Why is radian so common in maths?

Question

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 intrinsic advantage is that the limit $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ holds only if we use radians. This is very important in calculus. — MathematicsStudent1122, May 24 '16 at 08:42
Well. For a unit circle the actual length of an arc spanning $n\deg = \pi/180$ units. Wheres as the actual length of the arc spanning $n$ radians is $n$ units. — fleablood, May 24 '16 at 08:44
One advantage of using radians is that you don't have to bother writing that little circle that is the symbol for degrees. — bof, May 24 '16 at 08:45
+1 because after 3 years of studying math I don't know either — Prince M, May 24 '16 at 08:49
Related: http://math.stackexchange.com/questions/720924/why-do-we-require-radians-in-calculus, http://math.stackexchange.com/questions/1339540/why-does-the-derivative-of-sine-only-work-for-radians — Hans Lundmark, May 24 '16 at 09:25
An advanced alien race would know about radians, but surely not degrees, which are an artifact here on Earth inherited from the Babylonian fascination with the number 60. Our alien friends might have inherited their own strange system of angle measurement, perhaps related to the number on tentacles sprouting from their foreheads. But, being advanced, they would have moved on to radians, or blozzletons, as they might call them. — zhw., May 24 '16 at 21:38
Having "degree" as a unit is superficial. Since angles can be measured as the ratio of arc length to radius, they are a unitless quantity. — Superbest, May 24 '16 at 22:35
@zhw an advanced alien race might divide a circle into 168 'flugnorps' and make war on our puny planet for our degree/radian heresy. — David Conrad, May 24 '16 at 23:41
I want to write an answer to this but don't have time right now. In 4 weeks I will have time. Please send me a comment in 4 weeks if I've forgotten to write an answer. — goblin GONE, May 25 '16 at 07:18
Radians are the natural units of mathematics but degrees certainly have pragmatic value in the "real world". Having an integer number of slices add up to the whole is consistent with other common units (consider 100cm = 1m, 14lb=1st, 1hr=60m). And all the divisors of 360 come in handy when you're laying out a clock face, cutting a cake or building a cathedral dome (100 degrees would be far more restrictive) — ejrb, May 25 '16 at 10:21
Good thing with degrees: easy to write real-world values. Good thing with radians: easy to do trigonometry and calculus. Mathematicians are mostly interested in the latter, while the rest of the world won't care for anything else than the former. — tarulen, May 25 '16 at 13:59
The concept of 'Radians' is important in mathematics because it hides significant philosophical discussions behind an abstraction. It is a convenience. So much so that most people don't even know where the hidden scale factors are. https://xkcd.com/435/ — Philip Oakley, May 26 '16 at 14:57
@zhw Not necessarily. Why not measure degrees in "circulons," where $2pi rad = 1 circulon$? Or "diametrons," where the arc defined by an angle of 1 "diametron" is exactly the length of the circle's diameter? — Kyle Strand, May 27 '16 at 00:51
@KyleStrand Those would be a strange choices, but it's still arc length along the unit circle. — zhw., May 27 '16 at 01:06
@CalumGilhooley, thanks. As chance would have it, I still don't have time to write a proper answer. But maybe in the next few days, things will cool down a little. — goblin GONE, Sep 12 '16 at 13:20

tomasz · Answer 1 · 2016-05-25T09:55:46.713

109

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.

The length of a circle segment is $x\cdot r$, where $x$ is the measure and $r$ is the radius, instead of $x\cdot r\cdot \pi/180$.
The power series for sine is simply $\sin(x)=\sum_{i=0}^\infty(-1)^i{x}^{2i+1}/(2i+1)!$, not $\sin(x)=\sum_{i=0}^\infty(-1)^i(x\cdot \pi/180)^{2i+1}/(2i+1)!$.
The differential equation $\sin$ (and $\cos$) satisfies is $f+f''=0$, not $f+f''\pi^2/(180)^2=0$.
$\sin'=\cos$, not $\cos\cdot 180/\pi$.

You could add more and more to the list, but I think the point is clear.

edited May 25 '16 at 09:55

answered May 24 '16 at 08:48

tomasz

35,474

9

@Soke The real reason to use $\tau$ over $\pi$ in my opinion is that it would make the introduction to trigonometry (beyond right-angled triangles) a lot easier for those poor 17-year-olds (or however old they are where you're from). It would make math lessons a smidgen less frightening and complicated at no cost of factual accuracy or anything like that. The real cost is the rewriting and reprint of all the school books, plus teachers who don't see the point and teach $\pi$ anyways, which is even worse because now they're contradicting the book. – Arthur May 25 '16 at 19:09
9

@Soke, if you used $\tau$ you'd get random $\frac 12$'s everywhere instead. The only real benefit with $\tau$ is that it is easier to write. And while we are speaking of it, how come $\tau = 2\pi$ when $\pi$ has two legs, but $\tau$ only one? – Ennar May 25 '16 at 19:11
11

@Ennar Meh, I think $C = \tau r$ and $A = \frac{1}{2} \tau r^2$ is better. I would hardly call the $1/2$ in $\frac{1}{2} x^2$ to be a "random" half. – MT_ May 25 '16 at 19:48
5

@Soke, it is as random as $2$'s with using $\pi$. I'm not convinced. On completely unrelated note, fraktur should probably disappear altogether... – Ennar May 25 '16 at 19:57
3

@Ennar, The point is that the $1/2$ clearly comes from anti differentiation of $x$. Anyway, it's a conversation that's been had too many times. I don't think $\tau$ is better enough to warrant changing the standard, but I do think it's marginally better. – MT_ May 25 '16 at 20:13
11

@Soke, again, $\frac 12$ comes from antidifferentiation of $x$ as clearly as $1$ comes from antidifferentiation of $2x$. In my opinion, the whole $\pi$/$\tau$ discussion brings no mathematical benefit and lies into "popular mathematics" sphere. – Ennar May 25 '16 at 20:25
If you see $a^2 + b^2 = c^2$, do you think this comes from anti differentiation of $2a, 2b, $ and $2c$ as well? Of course not. – MT_ May 25 '16 at 21:17
I agree that τ is really much more reasonable/grounded number, than π. While π – was just a historical mistake. (But not so important mistake.) – Sasha May 26 '16 at 13:39
@Ennar because the "one leg" letter already exists in the computer, and the "three leg" letter (the original manifesto's character popularizing the idea) cannot be typed. – JDługosz May 27 '16 at 06:15
1

@Ennar, yes, $\frac{1}{2}$ comes from antidifferentiation of $x$ as clearly as $1$ comes from antidifferentiation of $2x$ -- if you already know calculus. If you're learning calculus, the visual similarity between $\frac{1}{2} \tau r^2$ and $\frac{1}{2} x^2$ makes the relationship much more obvious. – senderle May 27 '16 at 12:15
@Arthur May I refer you to the Tau Manifesto? As for the issue with rewriting, there is no reason both can't be taught. There is no ambiguity. – Solomon Ucko Apr 03 '19 at 11:53
1

@SolomonUcko First, this is three years old. Second, you see me advocating $\tau$ as the objectively better alternative here, did you really believe I hadn't seen the manifesto before? And third, as a math teacher, there is one big reason both can't be taught: It would confuse the heck out of the poor students. All but the brightest would probably go cross-eyed with questions of "Why?", and "I don't remember which was which", and "When do I use $\pi$ and when do I use $\tau$?" – Arthur Apr 03 '19 at 12:29
That doesn't mean it's now irrelevant... 2. I apologize; I misread your comment as implying that reduced confusion is the only reason to use $\tau$. 3. We could teach people to convert $\pi$ to $\tau$, so they can understand math written in terms of $\pi$, and tell them to use $\tau$, and not $\pi$, when writing. $\pi$ would be exclusively taught as $\frac\tau2$. One good way to remember which is which is that $\pi$ has 2 tallies in it's "denominator", whereas $\tau$ only has 1.

Solomon Ucko

Apr 03 '19 at 12:51

@SolomonUcko: That's still a whole lot of confusion for, let's say, uncertain improvement. – tomasz Apr 03 '19 at 13:05

@tomasz I personally believe that the small potential for confusion caused by having both is by far outweighed by the benefits of $\tau$. – Solomon Ucko Apr 03 '19 at 16:26

@SolomonUcko: Well, I strongly disagree - both about the small potential, and the supposed benefits. – tomasz Apr 03 '19 at 18:20

@tomasz Could you explain why? As for the benefits, for now, I will refer to https://tauday.org/ and the resources it links to to avoid repetition. – Solomon Ucko Apr 03 '19 at 18:29

@SolomonUcko: It's just my impression from my experience as a student and a teacher. There is not much to be said --- I just think it would be very confusing for inexperienced students. Also, Firefox tells me that the site you have linked is unsafe, so I won't look there. – tomasz Apr 03 '19 at 20:10

Let us continue this discussion in chat. – Solomon Ucko Apr 03 '19 at 20:11

@tomasz What subject(s) did you find $\tau$ confusing in and how? As for tauday.org, it is apparently actually a redirect to https://tauday.com/, which seems to work better (oops). – Solomon Ucko Apr 05 '19 at 02:27

score 69 · Answer 2 · answered May 24 '16 at 08:47

69

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.

answered May 24 '16 at 08:47

Eric Towers

67,037

1

Not related to just circles though, but why don't we measure all angles like in a triangle, or just angle between 2 lines in radians? and not degrees? Why is the protector defined in degrees? – MonK May 24 '16 at 10:07
25

@MonK: historical reasons. The Sumerian or Babylonian astronomers who invented degrees knew some geometry of course, but they didn't have power series and differential equations, and so they didn't do so much calculation with angles as we do now. They were mostly interested in expressing position along an arc, so like Eric says they wanted an even division of the circle into units they could handle, preferably as integers. And it's only been 3000 years since then, which isn't long enough to persuade the general public to revisit the decision ;-) – Steve Jessop May 24 '16 at 10:28
4

"Degrees are useless." - except when communicating with nonspecialists, that is. One doesn't see much use of radians in, say, maps or carpentry. Snipers still use grads/gons, last time I checked. – J. M. ain't a mathematician May 24 '16 at 11:11
17

"Degrees are useless." - except in the real world. I am building a CNC machine. Lots of trigonometry. No angle sensor, rotary encoder or any other method of measuring angle directly reads radians. (the most common digital angle measurement is in multiples of 1/(2^n) rotations.) – hildred May 24 '16 at 11:59
33

-1 for saying "degrees are useless". As an engineer, I can tell you that NO ONE ever describes an angle in a design in radians. While the mathematical side is defined in radians, degrees are what people use for historical reasons. In addition, the convenience of writing $90^o$ vs $\pi/2$ is not to be ignored. – FundThmCalculus May 24 '16 at 13:17
11

From an engineering perspective, I think the key staying power for degrees is the $90^\circ$ association that @FundThmCalculus mentioned. It's very common to want to specify exact right angles, which don't have a nice decimal representation in radians. For practical construction considerations, there are lots of tools that nominally yield 90 degree relationships, so it would require special conventions if using decimal angle values to specify "nominally a right angle" even though other angles are otherwise specified to some fixed degree of precision. – Dan Bryant May 24 '16 at 15:12
1

@J.M. : "grads" are gradiens, the other engineering angle, defined as 100 per quadrant. So to those who claim "degrees are what engineers use": Yeah. Sorta'. Sometimes. – Eric Towers May 24 '16 at 16:06
7

@MonK : My protractor is in radians. – Eric Towers May 24 '16 at 16:09
@hildred : This "Handbook for Machine Shop and Drafting Room" disagrees. And it is no challenge to find similar references through the subsequent century, up to the present. – Eric Towers May 24 '16 at 16:12
4

@EricTowers: Sweet, can I get one in terms of $\tau$? – Daniel R. Collins May 24 '16 at 16:30
5

@DanBryant "right angles, which don't have a nice decimal representation in radians. " If you consider $\pi$ a scalar, a right degree is 1/2$\pi$ or .5$\pi$ which is quite nice. Of course, equilateral angles are an inconvenient .333$\pi$. But that's nothing new-- everyone knows we'd be better off if we had the foresight to base our number system on 12. As the vast majority of what most people use angles for is dividing circles into even parts or looking trig values in tables (which are unit neutral) degrees aren't going away. Although radians are for superior in calculus and mathematics. – fleablood May 24 '16 at 17:23
3

For one thing, in teaching angle measurements to children radians is never going to work. Teaching gradiens might be .... no, it won't... can you imagine talking about the special properties a 33 1/3, 66 2/3, 100 gradien triangle? But +1 for "degrees are useless". A good teacher shocks a student to action to realize when things are arbitrary and when they are not; it is a good reason to shake them up. – fleablood May 24 '16 at 17:29
23

Would it not be better to wait until your students graduate before you tell them that degrees are useless ;) – user_of_math May 24 '16 at 17:32
3

Units which conveniently describe angles using whole numbers are useful. I use radians all the time in my software, but I'm continuously forced to be aware of floating point roundoff that I am obliged to have when describing a 90 degree angle in radians. There are plenty of problems that are not computable due to floating point roundoff issues if you choose a unit that is inconvenient for describing the quantities you need to describe. Consider, as a real life example, GPS, which eschews radians and degrees for "semicircles," which permit them to use fixed point math over longitudes. – Cort Ammon May 24 '16 at 18:55
3

We should measure turns obviously. One turn = 360 degrees. Now none of your facts listed become ridiculous. A factor of $\pi$ shows up which is no more ridiculous than the existence of $\pi$ is. – djechlin May 24 '16 at 21:06
Right triangles, equilateral triangles, the very common 30/60/90 triangle, squares, etc. all have angles that are rational in one degree (or 360 degrees). This isn't true for radians. No one ever asked for one over pi-th radians of a pie. Perhaps degree wasn't perfect, but it's at least rational compared to radian. So I think this is a losing line of argument. – djechlin May 24 '16 at 21:15
1

@djechlin : Since the invention of rational numbers, at least 800 years ago, the argument that common angles should have whole numbers of units of measure has been bunk. The nice fact about radians is that your facetious example, $\frac{1}{\pi}$ radians, is very unlikely to be a correct angle measure because your example angles are all rational multiples (with tiny denominators) of $\pi$. Seeing a $\pi$ in the denominator is a clear signal that the student has made an impressive arithmetic error. Degrees can't do that. – Eric Towers May 24 '16 at 21:47
@djechlin: or to put it another way, nobody ever asked for $\frac{1}{\pi}$ radians for the same reason nobody ever asked for $\frac{180}{\pi^2}$ degrees. Because they don't want that exact amount. If that's how hungry they were, I suppose they might ask for 20 degrees or 0.3 radians, whichever their protractor was marked in (see above). – Steve Jessop May 24 '16 at 21:56
@EricTowers right, I meant to say "give me $\pi$ of a pie." It was unintuitive to express that way since I was thinking of asking for a part of a pie and fractions are intuitive to me. When we think of a pie, a wheel, a tire, a yard, a sphere, a ball, how many times we have to turn a screw, etc. we often think of one of those things, not $2\pi$ circumferences worth of the radius of that thing. – djechlin May 24 '16 at 22:00
2

@djechlin: but you realise that if what's important to you is a rational number, then a factor of 360 doesn't affect that? Just say half a pie if that's what you mean. – Steve Jessop May 24 '16 at 22:01
@SteveJessop yeah I generally would. Degrees are rational to wholes which is part of their appeal or usefulness or whatever you want to call it. As is the fact that the important angles of 90 deg, 45 deg, 60 deg and 180 deg are whole in degrees. Radians suffer the same problem. Semi-facetiously we promote $\tau$ sometimes, and while radians win in defining $\sin$ and $\cos$, $2\pi$ appears in mathematical expressions way more often than $\pi$ does, lending credibility to the fact that we got our central unit off by a rational factor. – djechlin May 24 '16 at 22:04
@SteveJessop and you have the fact that "turn 90 degrees right" immediately conveys "exactly a right angle, sharply," in a way that "turn right" or "turn a quarter turn right" wouldn't. So it's of course useful in communication. It's probably why this answer caught my attention -- "degrees are useless" just makes mathematics sound out of touch with things like how people talk to each other or think in whole circles instead of $2\pi$ worth of circles or that "one degree" really is about the smallest unit of angle I can visualize at human scale, not more enlightened. – djechlin May 24 '16 at 22:09
@djechlin : $\pi r^2$, $r \theta$, and $\frac{1}{2} r^2 \theta$ disagree that we're off by a factor of 2. Far worse than being off by a factor of two is being off by a transcendental number near 57, which is the case if we use degrees. I see your most recent comment in my students -- they think it is easier to compute one-sixth of a circle as "$\frac{1}{6}$ of $360^\circ$ is $60^\circ$", which is false. It is much faster to write $\frac{1}{6} 2\pi$ and be done with it. (Cancelling the "$2$"s is a good idea, but is not essential.) – Eric Towers May 24 '16 at 22:11
Telling students what is and is not easy to them is not a winning argument. Sorry. – djechlin May 24 '16 at 22:12
1

@djechlin : Who's telling? I'm observing their error rates in degree computations in the first three weeks with their errors in radian computations at the end. Radians win out every time (for the students who are not wasting their time). – Eric Towers May 24 '16 at 22:13
Could also just write $\frac16 360$. Or $\frac16$ if I submit "whole" as a better unit than degree that is rational to degree. $\frac16$ is easier to be done with than $\frac162\pi$. – djechlin May 24 '16 at 22:14
4

Really I just think this answer reads like a needless proof by contradiction. Instead of listing 7 advantages of radians (IMO this most important is probably the limit of $\sin x / x$) you list 7 disadvantages of degrees. Now you have people like me, and I can only imagine a good lot of students, wondering why degrees have no advantages when they clearly have many extra-mathematical advantages and still some mathematical advantages. I'm not sure who wins from a hardline stance starting with a polemic like "degrees are useless." Which is clearly false, just saying. – djechlin May 24 '16 at 22:16
1

@EricTowers - If in the handbook you linked to in replying to hildred, you back up to the start of the section on radians (prev page), it begins with the sentence "While in practical work angles are always measured in degrees and minutes ...[radians are] often employed in theoretical work and in formulas relating to revolving bodies". I.e., that handbook supports hildred's position. There is no disagreement. – Paul Sinclair May 24 '16 at 23:53
@EricTowers, yes radians make the math easier, show me a sensor that reads them. – hildred May 25 '16 at 01:42
@hildred : I'm not aware of an encoder that reads radians or degrees. Internally they read fractional turns and report the result in binary. This is another way that degrees are not practical since they are not a binary system relative to full turns. (Why would that be relevant? Imperial wet volume measurements (gallon, quart, pint, cup, et c.) are (almost) a binary system, so clearly such systems predated binary position encoders.) First Google hit example. – Eric Towers May 25 '16 at 01:57
Here is the task. Write right angle value using numbers only. Is it even possible to solve with raidans? Is there such common and hard task for degrees? What is realy useless: something that makes common tasks impossible to solve or something that makes some calculations a bit harder? – Atomosk May 25 '16 at 01:59
2

@EricTowers you are lousy at picking examples to argue your point. Is far as I knew they quit making degree calibrated rotary encoders years ago, and you found one. Yes they also make the more common binary fractional rotation ones which I use on a regular basis, and for which for any single bit change I can tell you exactly what fraction of a degree it is, but I cannot give you an exact rational radian equivalent as they only make them available in integer number of pulses per second. – hildred May 25 '16 at 02:11
@Atomosk : First: "Why is radian so common in maths?". This is the question being answered. Second: $\pi/2$. It's a very short number. It involves exactly as many numerical symbols as "$90^\circ$" and doesn't require bizarre symbols like "${}^\circ$". Finally, if you want to talk about exactness in engineering and/or machining (i.e., "common tasks"), I refer you to very common metric to English tap and die conversion posters (example) where apparently 4 decimals is enough for modern machinists. – Eric Towers May 25 '16 at 02:12
@hildred : Nope. That's a turn calibrated encoder. It returns binary fractional turns. Now my question to you: In what way is your string of comments relevant for improving my answer to address the OP's 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?" If you're not going to be on point, I don't see why I have to be. – Eric Towers May 25 '16 at 02:14
@EricTowers, 1: Look closer, they offer both. 2: For any model I can give you an exact conversion ratio for any reading from any device they sell to degrees using only rational numbers. 3: find any encoder that gives an exact rational conversion factor to radians and i will give your answer a bounty. – hildred May 25 '16 at 02:24
4

+1 for saying that degrees are useless. I'm not an engineer, but I have seen many designs that use radians rather than degrees. Besides, this is about math; common usage in non-mathematical applications doesn't make it less silly. An acre, for example, derives from the extraordinarily useful measure of the amount of land that can be plowed in one day with a pair of oxen. That's a bit vague, but in modern times it's fixed at the area of one chain by one furlong, or simply 43560 square feet. Convenient! – anomaly May 25 '16 at 04:49
2

@EricTowers $\pi/2$ isn't the value, it's expression which evaluates into right angle value. That divide by 2 means instead of result you telling me how to get result. Like when you order a cup of coffee, waiter tells you that you need to grow a coffee tree, collect beans, make cup etc. – Atomosk May 25 '16 at 08:26
3

Degrees are NOT useless. I have just started flying after a 20 year layoff. I got out my dad's E6-B computer, and solved some wind triangle problems. But I don't really need it. I easily estimate my cross wind component by multiplying by the sine of the wind angle. With degrees, I use sin(x) = x/60. To find the crab angle, use arcsine(y) = 60*y. This approximation is within a degree for angles less than 30 degrees-- good enough for navigating an airplane by dead reckoning. And if the angle is greater than 30 degrees... I don't want to be flying in an 80 mph crosswind! – richard1941 May 25 '16 at 20:51
1

Re, "Dividing circles into even numbers of parts." If we couldn't do that, then we never would have built any machines more complicated than the lever, the screw, the wedge, ... If we couldn't divide circles into equal parts, we would not be able to manufacture gears or sprockets. – Solomon Slow May 25 '16 at 22:20
1

@richard1941 , with radians the estimations are even easier! For small angles, you get sin(x) = x and arcsin(y) = y. You should try it, really! – TonioElGringo May 26 '16 at 08:08
I think that at my uni, a lecturer telling their students that "degrees are useless" would be a little questionable :P – Sebi May 28 '16 at 13:17
@fleablood What if we express the angles as $\frac\tau{12}$, $\frac\tau6$, $\frac\tau4$ or $\frac1{12}\tau$, $\frac2{12}\tau$, $\frac3{12}\tau$? – Solomon Ucko Apr 03 '19 at 12:07
That's .... fractions. And we all know how much elementary school children LOOOVE fractions. – fleablood Apr 05 '19 at 14:47
@SolomonUcko That's just replacing on basic unit of a "degree" with a different unite a "tau-12th". If you are going to consider that you can get finer accuracy with a "tau-360th". – fleablood Apr 05 '19 at 14:50
@fleablood The thing is, radians are flexible. If we need to divide it into a different number of pieces, we can easily do that by changing the denominator. And with $\tau$, the ratio of the numerator to the denominator is actually directly meaningful. – Solomon Ucko Apr 05 '19 at 15:01
Yawn... who cares? This thread is two years old and we all know how use both. – fleablood Apr 06 '19 at 05:24

robert bristow-johnson · Answer 3 · 2016-05-25T04:37:48.607

7

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.

edited May 25 '16 at 04:37

answered May 25 '16 at 03:58

robert bristow-johnson

721

8

It's such a great honor meeting someone who has visited planet Zog and worked with their scientists! Pray tell, good sir, kind sir, did they reveal to you their secrets of interstellar travel? – Asaf Karagila May 25 '16 at 04:40
6

i never said i visited Zog. but i am still quite sure that any aliens that do calculus understand the concept of the dimensionless measure of angle in radians. – robert bristow-johnson May 25 '16 at 04:46
7

It's very easy being certain that what you do is the universal way. If history taught us any lessons, is that whenever you're sure that the way you do it "is the reasonable way, and anyone should understand it", you're almost guaranteed to be wrong about it (not about how you do it, rather about how others are doing it). I only assumed, in that case, that confidence came from visiting unearthly planets. – Asaf Karagila May 25 '16 at 04:49
2

well, i would say that the concept of minimum is universal. and that mathematical truth (that means the axioms and theorems) are universal. the aliens that do calculus will discover, as did humans, that the minimally complicated expression of angles in analytical geometry is in radians. the arc length divided by the radial arm. no scaling numbers that any aliens or humans have pulled outa their ass. that is minimal complexity. sure we (or they) could do calculus with a base other than $e$ and angles of some other measure, but there would be scaling constants that would be added to the math. – robert bristow-johnson May 25 '16 at 04:58
in terms of natural physical units (that we would need if we were to communicate with an alien culture somehow), you might look up Planck units. that's virtuallythe only way we can have a common reference if either were to ask the other "how tall are you beings?" but i think that it would be more natural if Plank units normalized $4 \pi G$ instead of just $G$ and if they normalized $\epsilon_0$ instead of $\frac{1}{4 \pi \epsilon_0}$. but $c$ and $\hbar$ are normalized to 1. the aliens may very well be doing that. – robert bristow-johnson May 25 '16 at 05:05
hey @AsafKaragila, do you think that aliens that can do calculus and analytical geometry know about $\pi$ or $\tau = 2 \pi$? and about $e \approx$ 2.718281828... ? and do you think they might understand the minimal complexity of a numbering system based on base 2? that their early digital computers would have two logic levels? – robert bristow-johnson May 25 '16 at 05:17
You do know that just the fact our current understanding of physics suggests such units does not mean that (1) other aliens have the same understanding; or even that (2) we're even remotely correct about it. Right? You're falling back into my previous comment. You're sure that we've got it right, and therefore everyone else has to get it the same way. Not to mention that mathematical truths also depend on your inference rules and so on, and even different schools of mathematicians will disagree on "fundamental truths", if they disagree on the underlying logic. – Asaf Karagila May 25 '16 at 05:24
1

i don't know if aliens will canonize the value $\pi \approx$ 3.14159... or instead $\tau = 2 \pi \approx$, 6.28318... but i am pretty sure it will be one of the two. and, if they do calculus, they will understand $e \approx$ 2.71828... as the natural base of logarithms and exponentials. those numbers, in whatever numbering system they use, will be "special" to the mathematicians and scientists on the planet Zog. – robert bristow-johnson May 25 '16 at 05:32
2

There's no reason to think there are any pure mathematicians at all on Zog: maybe they are all engineers. And maybe for historical reasons they divide their circles into 273 parts. It might be somewhat inconvenient to do calculus, as factors of $2 \pi/273$ keep popping up, but they manage to deal with it. – Robert Israel May 25 '16 at 19:23
i truly doubt that Zoglings that become advanced enough to send and/or receive electromagnetic communication from distant exo-planets, ain't gonna understand calculus as well as Earthlings do. that Zoglings divide $2 \pi$ radians (which is a turn) into 273 parts is as irrelevant as humans dividing the turn into 360 parts. but the Zoglings will understand the natural measure of angles in radians, because they're pretty good at concepts of calculus. and they're pretty good at calculus because they're pretty good at technology involving electromagnetic waves coming from far away. – robert bristow-johnson May 26 '16 at 06:10
1

Surely the planet Zog uses the cycle as the natural unit, with that cy-bar symbol (see h-bar) being used for cyc/2.pi. The choice of unit is a convenience for the user, which is why other units are in play in other places (see also mils, grads, and right angles) – Philip Oakley May 26 '16 at 15:08
I was originally going to comment in robert's defense, but the arguments above have swayed me- of course we can't rely on aliens understanding what the ratio of a circle's circumference to its diameter is, for all we know circles are actually triangles on other planets. And maybe ratios are really exponents- to think otherwise would be the height of hubris, assuming we know math better than they do. Heck, we can't even be sure that addition works! Maybe on Zog, when you eat an apple you get two apples! Physics hasn't given us an answer yet, so we just can't know. – P... May 26 '16 at 16:42
sure the cycle (or "turn") is a natural unit of angle. and when the Zoglings use it in their calculus, they will find factors of $2 \pi$ popping up all sorts of places and they will eventually get it. i am convinced that any culture sufficiently technologically astute to be able to send or receive communications from beyond their solar system using electromagnetic radiation, that they will be very very good at calculus. – robert bristow-johnson May 26 '16 at 17:04

score 6 · Answer 4 · answered May 24 '16 at 08:48

6

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.

answered May 24 '16 at 08:48

Matias Heikkilä

2,542

6

The reason might be that 360 as a lot of divisors (24 exactly) so it can be cut easily. (Same reason for the 24 hours day and 60 minutes hour) – luxcem May 24 '16 at 12:57
3

For your information, the Babylonians liked the sexagesimal (base-60) system: $6*60=360$ degrees in a circle, 60 minutes/hour, 60 seconds/minute, etc. – FundThmCalculus May 24 '16 at 13:18
3

@FundThmCalculus - Which is of course itself a bi-product of us using a decimal numbering system in the first place. There's nothing special about base-10 other than that's how many fingers our species is typically born with. Things would be much easier if we were born with one less finger on each hand, then we'd all be using base-8, which is a natural extension of binary, and that at least is fundamentally significant. Base-10 is just arbitrary, really. – Darrel Hoffman May 24 '16 at 18:42
There's not really an answer to the question in here. – djechlin May 24 '16 at 21:07
In my opinion, this is the most important point: It is a dimensionless unit (it is not a real unit like degrees, it is merely just a scalar). You do not have to mess with extra multiplications when using radians. – rexkogitans May 25 '16 at 08:14
More on the history of 360: http://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/ – mattdm May 25 '16 at 20:59
@rexkogitans, the failure to include at least an allowance of Angle as a dimension in SI is a major failure. Dimensional analysis is an important tool for spotting mistakes, and for any analysis that covers angles, the lack of an angle dimension does cause lots of errors (see 4.pi arguments in electromagnetics). Ultimately the problem is that Length is not a dimension but the norm of a 3-space, and Angle is an indicator of a change of direction in that space. Otherwise it's all Torque and no Work. ;-) – Philip Oakley May 26 '16 at 15:03

score 5 · Answer 5 · answered May 26 '16 at 00:20

5

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.

answered May 26 '16 at 00:20

Mark Joshi

5,604

score 4 · Answer 6 · answered May 24 '16 at 21:45

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.

the rotation is indeed a great unit to measure angle in particularly for cnc applications. look for absolute rotary encoder which does exactly that (although math with grey code is a barrel of something unprintable) — hildred, May 25 '16 at 01:56
Revolutions are implicitly used with frequency (typically in Hz, rpm) while radians are used with angular frequency. — , May 25 '16 at 13:43

score 4 · Answer 7 · answered May 26 '16 at 18:31

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.

Sort of but not really, $\tau$ is just $2\pi$, so you're still using radians, you're not using a different unit, if you were you wouldn't have to multiply by a constant every time. — Michel Rouzic, Feb 24 '24 at 11:59

score 2 · Answer 8 · answered May 24 '16 at 16:23

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.

score 2 · Answer 9 · 2016-05-27T00:35:26.573

I don't know if this answer is good or not.

Reason 1

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.

Reason 2

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.

The skinny triangle rule which is a cool application in astronomy relies on the fact that $\sin\theta\approx\theta$ when $\theta$ is a small angle. The formula would thus b $180/\pi\cdot l/\theta$ if the angle were measured in degrees. This goes along with the other answers showing that using radians results in trig functions with nicer properties (e.g. no need to the $\pi/180$ factor). — jdods, May 27 '16 at 00:43
@jdods Absolutely. I just showed the practical use of radians in favour of degrees. — , May 27 '16 at 00:48

score 2 · Answer 10 · answered May 26 '16 at 16:04

2

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

answered May 26 '16 at 16:04

G Scruton

21

score 1 · Answer 11 · edited Jun 23 '16 at 11:07

1

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?

edited Jun 23 '16 at 11:07

TRiG

203

answered May 24 '16 at 18:28

John Cobb

11

No, I don't. I answer $pi/2$ when asked the measure of a right angle. – BudgieJane May 24 '16 at 20:26
5

@BudgieJane: I answer $\tau/4$ ;) – TRiG May 24 '16 at 20:29
Really proves the point that it's much easier to remember a right angle in degrees. – djechlin May 24 '16 at 21:08
So why not "diametrians"? – Random832 May 25 '16 at 03:07
1

"diametrians"? a new one to me. Thanks. Maybe that will be "natural" for future generations in that 2 \pi can get replace by \tau with no leading constant. Alas, my brain is already too ossified to internalize that innovation completely. As the old quote goes, academic revolutions occur apace with caskets. That said, both are objectively more "natural" than degrees or grads. – John Cobb May 25 '16 at 21:07

score 1 · Answer 12 · answered May 25 '16 at 18:59

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.

score 1 · Answer 13 · answered Jun 29 '19 at 10:14

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 $.

score 1 · Answer 14 · answered May 04 '21 at 23:57

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.

score 0 · Answer 15 · answered May 24 '16 at 21:05

0

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.

answered May 24 '16 at 21:05

Engineer

119

You said "in circles it is radians" but gave the final answer of "0.8 miles." So which is it? – djechlin May 24 '16 at 21:10
It's r * pi / 4 radians. The radius of the radians is 1 mile and if you traverse it, you will travel .8 miles in an arc. – Engineer May 24 '16 at 23:25
Let me put it another way. The radian measurement is pi / 4 times the radius. Google says this is 0.78539816339; call it 0.8. You multiply that by the radius of the circle. My circle is 1 mile around me, and the two places I mentioned are on that circle. From my viewpoint, they are 45º apart. My friend needs to traverse the arc for 45º, which will mean he has to walk 0.8 miles to get to the arena. If I change the example and say those places were 2 miles away, then 45º for me would mean he has to walk 1.56 miles (2 * pi / 4). – Engineer May 27 '16 at 02:24
In private pilot ground school we use degrees. With degrees, it is easy to find the approximate sine of an angle, by sin(x)=x/60 ( the "clock rule"). However, that rule breaks down beyond 33 degrees. For 33 to 55 degrees, subtract a correction factor, (x-33)/250. All of this is easy to calculate in your head without resort to an E6-B computer (circular slide rule). I guess I am on a little project to refine these approximations. We could memorize a few sines and do linear interpolation. Two digits of accuracy is all we need for airplane navigation. – richard1941 Jun 19 '16 at 02:05
Well a 30 degree angle is pi/6. Pi is about 3.1, so it's 31/60 or about .52. – Engineer Jun 20 '16 at 06:25

jdods · Answer 16 · 2016-05-25T18:46:39.597

(Note: See edits below original post.)

Here is another reason why:

Radians are unitless (a.k.a. dimensionless).

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.

I think this is entirely missing the point. As you mention in the last paragraph, radians are only "unitless" because we've decided to make them the standard unit of angle. — Eric Wofsey, May 25 '16 at 04:19
That's not correct. They are unitless because they are a ratio of distances. — jdods, May 25 '16 at 05:51
Careful: "Dimensionless" is not the same as "unitless". Radians are dimensionless units, as are degrees. -- https://en.wikipedia.org/wiki/Dimensionless_quantity#Properties — Daniel R. Collins, May 25 '16 at 06:16
Maybe I was incorrect with the language. I refer, perhaps incorrectly, to meters and inches add two units of measurement. The point is that degrees need to be declared as the unit, radians do not. As far as OP's question is concerned, this seems to be an appropriate point. The exposition on calculus etc is included in other answers. — jdods, May 25 '16 at 07:04
@DanielR.Collins, I did a little looking around and it seems normal to use "unitless" they way I have. Do you have any "authoritative" references that discuss this? Otherwise, this seems like a semantic quibble. However, the comments are welcome and add good content. Cheers. — jdods, May 25 '16 at 07:26
I don't have a textbook source for that. However, consider the comments when this has come up several times on Physics.SE with the tag of "dimensional-analysis", e.g., here: http://physics.stackexchange.com/questions/252288/are-units-of-angle-really-dimensionless — Daniel R. Collins, May 25 '16 at 16:10
And here: http://physics.stackexchange.com/questions/251762/do-all-equations-have-identical-units-on-the-left-and-right-hand-sides — Daniel R. Collins, May 25 '16 at 16:10
Radians are NOT dimensionless. By definition they must span (at least) a 2d space, so they do indicate a change of direction and a change of dimension (Angle). The problem is the lack of distinction between the 2d dimensions, so we have what is known as the Dimension of Length actually being the norm of the space. If one axis is Time, and the other axis is Mass, is the local slope of a curve still in Radians? — Philip Oakley, May 26 '16 at 15:14
@PhilipOakley, read the links in the comments above about the concept of "dimensionless". The area an angle sweeps out depends on the radial extent. As that radial extent goes to zero, so does the size of the area, hence the area has the property of "dimension" only because of the radial extent. The angle is irrelevant. — jdods, May 27 '16 at 02:10
@jdods, I have read the various comments. I do appreciate that it has become a convention that angles are not considered a dimension, and that in many ways that does contradict the original intent of dimensional analysis. It's further excacerbated because of the maths vs physics distinctions where the maths abstractions means that no dimensions are actually named. Metrologia 47 (2010) R41–R51, Marcus P Foster, is a worthwhile read on the issues. — Philip Oakley, Jul 06 '16 at 22:47

score 0 · Answer 17 · answered Sep 05 '19 at 04:00

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

$\cos(0) = 1$
$\sin(0) = 0$
$\forall x \in \mathbb{R}\cos'(x) = -\sin(x)$
$\forall x \in \mathbb{R}\sin'(x) = \cos(x)$

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.

score 0 · Answer 18 · answered Feb 24 '24 at 05:50

The radian has certain special properties that facilitate analyzing trigonometric functions.

$$\frac{d}{dx} \sin(x rad)=\cos (x rad)$$

Now let us look at quadrants. (1 quadrant is a right angle)

$$\frac{d}{dx} \sin(x quad)=\frac{\pi\cos(x quad)}{2}$$

A generalized version is

$$\frac{d}{dx} \sin(x)=\frac{\pi\cos(x)}{H}$$

Where $H$ is the measurement of a half-circle in the same units as $x$

For radians, $H=\pi$, which simplifies the derivative. Repeated derivation actually cycles back to $\sin(x rad)$, but not for other units.

Why is radian so common in maths?

18 Answers18

Reason 1

Reason 2

Linked

Related