + i nsin x
This has to be
cos 2πx + i sin 2πx
which is then
2πix ( 2π) ix ix
e = ( e ) = f
2π
Where f = e is a weird number like 535.4916. This f doesn't have nice properties. E.g.:
d x x
- f /= f
dx
Otherwise it works; for instance 90 degrees is 0.25 and surely enough
0.25i
f = i
In situations not involving e in relation to angular representations via Euler, f cannot replace e.
I'm all for having parallel trig functions in libraries that work with turns, though.
The annoying 2π factor shows up in lots of places though. Should way, say, in electronics, redefine a new version of capacitive reactance which doesn't have 2πf in the denominator, but only f?
I see where you're coming from, if the formulas end up having weird numbers like 535.4916 or numbers like 2.718 or 6.28318 then obviously there's something suspicious about the equation. But small correction though. You got the number wrong, it's actually much more weird than any of those mentioned. The actual equation you come to for ncos an nsin is:
(-1)^(2x) = ncos(x) + i nsin(x)
And yes, -1 is a very weird number. If you take it to the power of something divisible by 2 you get itself raised to zero. What's up with this spooky periodicity? Also if you have x=1/4, then we get weird numbers like sqrt(-1) what on earth is that all about? No way that will fly, no way. No I'll take my 2.718^((-1)^(1/2)) and multiply through with 6.28318 that way I don't have to bother understanding what I'm doing I can sleep comfortable at night knowing that someone else has done all the thinking that needs to be done on the matter, and that turns or rotations are a blasphemous concept that breaks the very concept of math through scaling of an axis. You'd think math was strong enough to withstand such a minor change, but the textbooks do not mention it thus it must not be contemplated!
This is a very good point, but it took me a minute to get what you were saying beneath the snark. Translating without the snark:
There's a famous equation relating sin and cos to complex exponentiation. It also helps explain the Taylor expansions of sin and cos, which is one way to compute them and to find properties about them. It's a very important equation. It is:
ix
e = cos x + i sin x
kazinator's point was that this equation relies on cos and sin taking radians as arguments. If they take turns instead, then you need to insert messy extra constants to state this equation!
jVinc's counter-point, made with lots of snark, is that there's an equation that's even nicer if you just instead measure angles in turns with ncos and nsin:
(-1)^(2x) = ncos(x) + i nsin(x)
It's similar, but doesn't require the magic constant e.
A proof sketch that these are equivalent:
(-1)^(2x) = e^ln((-1)^(2x)) = -e^(2x) = e^(i * (2 pi x)) using e^(pi i) = -1
That's a nice result. If we rearrange the products in the exponent we get
2πix πi2x ( πi ) 2x
e -> e -> (e )
Where e^(πi) is -1. That shows there is something to the turns units; we can express the analog of the Euler identity using exponentiation using a base and factor which are integers.
That version of euler's formula might make a nice case for half turns. Then it's just
(-1)^x = ncos(x) + i nsin(x)
It's obvious how to handle it for integers (an even number of half turns is 1, an odd number is -1), and the extension to real numbers aids the intuition.
Or, depending on your focus, quarter turns are very clean too:
i^x = ncos(x) + i nsin(x)
Either way, turns > radians (it's what I think in when doing most fourier kinds of work anyways!).
>The actual equation you come to for ncos an nsin is:
>(-1)^(2x) = ncos(x) + i nsin(x)
Try to formally define this procedure, though. You end up going in circles.
Here's another version:
lim[N->infinity] (1 + ix/N)^N = cos(x) + i sin(x)
Now there are no "weird numbers", and both sides of the equation can be calculated directly, even by hand if you wanted.
If all you're teaching students is a bunch of formulas to be memorized, the (-1)^x notation is kind of cute. But usually when teaching math, we want to build some kind of understanding.
> if the formulas end up having weird numbers like 535.4916 or numbers like 2.718 or 6.28318 then obviously there's something suspicious about the equation.
Well, 2.718 is different than those numbers, because the derivative of 2.718^x is 2.178^x, which is a very interesting property of 2.718. The same cannot be said about 535.4. (6.283 is the ratio of a circle's, diameter to radius, which is just something intrinsic to the universe. I think it even transcends the universe, but that's hard for me to reason about. But basically, both 2*pi and e are fundamentally interesting.)
If you're not using derivatives, integrals, or complex numbers, maybe you'd be better off using Wildberger's "rational trigonometry" with quadrances and spreads instead of angles? I haven't actually tried it myself. Wildberger's motivation is a sort of ultra-strict Platonism* mixed with the desire to extend analytic geometry to fields other than the real numbers, though, so it wouldn't be surprising if it wasn't actually a simpler way to write Asteroids. Doing trigonometry in Galois fields sounds super cool though and I hope I understand it one day.
Alternatively you can just directly represent angles as unit vectors in the desired direction, which is pretty much the same as using complex numbers. Angle addition is complex multiplication, angle bisection is complex square root, and computing the sine and cosine is simplicity itself. (This takes twice as much space. If you choose to store only the real part of the complex number, you can only represent angles up to half a turn, same as in Wildberger's approach, you lose some precision near the limits, and the other operations require some extra computations.) I have tried this, for example in http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytra... and https://gitlab.com/kragen/bubbleos/-/blob/master/yeso/sdf.lu..., and in the cases I've tried it, it works great.
I'm interested to hear other people's experiences on this count!
______
* His main concern is that irrational numbers don't, in some sense, really exist, so they're a bad basis for trigonometry. As I understand it, not only is Platonism now a minority among foundations-of-mathematics types, but even Platonists generally believe that irrational numbers are just as real as rational ones, so as I understand it, Wildberger's viewpoint is held by quite a small minority. That doesn't, of course, imply anything about whether it's correct.
It's simpler than that. Draw a circle of radius 1, and draw two lines through the center. The distance along the circumference between those lines is the angle between them in radians. If you really want to remove the multiplicative identity as a magic number, you can define the radian angle as the ratio of the subtended circumference over the radius.
IMO this is at least the most accessible argument for why radians are special, and while I don't pretend to understand complex exponentiation, I expect it's the root of why other math involving radians turns out nicely.
The author makes the point that turns allow for exact representation of many commonly used angles, but with binary floating point, many common angles (1/6 of a turn, for example) are inexact.
This could be addressed by using a whole number other than 1 to represent a turn ... one that is a multiple of 3 (or 3x3) and 5, and while we're at it, 2 (or 2x2x2), so most commonly-used angles are whole numbers! That gives us 360 as the value representing a whole turn.
I just want to point out that that is an issue with radians too (pi/3). Whenever this happens I just use that same integer representation (or rational as some poster said) and then remember to multiply by tau before using a math library. With a turns-based library it would only make my life (very slightly) easier
There's no need to represent fractions of a turn as binary fractions, since you don't ever need more than 1 turn. You can represent fractions of a turn as (pair of integer) rationals, and round on the rare occasion that the denominator gets too big.
Indeed, maths never "decreed that sine and cosine have to take radian arguments". But thinking that makes any sort of point is a fundamental misunderstanding of maths.
There are infinitely many sinusoidal functions out there. You can just adjust amplitude, frequency and phase to your heart's content.
Trigonometry basically requires that sine and cosine have specific amplitudes and phases, but gives not one shit about how you map angles to frequency. Degrees are completely arbitrary, but both radians and turns have pretty natural definitions, with turns indeed being the easiest to work with. So far so good.
Calculus does have an opinion on frequency, though. There is exactly one non-trivial pair of sinusoids s(x) and c(x) where c'(x) = - s(x) and s'(x) = c(x), among a bunch of other very useful properties.
When you put calculus and geometry together, s and c are have the same amplitude and phase as sine and cosine from geometry, and the two pairs are exactly the same if you match the frequencies such that the argument is the angle measured in radians. It's just so damned useful to use angles in radians and make everything play together nicely.
Degrees are very natural in the context of ancient astronomy/astrology, where you have (1) ~365 days in a year, so that if you look at the path of something that takes a year you get about one degree change per day but with a number that is more easily divisible. (2) approximately 4y, 10y, 8y, 15y, 12y, 30y cycles for the moon and various planets. (3) A calendar with 12 months, 12 zodiac signs. (4) A timekeeping system which breaks days into 24 hours and then uses divisions by sixty for smaller units. (4) A base-sixty number system – from ancient Mesopotamia, which persisted as the standard for astronomical calculations for millennia, only displaced in the very recent past.
All those approximations have error and the behaviour you describe depends on where you are on Earth. Moreover, degrees are not natural from a mathematical perspective.
My favourite way of handling angles was always with either unsigned char or 16bit unsigned int that was treated as 1/nth of turn. Usually in these cases cos/sin tables were pre-calculated for speed, although that need went away to an extent. As long as as the calculations wrap around on the underlying system, it makes angles much easier to manage, because angle1 + angle2 = angle3 is always within 0 to 255 or 0 to 65535. Unfortunately I mostly work with higher level languages now that have mostly dropped integer types.
If anybody knows how similar calculations can be easily achieved in JS for example, I'd love to hear about it. I'm sure there must be a better way than boundary checks and manual wrap-around.
> If anybody knows how similar calculations can be easily achieved in JS for example
Simply "a = (a + 0x1234) & 0xffffffff;". Or whatever width you require, 0xff or 0xffff. JIT is going to optimize that and-operation away (at least for 32-bit mask 0xffffffff) and keep the integer value internally.
You can also "cast" a var to int by "ORring" 0 with it, like "a |= 0;"
Thank you! This does exactly what I meant. I think this is the best solution for my use-cases. It even handles floating point operations to a correctly, something I didn't expect.
For the same number of bits, an integer representation of angle is always going to be more precise than a floating point one for angles away from 0. It's also going to be equally precise for the whole circle.
In JavaScript, I’d stay with floating-point (don’t fight the language if you don’t have to) and use something like x => x - Math.floor(x) to normalize.
> PICO-8 uses an input range of 0.0 to 1.0 to represent the angle, a percentage of the unit circle. Some refer to these units as "turns". For instance, 180° or π (3.14159) radians corresponds to 0.5 turns in PICO-8's representation of angles. In fact, for fans of τ (tau), it's just a matter of dropping τ from your expression.
Back in the early 80's a common thing to do in games on 8 bit computers was to implement sin and cos as lookup tables with the angles being 0-255 or 0-128 or something like that and the result also an integer that was some fixed point representation, so you'd do something like:
I agree that this makes sense for the kind of situations that the article talks about. If you only need to express common angles like 90 degrees, 45 and so on, radians are just messy (though in physics, you get used to it).
But in other cases, radians are useful. For example consider the case of small deviations from a direction. If you give it in radians, let's say three mrad (milliradians), it's very easy to estimate how large the error will be over the course of a meter; three mm.
This is just to say: choose the right unit for the job.
That's because of the equality relationship between 2π radians and the length of the unit circle perimeter. If one is working with a sine taking in turns, one can just adjust by saying sin(q) ≈ 2π * q for small q.
As already mentioned by others, radians are not arbitrary units for angles; in fact, they are the "natural" "units", so to speak.
By definition, an angle is just the ratio of a circular arc (s) to its radius (r), θ = s/r (as an exercise, imagine how to apply this definition to the angle between two intersecting lines). When the length of the circular arc equals its radius (s = r), the angle subtended is exactly 1 radian; of course, since this is just a ratio, 1 radian is exactly the same as 1 numerically, which is why I put "unit" in quotes earlier -- a radian is not really a unit at all!
A degree, in contrast, equals pi / 180 radians. Of course, since 1 radian = 1, that really just means that 1 deg = pi / 180, similar to how 1%=0.01. Putting this all together, it is perfectly parsable (although not recommended) to say that a $5 burger costs roughly $29000% deg.
The fact that it is natural doesn't make it performant and straightforward for all applications.
For example linear algebra is the natural and general way to handle vectors. However game developers still find quaternions faster and more performant.
>The fact that it is natural doesn't make it performant and straightforward for all applications.
How does changing the scale make anything more or less performant?
If anything, it makes things less performant since to use any hardware supported trig functions you now have to convert your weird angle representation into radians. For simple addition or fractions of your angle, it is just as performant as using angles in any scaling.
> For example linear algebra is the natural and general way to handle vectors. However game developers still find quaternions faster and more performant.
They only use quaternions for a few things, like slerp, and mostly because of gimbal lock.
For everything else they still use linear algebra, and linear algebra is used much, much more than quaternions for nearly any 3d program.
Radians are a natural unit to measurements that are based on the radius. Radians are notably not natural to measurements that are based on the circumference, or to any equally divisible arcs of a circle.
Saying 1radian=1 is just as senseless as saying 1m=1=$1.
It's true that abstract math often drops units because some things (like Taylor series) work nicely in certain units. That doesn't make the unit meaningless.
Street-Fighting Mathematics, thesis/book by Sanjoy Mahajan, shows what amazing things you can die in abstract math if you don't forget units.
Sorry, but this is incorrect. An angle is defined through the ratio of two objects with common units; it is dimensionless for the same reason that 5m / 5m is dimensionless. You could argue that radians should only refer specifically to angles, but your own example demonstrates how impracticable that would be: you can't sensibly Taylor expand a trigonometric function (eg, sin(x) ~ x) if the left-hand side and right-hand side have incompatible units.
It's a good question since "radius" seems somewhat arbitrary, but the reason for defaulting to radius is that the unit circle (radius 1) produces convenient numbers in general in a way that a circle of diameter 1 doesn't. However, that doesn't mean that it's the most parsimonious choice for your particular application.
That was quite convincing actually. I guess we all have this realization at some point in early math education.
Why is it 360 degrees? Mainly because that's a nicely divisible number, no other good reason. Sometimes you find a 400 degree system on calculators but it doesn't seem to be taught anywhere (is it a French thing?)
Then at some point you get shown radians, which relates the arc length to the radius. That somehow seems natural, but it does mean there's going to be this constant lying around somewhere in your calculations.
Parameterizing the angle as a proportion of how big it can be (number of full circles) seems pretty sensible. I mean if you can avoid the constant for at least some of your geometry, then why not?
360 comes from the Babylonians, who used base-60 for numbers much for the reasons you describe (and who gave us the 24-hour day, 60 minute hour and 60-minute second, not to mention the 7-day week).
NATO forces have compasses labelled in mils or milliradians, which are not actually 1/1000 of a radian but as an approximation 1/6400 of a full turn. I still have my Silva military compass from 1989 graduated thus.
The 400 system is the grads or gradians, indeed originating from the French revolution.
Nowadays I don't think they're used as the principal unit in any country. Wikipedia does mention it gets some use in specialized fields such as surveying, mining and geology.
They are used indirectly through distance. At the time, the meter was defined as one ten millionth of the distance between the north pole and equator through the Paris meridian. That means that the meter corresponds to 1/100000 of a grad of latitude -- which is better read as "a kilometre is 1/100 of a grad".
This is symmetrical to the nautical mile, which is one minute of arc.
Regarding the 400 "degree" system: they're called gradians and it's part of the centesimal system of angular measures according to Wikipedia. And your French guess was right, they have their origins in the French Revolution. More here: https://en.wikipedia.org/wiki/Gradian
I was going to look that up to confirm it, but then I realized I could prove that statement true using some simple logic I already know.
Earth does one cycle around the sun in 365 days.
So at midnight looking straight up on a specific star (that is angled perpendicular of the rotating poles of earth) in the sky, the star you would have spotted on that day would appear slightly off the next day at midnight.
It would only end up on the same spot on midnight after 365 days.
So we are 5 days off, but I am going to believe it is true until someone is correcting me.
Readit News
That is not entirely true. It comes from the relationship between those functions and the complex numbers via the Euler formula.
There may be arithmetic/numerical inconveniences, but that's not all there is to "math".Let's define ncos and nsin ("nice cos, nice sin") as follows:
So then what do we make of: This has to be which is then Where f = e is a weird number like 535.4916. This f doesn't have nice properties. E.g.: Otherwise it works; for instance 90 degrees is 0.25 and surely enough In situations not involving e in relation to angular representations via Euler, f cannot replace e.I'm all for having parallel trig functions in libraries that work with turns, though.
The annoying 2π factor shows up in lots of places though. Should way, say, in electronics, redefine a new version of capacitive reactance which doesn't have 2πf in the denominator, but only f?
(-1)^(2x) = ncos(x) + i nsin(x)
And yes, -1 is a very weird number. If you take it to the power of something divisible by 2 you get itself raised to zero. What's up with this spooky periodicity? Also if you have x=1/4, then we get weird numbers like sqrt(-1) what on earth is that all about? No way that will fly, no way. No I'll take my 2.718^((-1)^(1/2)) and multiply through with 6.28318 that way I don't have to bother understanding what I'm doing I can sleep comfortable at night knowing that someone else has done all the thinking that needs to be done on the matter, and that turns or rotations are a blasphemous concept that breaks the very concept of math through scaling of an axis. You'd think math was strong enough to withstand such a minor change, but the textbooks do not mention it thus it must not be contemplated!
There's a famous equation relating sin and cos to complex exponentiation. It also helps explain the Taylor expansions of sin and cos, which is one way to compute them and to find properties about them. It's a very important equation. It is:
kazinator's point was that this equation relies on cos and sin taking radians as arguments. If they take turns instead, then you need to insert messy extra constants to state this equation!jVinc's counter-point, made with lots of snark, is that there's an equation that's even nicer if you just instead measure angles in turns with ncos and nsin:
It's similar, but doesn't require the magic constant e.A proof sketch that these are equivalent:
Huge selling point for turns, IMHO.
(-1)^x = ncos(x) + i nsin(x)
It's obvious how to handle it for integers (an even number of half turns is 1, an odd number is -1), and the extension to real numbers aids the intuition.
Or, depending on your focus, quarter turns are very clean too:
i^x = ncos(x) + i nsin(x)
Either way, turns > radians (it's what I think in when doing most fourier kinds of work anyways!).
>(-1)^(2x) = ncos(x) + i nsin(x)
Try to formally define this procedure, though. You end up going in circles.
Here's another version:
lim[N->infinity] (1 + ix/N)^N = cos(x) + i sin(x)
Now there are no "weird numbers", and both sides of the equation can be calculated directly, even by hand if you wanted.
If all you're teaching students is a bunch of formulas to be memorized, the (-1)^x notation is kind of cute. But usually when teaching math, we want to build some kind of understanding.
Well, 2.718 is different than those numbers, because the derivative of 2.718^x is 2.178^x, which is a very interesting property of 2.718. The same cannot be said about 535.4. (6.283 is the ratio of a circle's, diameter to radius, which is just something intrinsic to the universe. I think it even transcends the universe, but that's hard for me to reason about. But basically, both 2*pi and e are fundamentally interesting.)
In cases outside of that, radians lose their advantage over turns.
Alternatively you can just directly represent angles as unit vectors in the desired direction, which is pretty much the same as using complex numbers. Angle addition is complex multiplication, angle bisection is complex square root, and computing the sine and cosine is simplicity itself. (This takes twice as much space. If you choose to store only the real part of the complex number, you can only represent angles up to half a turn, same as in Wildberger's approach, you lose some precision near the limits, and the other operations require some extra computations.) I have tried this, for example in http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytra... and https://gitlab.com/kragen/bubbleos/-/blob/master/yeso/sdf.lu..., and in the cases I've tried it, it works great.
I'm interested to hear other people's experiences on this count!
______
* His main concern is that irrational numbers don't, in some sense, really exist, so they're a bad basis for trigonometry. As I understand it, not only is Platonism now a minority among foundations-of-mathematics types, but even Platonists generally believe that irrational numbers are just as real as rational ones, so as I understand it, Wildberger's viewpoint is held by quite a small minority. That doesn't, of course, imply anything about whether it's correct.
IMO this is at least the most accessible argument for why radians are special, and while I don't pretend to understand complex exponentiation, I expect it's the root of why other math involving radians turns out nicely.
This could be addressed by using a whole number other than 1 to represent a turn ... one that is a multiple of 3 (or 3x3) and 5, and while we're at it, 2 (or 2x2x2), so most commonly-used angles are whole numbers! That gives us 360 as the value representing a whole turn.
I just want to point out that that is an issue with radians too (pi/3). Whenever this happens I just use that same integer representation (or rational as some poster said) and then remember to multiply by tau before using a math library. With a turns-based library it would only make my life (very slightly) easier
Helix.
There are infinitely many sinusoidal functions out there. You can just adjust amplitude, frequency and phase to your heart's content.
Trigonometry basically requires that sine and cosine have specific amplitudes and phases, but gives not one shit about how you map angles to frequency. Degrees are completely arbitrary, but both radians and turns have pretty natural definitions, with turns indeed being the easiest to work with. So far so good.
Calculus does have an opinion on frequency, though. There is exactly one non-trivial pair of sinusoids s(x) and c(x) where c'(x) = - s(x) and s'(x) = c(x), among a bunch of other very useful properties.
When you put calculus and geometry together, s and c are have the same amplitude and phase as sine and cosine from geometry, and the two pairs are exactly the same if you match the frequencies such that the argument is the angle measured in radians. It's just so damned useful to use angles in radians and make everything play together nicely.
Degrees are very natural in the context of ancient astronomy/astrology, where you have (1) ~365 days in a year, so that if you look at the path of something that takes a year you get about one degree change per day but with a number that is more easily divisible. (2) approximately 4y, 10y, 8y, 15y, 12y, 30y cycles for the moon and various planets. (3) A calendar with 12 months, 12 zodiac signs. (4) A timekeeping system which breaks days into 24 hours and then uses divisions by sixty for smaller units. (4) A base-sixty number system – from ancient Mesopotamia, which persisted as the standard for astronomical calculations for millennia, only displaced in the very recent past.
Deleted Comment
If anybody knows how similar calculations can be easily achieved in JS for example, I'd love to hear about it. I'm sure there must be a better way than boundary checks and manual wrap-around.
Simply "a = (a + 0x1234) & 0xffffffff;". Or whatever width you require, 0xff or 0xffff. JIT is going to optimize that and-operation away (at least for 32-bit mask 0xffffffff) and keep the integer value internally.
You can also "cast" a var to int by "ORring" 0 with it, like "a |= 0;"
This is from asm.js which had to emulate integers so they looked through what it would take
http://asmjs.org/spec/latest/
For the same number of bits, an integer representation of angle is always going to be more precise than a floating point one for angles away from 0. It's also going to be equally precise for the whole circle.
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Deleted Comment
https://pico-8.fandom.com/wiki/Sin
> PICO-8 uses an input range of 0.0 to 1.0 to represent the angle, a percentage of the unit circle. Some refer to these units as "turns". For instance, 180° or π (3.14159) radians corresponds to 0.5 turns in PICO-8's representation of angles. In fact, for fans of τ (tau), it's just a matter of dropping τ from your expression.
Also, since PICO-8 coordinates have their origin in the top-left corner (y goes down), sin()'s result is also inverted.
But in other cases, radians are useful. For example consider the case of small deviations from a direction. If you give it in radians, let's say three mrad (milliradians), it's very easy to estimate how large the error will be over the course of a meter; three mm.
This is just to say: choose the right unit for the job.
By definition, an angle is just the ratio of a circular arc (s) to its radius (r), θ = s/r (as an exercise, imagine how to apply this definition to the angle between two intersecting lines). When the length of the circular arc equals its radius (s = r), the angle subtended is exactly 1 radian; of course, since this is just a ratio, 1 radian is exactly the same as 1 numerically, which is why I put "unit" in quotes earlier -- a radian is not really a unit at all!
A degree, in contrast, equals pi / 180 radians. Of course, since 1 radian = 1, that really just means that 1 deg = pi / 180, similar to how 1%=0.01. Putting this all together, it is perfectly parsable (although not recommended) to say that a $5 burger costs roughly $29000% deg.
For example linear algebra is the natural and general way to handle vectors. However game developers still find quaternions faster and more performant.
How does changing the scale make anything more or less performant?
If anything, it makes things less performant since to use any hardware supported trig functions you now have to convert your weird angle representation into radians. For simple addition or fractions of your angle, it is just as performant as using angles in any scaling.
> For example linear algebra is the natural and general way to handle vectors. However game developers still find quaternions faster and more performant.
They only use quaternions for a few things, like slerp, and mostly because of gimbal lock.
For everything else they still use linear algebra, and linear algebra is used much, much more than quaternions for nearly any 3d program.
It is not dimensionless.
A radian, or a turn, has a dimension: angle.
Saying 1radian=1 is just as senseless as saying 1m=1=$1.
It's true that abstract math often drops units because some things (like Taylor series) work nicely in certain units. That doesn't make the unit meaningless.
Street-Fighting Mathematics, thesis/book by Sanjoy Mahajan, shows what amazing things you can die in abstract math if you don't forget units.
https://en.wikipedia.org/wiki/Radian#Dimensional_analysis
https://en.wikipedia.org/wiki/Radian#As_a_SI_unit
I am not sure that the current definitions are consistent or useful, but I myself don't know better.
Why is it 360 degrees? Mainly because that's a nicely divisible number, no other good reason. Sometimes you find a 400 degree system on calculators but it doesn't seem to be taught anywhere (is it a French thing?)
Then at some point you get shown radians, which relates the arc length to the radius. That somehow seems natural, but it does mean there's going to be this constant lying around somewhere in your calculations.
Parameterizing the angle as a proportion of how big it can be (number of full circles) seems pretty sensible. I mean if you can avoid the constant for at least some of your geometry, then why not?
NATO forces have compasses labelled in mils or milliradians, which are not actually 1/1000 of a radian but as an approximation 1/6400 of a full turn. I still have my Silva military compass from 1989 graduated thus.
https://en.wikipedia.org/wiki/Milliradian
(It is off by 1.86%. That much error matters, nowadays, though it wouldn't have, back when.)
Nowadays I don't think they're used as the principal unit in any country. Wikipedia does mention it gets some use in specialized fields such as surveying, mining and geology.
This is symmetrical to the nautical mile, which is one minute of arc.
I was going to look that up to confirm it, but then I realized I could prove that statement true using some simple logic I already know. Earth does one cycle around the sun in 365 days. So at midnight looking straight up on a specific star (that is angled perpendicular of the rotating poles of earth) in the sky, the star you would have spotted on that day would appear slightly off the next day at midnight. It would only end up on the same spot on midnight after 365 days. So we are 5 days off, but I am going to believe it is true until someone is correcting me.