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I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)
My question: Is it just mathematical convention that radians are much more used in higher maths than degrees or do radians have some intrinsic advantage over degrees?
For me personally it doesn't matter if I write $\cos(360°)$ or $\cos(2\pi)$. Both equals 1, so why bother with two conventions?
The reasons are mostly the same as the fact that we usually use base $e$ exponentiation and logarithm. Radians are simply the natural units for measuring angles.
You could add more and more to the list, but I think the point is clear.
As I teach my trigonometry students: "Degrees are useless."
You want to know the length of a circular arc? It's $r \theta$ where $r$ is the radius of the circle and $\theta$ is the angle it subtends in radians. If you use degrees, you get ridiculous answers.
You want to know the area of a sector? It's $\frac{1}{2} r^2 \theta$, with $r$ and $\theta$ as above. Again, if you use degrees, you get ridiculous results.
To really understand this, move on to calculus and study arc length. The arc length of the graph of the circle gives radian results. Or, look at the power series expansion of the circular trigonometric functions: if you use radians, everything works with small coefficients; if you use degrees, extra powers of $\frac{\pi}{180}$ scatter around.
What are degrees any good for? Dividing circles into even numbers of parts. That's it. If you want to actually calculate something, degrees are useless.
Radians naturally arise when you look at some circles (note that they are a dimensionless unit). On the contrary, full circle being $360^\circ$ is due to some dude dividing the circle to as many pieces as there are days in the year (for some historical reason this resulted in $360$).
Why people think in radians then? My personal guess is that the reason is simply that mathematicians prefer to work with things that are somehow intrinsic to the object in consideration.
radians are the natural unit of measure for angles. it's no anthropocentric convention. aliens on the planet Zog that do calculus and solve physics problems will also be understanding the naturalness of describing angles in radians.
as mentioned previous, the angle, expressed in radians, is the amount of circular arc swept by the angle divided by the radial arm with both lengths expressed in the same units. so radians are dimensionless. they're just a number. no units.
if you have a wheel of radius $r$ on a slip-free surface, the distance the wheel moves on the surface $x$ is equal to the angle turned, $\theta$ (in radians) times the radius of the circle of the outer rim of the wheel, $r$.
$$ x = r \cdot \theta $$
if the wheel (of radius $r$) is spinning at a rate of $\omega = \frac{\text{d} \theta}{\text{d} t}$, the speed that the wheel moves relative to the surface is $$v = \frac{\text{d} x}{\text{d} t} = r \cdot \omega \ .$$
if the angle was measured in any other manner, there would be constants of proportionality necessary for those equations to be true but those constants of proportionality are equal to 1 (and go away) if the angle is measured in radians.
radians are as natural to angles as $e \ \approx \ $ 2.718281828... is natural as a base for logarithms and exponentials in calculus. Euler's equation
$$ e^{i \theta} = \cos(\theta) + i \sin(\theta) $$
would need more nasty constants of proportionality if the base was not $e$ or the angle $\theta$ was not in radians.
so it's the opposite of human convention that in calculus the measure of angles are expressed in terms of radians.
Degrees are a mistake of history (not speaking of minutes and seconds). Division in four quadrants of ninety degrees is quite arbitrary and inconvenient, but for one thing: it allows an easy representation of the remarkable angles, $30°$ and $45°$. In this respect, it is a little better than the $4\times100$ subdivisions in grades.
As explained by many others, radians are a natural unit as they avoid a constant factor when taking derivatives and make the formulas for the arc length or sector area the simplest.
IMO, the opportunity to use an interesting alternative is gone forever: the revolution. Counting angles in revolutions is pretty convenient as you can handle them modulo $1$, i.e. just using the decomposition in integer and fractional part.
it's because calculus would be really annoying using degrees.
$$ (\sin x)' = \cos x. $$ in radians but not in degrees.
Also, the related fact $$ \cos x = \frac{e^{ix}+e^{-ix}}{2} $$ holds in radians.
Everybody's talking about radians vs degrees. What about radians vs turns, a turn being $2\pi$ radians? I'm asking because it seems logical to think of a measure of angle as the fraction of a full circle, also I end up putting so many $2\pi$'s everywhere (mostly in sound processing) that it makes me wonder why not go the other way and use turns. The worst it can do is require you to add $\tau^{-1}$ here and there. I wonder why it's not an option discussed more often, if the world was to divide itself in two camps you'd think it would be between pro-radians and pro-turns.
In computer programming using turns instead of radians also has the advantage that integers become perfect for storing angles, they give you fixed angular precision and wrap around perfectly when they overflow.
The simple reason is that radians incorporate pi as part of the ratio which tends to be more convenient for arbitrary calculations and more complex mathematical functions.
For strictly practical manufacturing type applications degrees are often preferred because they are easier to visualise and subdivide.
In engineering radians tend to be used for kinematic calculations where you are dealing with lots of trig functions as it effectively skips a step in approximating for pi and angular velocities are generally quoted in radians per second. For example if you know the angular speed of a disk in radians per second finding the tangential speed at the circumference is a simple multiplication by the radius without any messing about with irrational constants.
However angular dimensions are generally quoted in degrees as this gives more convenient units and you are mostly working with integers or a small number of decimal places.
Radians tell you the arc length. If you have 60° you then have a bit of work to figure out the arc length: Al = n°/360 * 2πr but if you have π/3 radians, you know that the arc length is π/3 radii or π/3 * r
I don't know if this answer is good or not.
Lets say you want to measure the distance of a very distant star from earth like in the below image.
Consider that small circle earth and the big one, some distant star.
Say you observe the star from two different points on earth.
Now you can find the angle $\alpha$ and $\beta$ approximately and you can also find the distance between those two places, say $l$.
now you know $\alpha$ and $\beta$ and you can find $\theta$.
Thus, the distance of that star from earth would be $\frac{l}{\theta}$, considering $\theta$ is in radians.
not the most important use of radians but nevertheless a use of it.
Lets say i defined the area of that red rectangle $1 z$. Then i can defined area of any polygon (i took polygons only we can take any closed curve) as $x z$ where $x$ is some real number.
now what would you prefer, having area in some relation with the side or $x z$, where $z$ is just some randomly chosen area ?
This is the difference between radians and degrees.
(Note: See edits below original post.)
Here is another reason why:
This means that "$2\pi$ radians" actually equals $2\pi \ $ (the number) without any need to signify a unit of measurement. With $360^\circ$, it is absolutely necessary that we include the "$^\circ$" symbol to signify that it is in the unit of "degrees".
This is important because in dimensional analysis, you can insert radian measurements everywhere without affecting the overall units.
E.g. consider the gravitational acceleration constant: $$g=9.8 \ \text{m}/\text{s}^2. $$
If we, for whatever odd reason, needed to multiply $g$ by $\pi$ radians, the units of measurement are not affected:
$$\pi \text{ radians} \times g = 9.8\pi \ \text{m}/\text{s}^2. $$
If you felt compelled to do so, you could include the declaration of radians as a unit of measurement used:
$$\pi \text{ radians} \times g = 9.8\pi \ \text{m $\cdot$ rad}/\text{s}^2, $$
but it isn't necessary.
Of course, one might argue that this isn't a reason why we use radians, but it is simply a result of using radians. That may be like a chicken and the egg question though. However, I bet some historical research could answer the question.
Edits: Details added to avoid possible confusion.
After some feedback, here is an addendum. I was using the term "unitless" to mean "identical to pure quantity". This is not technically incorrect, per se, but is not a rigorous concept.
1) An angle $\theta$ is a dimensionless physical extent as opposed to distance $d$ which has dimensions of length.
2) Units of measurement of angles are therefore all dimensionless. However, measurements of length will always have dimension.
3) Radians are a unitless measure of angle, however, degrees are not. I.e. measurements in radians are identical to pure quantity, but this is not true for measurements in degrees. (Details below.)
Due to the way our system of mathematics has been historically constructed, measurements in radians are identical to pure quantity and therefore, the declaration of radians as the unit of measurement is redundant. (Of course, it is helpful sometimes.) For this reason, radians are unitless as they are identical to pure quantity. I.e., in the same way that Euler's Constant $e$ requires no unit attached to it (unless it represents a specific dimensional quantity), $1 \ \texttt{rad}$ does not actually need the text "$\texttt{rad}$" attached to it as it is identical to the number $1$.
$$\theta \ \texttt{rad} = \theta = \frac{\text{length of arc on a circle subtended by angle $\theta$}}{\text{radius of the circle}}$$ which has dimensions $$[\theta \ \texttt{rad}] = [\theta] = 1 = \frac{[\text{length}]}{[\text{length}]}$$ and units $$[\theta \ \texttt{rad}] = [\theta] = 1 = C \frac{[\texttt{u}_{\text{arc}}]}{[\texttt{u}_{\text{radius}}]}$$ where $\texttt{u}_x$ are the units used to measure $x$ and the conversion factor is $C[\texttt{u}_{\text{arc}}]=[\texttt{u}_{\text{radius}}].$ For example, if we measure the arc in $\texttt{cm}$ and the radius in $\texttt{m}$, then we get that $1 \ \texttt{rad} = 100 \ \texttt{cm}/\texttt{m} = 1$.
However, if we were to use degrees, then $$\theta^\circ = \theta \ \texttt{degrees} \cdot 1 = \theta \ \texttt{degrees} \cdot \frac{\pi \ \texttt{radians}}{180 \ \texttt{degrees}} = \theta\frac{\pi}{180} \ \texttt{radians} = \theta\frac{\pi}{180}.$$
Hence, $\texttt{degrees}$ are not unitless, as converting them to pure quantity requires a conversion factor.
We could have constructed our system of mathematics such that $\texttt{degrees}$ were unitless instead though. However, that would lead to more complicated formulae as noted by others in comments and answers.
As others noted, radians are a natural choice in mathematics. However, the same reason Babylonians chose 360 as the number of degrees in a circle (nice subdivisions of the whole circle) makes 360 a better choice for a full circle in numeric applications intended for graphics than $2\pi$: every different floating point format has a different numerical approximation of $2\pi$, and it is a major nuisance if $\pi/2$ may not be representable "exactly" when changing precision and/or be not exactly a third of $3\pi/2$ and/or be inequal to $2\pi-\pi/2$ even while keeping full precision.
PostScript, PDF, and METAFONT/METAPOST use trigonometric functions based on 360°/circle. Similarly the Cairo graphical library.
Radians are in some sense the “natural” units in which to measure angles. For a circle of radius $r$, an angle of $A$ radians will subtend an arc on that circle of length $rA$.
The use of degrees or grads is just a change of units for measuring angles, but if one uses units other than radians, one must always carry around conversion factors like $180/\pi$ all over the place. This pain goes beyond simple geometry since the relationship of arc-length to angle permeates so much of math, including the circular functions (sin, cosine) and other trig. functions, etc. The further one goes in math the more radians become natural. So much so that I now naturally answer “$\pi/2$” when I am asked the measure of a right angle — doesn’t everyone?
The answer is simple, it's a distance measure. When you move in a straight line you use inches or metres, in circles it is radians.
If you are at Disneyland and ask how far it was to Anaheim Stadium [go, Angels!] and I tell you that from my house it's about 45º, you are probably not going to be happy.
You want the distance traveled, at 1 mile out from my house, from one point to another. This distance is pi/4 * 1 mi = about 0.8 miles. Enjoy the game.
It is natural to define $$ \sin x=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}\tag1 $$ rather than $$ \sin x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}\left (\frac {2\pi x}{360}\right)^{2n+1}\tag2. $$
The function defined by (1) has many nice properties, for example it is the solution of the ODE: $$f''(x)=-f (x), \quad f (0)=0,f'(0)=1. $$
The same reason is the background of a special role of the Euler number $e $.
I don't know what the real reason is. I see one possible good reason to use radians. Some people define $\cos$ and $\sin$ of any number of degrees in terms of a circle. Other people define $\cos$ and $\sin$ purely as a function from $\mathbb{R}$ to $\mathbb{R}$ where the operand of $\cos$ or $\sin$ is just a real number with no units according to the differential equations
A radian is defined to be $\frac{180}{\pi}°$. It can be shown that $\forall x \in \mathbb{R}$, $\cos(x \text{ radians})$ using the first definition of $\cos$ is equal to $\cos(x)$ using the second definition of $\cos$ and $\sin(x \text{ radians})$ using the first definition of $\sin$ is equal to $\sin(x)$ using the second definition of $\sin$.
Radians are also used in one of the formulae for centripetal acceleration. The angular frequency of a uniformly rotating object denoted $\omega$ is the number of revolutions per amount of time divided by $2\pi$. That formula is $|a| = \omega^2r$. Finally when you take the Taylor series of $\cos$ and $\sin$ in radians, it's not that hard to compute that $\forall x \in \mathbb{R}\cos(x) = \sum_{i = 0}^\infty\frac{(-1)^ix^{2i}}{(2i)!}$ and $\forall x \in \mathbb{R}\sin(x) = \sum_{i = 0}^\infty\frac{(-1)^ix^{2i + 1}}{(2i + 1)!}$. That makes it so easy to compute given a number its $\cos$ and $\sin$ in radians because the $\pi$ does not appear anywhere in the Taylor series' of $\cos$ and $\sin$ in radians.
The author of the question https://academia.stackexchange.com/questions/78068/is-there-a-place-in-academia-for-someone-who-compulsively-solves-every-problem-o wants to solve every problem on their own. There might be some people like that in a job who refuse to accpet that the first 16 digits of $\pi$ are what they were told they are and use the value of $\pi$ to build something when they cannot afford to get it more than a tiny bit off until they figure out its value to a high enough accuracy on their own because people sometimes make a mistake and spread false rumours in research. In fact, the Taylor series of $\sin$ can be used to calculate $\pi$. $\pi$ is $6 \times \sin^{-1}(\frac{1}{2})$. Given that $\sin^{-1}$ has a power series centered around 0 that has a nonzero radius of convergence and is equal to $\sin^{-1}$ in its radius of convergence, there is an obvious way to use the rule of composition for a power series and the fact that $\forall x \in [-\frac{\pi}{2}, \frac{\pi}{2}], \sin(\sin^{-1}(x)) = x$ to compute the power series for $\sin^{-1}(x)$. Dropping the assumption that $\sin^{-1}$ has a power series with a nonzero radius of convergence that is equal to $\sin^{-1}$ in the radius of convergence, if we then see that that series has a nonzero radius of convergence, then for all numbers in the radius of convergence, if you first apply that power series then apply $\sin$, you will get back the original number because of the rule for composition of two power series'. From this, we can deduce that that power series is $\sin^{-1}$ in its radius of convergence. If its radius of convergence is more than $\frac{1}{2}$, that can be used to calculate $\pi$. I will not however give the power series of $\sin^{-1}$ here because in my opinion, it's better when people who want to find out the power series for $\sin^{-1}$ have to either figure it out on their own or do without knowing. Then maybe research groups that want to will be able to actually get people who for real are able to solve that problem on their own.
First I want to clear up some confusions that appear in other answers to this question.
Anything that you use to count with is a unit. That's why it's called a "unit"; it is "one" of whatever you are counting. So angle is a quantity that can be measured in various units: degrees, radians, revolutions, gradients, and I'm sure there are others.
Dimensions are things like "length" or "length$^2$" or "mass$^3$" etc. A basic rule of quantities with dimensions is that you cannot combine (add) two quantities unless they have the same dimensions. You can multiply and divide any dimensions you want, but not add. A meter plus a second doesn't equal $2$ of anything, for example. It's adding apples and oranges: undefined. (Which means it could be defined in some way if anyone thinks of a reason why you would ever want to, but nobody has yet. Anything you can do with that can be done more easily in other ways, so far.)
While angle has units, it IS dimensionless. An easy way to prove that is to consider trigonometric functions: you can't take the cosine of a kilogram, or the sine of a second. Thinking about the power series for sine, it becomes obvious that if angle had dimension D, you would have to add $D + D^2 + D^3 +...$ which is adding apples and oranges and orangutans and so on. So anything inside a trigonometric function (or an exponent, or a logarithm for that matter) must be dimensionless. This is very handy in physics class!
Now, what is the "natural" unit for angle? There's no such thing.
You can use any unit for angle you want, even make up your own, so long as it makes your life easier. Math is a tool for thinking; don't ever let it constrain your thinking needlessly instead of helping it.
If you tried to get an architect to measure all their angles in radians, they'd look at you as if you were an idiot and you'd deserve it. Degrees are far superior for basic geometric measurements. They were invented specifically because $360$ has a lot of integer divisors. If you are cutting a circle into an integer number of pieces, degrees are generally what you want to use.
Radians were invented to make different math easier. It turns out that for calculus and many aspects of higher math, radians are easiest to use. But let's start with the probable reason radians were invented in the first place: to help measure arc-length more easily. Here is how I present the arc-length formula:
Suppose you don't know how to measure arc of a circle and want to figure it out. One way to think of it is to compare a given arc to the one arc length we already know: the circumference of the circle. Let's say for simplicity that you have an angle less than a full revolution. How much length will that angle subtend when it is in the center of a circle? The easiest way is to consider what fraction of the circle is filled by the arc.
$$\frac{\theta \text{ revolutions}}{\text{one revolution}} = \frac{x}{2\pi r}$$ or $$\frac{\thetaº}{360º} = \frac{x}{2\pi r}$$ Solving for $x$, we get $$x = r \cdot \thetaº \cdot \frac{2\pi}{360º}$$
At this point a beginning physicist might say, "That's great, but I want to use this formula to derive lots of other formulas, so is there any way to make it simpler?" Yes, of course. You simply measure $\theta$ with different units, so that numerical factor goes away, and you are left with $$x = r\theta$$ Ta-da! You have just invented radians. If you like them best, you might even set them as your "default" so that if no units are mentioned, you assume that it is radians! This is what many mathematicians do, but it is a choice.
Now for different applications, different units will be easier. It turns out that many interlinked formulas based on trigonometry are shorter to write in radians, which is why people keep mistaking it for "THE natural" unit. It so happens that for calculus, radians make derivatives of trigonometric functions simpler, which has a lot of consequences.
I hope that clears up what is optional and how terms are used.
