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This question already has an answer here:

What is the advantage and use of measuring an angle is radian(s) compared to degree(s)? My book suddenly switched to radian(s) for measuring an angle in this grade and I do not know why.

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marked as duplicate by Carsten S, kingW3, hardmath, Adam Hughes, E. Joseph May 12 '17 at 16:13

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    $\begingroup$ One thing to note after reading @The Dead Legend 's comment: Degrees were only invented because the Babylonians approximated the number of days in a year as 360 for convenience in their base-60 number system. Because of this, degrees are used for cultural, not mathematical significance. $\endgroup$ – Toby Mak May 11 '17 at 9:32
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    $\begingroup$ If I may suggest : as early as possible, forget degrees ans switch to radians. It will make your life much easier in your studies. $\endgroup$ – Claude Leibovici May 11 '17 at 9:34
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    $\begingroup$ In one word: calculus $\endgroup$ – Marc van Leeuwen May 11 '17 at 14:33
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    $\begingroup$ Three related questions. $\endgroup$ – J. M. is a poor mathematician May 11 '17 at 15:25
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    $\begingroup$ @Claude, if the OP is doing carpentry, astronomy, navigation, or geodesy, maybe not so much. $\endgroup$ – J. M. is a poor mathematician May 11 '17 at 15:27

10 Answers 10

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Look at the following picture. It is a circle of radius $r=1$ and there is an angle $\alpha$ that cuts out an arc $c$ from the circumference of the circle.

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

How long is this arc? Well, this is where radians are immensely beneficial. If you express the angle in radians, then the length of the arc is exactly $\alpha$. If $\alpha=0.123$, then the arcs length is $0.123$ too. Easy.

You know that the full angle in radians is $2\pi$. If you choose $\alpha=2\pi$ in the above image, you would describe an arc $c$ that is actually just the full circle. An you immediately see, that its circumference (here also the length of the arc) is $2\pi$ too. This is exactly what the formula for the circumference $U(r)=2\pi r$ would give you anyway.

Of course this is also useful for circles of radii other than $1$. If your circle has a radius $r$ and you look at an arc cut out by an angle $\alpha$ in radians, then the length of the arc is $\alpha r$. It just scales linearly.


This is the visual aspect. But there are more (mathematical) reasons. Ask yourself, what would be a natural unit to measure an angle in? Degree is not very natural. There is nothing special about the number $360^°$ but its high divisibility. You could measure an angle in the interval $[0,1]$. Or you can use its natural connection to the circle that I explained above.

However, there are many mathematical functions that take angles as inputs, e.g., $\sin(x),\cos(x), etc$. But have you ever asked how to actually compute $\cos(x)$ without a calculator? There are formulas that give very good approximations, e.g.,

$$\cos(x)\approx 1-\frac12 x^2+\frac1{24}x^4.$$

But they only work for radians. You can write them down for other units, but they will never look so natural, not even for angles in the units $[0,1]$.

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    $\begingroup$ What do you mean by "You could measure an angle in the interval [0,1]"? $\endgroup$ – MrAP May 11 '17 at 16:52
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    $\begingroup$ @MrAP 0 means no angle. 1 means all the way around. 0.5 means half way around. 0.25 means 1/4 of the way around. Etc. $\endgroup$ – Yakk May 11 '17 at 17:25
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With radians, the arc length and sector area are especially $r\theta$ and $\frac{1}{2}r^2\theta$. We also have trigonometric formulae $$\sin\theta=\sum_{n\ge 0}\frac{(-1)^n\theta^{2n+1}}{(2n+1)!},\cos\theta=\sum_{n\ge 0}\frac{(-\theta^2)^n}{(2n)!}.$$To work with degrees instead, you'd have to replace $\theta$ in each result with $\frac{\pi\theta} {180}$. Which would you rather use?

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Degrees is the usual measure unit for angles, radians is the mathematical one. Degrees come from the historical base 60 operations. This base was chosen because it is divisible by the first six positive integers ($1$ to $6$). As it makes it easy to express common angles in degrees, its usage has persisted through centuries.

The radian on the other hand is a mathematical unit. An angle in radians is $ \frac {\operatorname{arc}} {\operatorname{chord}}$. You also have nice trigonometric functions with angles in radians: $\sin' = \cos$ and $\cos' = -\sin$.

That's the reason why we use degrees in daily life and radians in mathematics.

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    $\begingroup$ Radians - provided you never take out the $\pi$ term - express more fractions naturally than degrees. Normal usage would be considerable more flexible if we worked in radians. The use of degrees is simply historical. $\endgroup$ – Jack Aidley May 11 '17 at 14:58
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    $\begingroup$ @JackAidley nah, gradians all the way $\endgroup$ – theonlygusti May 11 '17 at 15:02
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    $\begingroup$ Why the downvotes? I think the point he's trying to make is clear, degrees have been more useful in common usage, radians are more useful in higher math. $\endgroup$ – Anonymous Pi May 11 '17 at 15:25
  • $\begingroup$ @JackAidley: radians without the $2\pi$ term are another unit: the turn. Ok it make sense to speak of that when cutting an apple pie (1 quarter), but I'm not fond of a $\frac \pi 6$ part... $\endgroup$ – Serge Ballesta May 11 '17 at 16:03
  • $\begingroup$ I'm not taking about the term, I'm saying so long as you say $\pi/2$ rather than 1.whatever, they're highly intelligible. Of course, it'd be better if we used tau but that's a different discussion. The point is that radians are no less intelligible in common use. $\endgroup$ – Jack Aidley May 11 '17 at 16:56
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I want to offer another perspective, which doesn't contradict the other answers but looks at them from a different angle...

The short answer

Defining angles in radians has the advantage that it is more consistent with how we tend to define other quantities.

The long answer

Lets put angles aside for a moment, and think about how we defined types of quantities in general. Lets take an example from the natural sciences. We want to define the "density" of a substance, that would represent our intuitive perception of the concept. Density means how much stuff is present in a some volume, so we define density as the ratio between number of particles of the substance and its volume:

$\mathrm{density=\frac{number\: of\: particles}{volume}}$ .

We could have also defined it as

$\mathrm{density=62.112\times\frac{number\: of\: particles}{volume}}$ ,

and it would still be useful for us, but since we don't have any reason to add the number $62.112$ we just don't. It seems simpler to us to just use $1$. The same applies to most of our definitions of quantity types. (We could argue that the number $62.112$ is just as arbitrary as the number $1$, but although it might be true in some philosophical sense, the fact is that us humans think of $1$ as being simpler. We like $1$ better).

Now, this applies also to mathematical definitions. For example, the function $\sin\alpha$ was defined originally as as the ratio between two sides of a right triangle with angle $\alpha$: the side opposite to that angle and the hypotenuse.

$\sin\alpha=\mathrm{\frac{opposite\: length}{hypotenuse\: length}}$

Why was $\sin$ defined in the first place? And why was this specific definition chosen? I don't know for sure, but I can guess that the answer to the former question is that someone (a few millennia ago) was interested in quantifying shapes of different right triangles, and $\sin$ as defined above is indeed a good such quantity. What about the latter question? Why didn't they choose the definition to be

$\sin\alpha=\mathrm{62.112\times \frac{opposite\: length}{hypotenuse\: length}}$ ?

Again, because it's less simple (and also they probably did not work with such numbers at all back then).

Now, let's try to define a quantity that would characterize the magnitude of an angle. We have an intuitive understanding of an angle, and we should try to quantify it. So what is an angle? I think about it as an opening between two lines:

enter image description here

How can we quantify this? Two options immediately come to mind. We can take the area that is "bounded" between the two lines, or we can take the length of the line that joins the line ends. Lets look at the latter option:

enter image description here

Is that a good definition of the magnitude of an angle? No, because in our intuitive concept, the magnitude of the angle should not depend on the length of the sides, and here we get that the longer either of lines, the bigger the angle. This could be solved by using ratios of lengths instead of absolute length. For example, the ratio of the dashed line to the ratio of the bottom line. But the result still depends on the ratio between the lengths of the top and bottom lines. So, let's decide that when measuring an angle, we also make sure that the lines have a certain fixed predefined length ratio. This ratio would be the "standard" ratio.

Which ratio should we choose? it can really by any number, for example $62.112$... You get the point already - it is simpler to use $1$.

enter image description here

There is still a problem with this definition that makes it less useful - it is not additive. That it, the magnitude of an angle which can be divided into two non-overlapping angles does not equal the sum of the magnitudes of the individual angles:

enter image description here

How can we improve our definition to circumvent this problem? we can use instead the length of the arc of the circle whose center is the intersection of the two lines.

enter image description here

We're almost done defining the angle. We said that we'll use the ratio between the arc length and the line length. But which ratio should we use? $\mathrm{angle=62.112\times\frac{arc\: length}{line\: length}}$? No. we don't like that. How about $\mathrm{angle=57.29577951308\times\frac{arc\: length}{line\: length}}$? Also ugly. But in fact that is (approximately) how angles were defined! This definition would make a right angle be $90^\circ$. (I'll call this the "degrees" definition). True, this definition has some advantages. It makes the maximum attainable angle be divisible by many integers. But, on the other hand, we usually define quantities with a $1$ when we don't need any number. That is how we defined $\sin\alpha$. And since that is how we usually define quantities, defining a new quantity otherwise would induce a whole lot of numbers which are not $1$ into our calculations. As an example, let's look at $\sin\alpha$ and at $\alpha$. Their definition are quite similar - ratios of two lengths. In fact, for very small angles, the right triangle and the straight lines + arc are almost indistinguishable:

enter image description here

If we define the values of $\sin\alpha$ and $\alpha$ in compatible ways, we can make this observation into a nice mathematical statement that $\sin\alpha\approx\alpha$. So let's make them compatible! One way is defining

$\sin\alpha=\mathrm{57.29577951308\times \frac{opposite\: length}{hypotenuse\: length}}$

and using the "degrees" definition for angles. But then again, we are not compatible with many other definitions we make, such as the other trigonometric functions, or areas. The area of a triangle would have the formula $\frac{1}{2\times57.29577951308}ab\sin\alpha$ instead of the nicer formula $\frac{1}{2}ab\sin\alpha$. The better solution is to just use $1$ in the definition of an angle. This is compatible with everything else we have defined, and it makes a lot of things simpler.

As an aside, I'd like to add that I don't like to think about radians and degrees as units, but rather as different definitions of the magnitude of an angle. They are not different methods of measurement, they are different definitions. Likewise, defining density as the number of particles divided by twice the volume they occupy would be a different definition of density, not a different unit system. But I won't go into the full discussion here...

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    $\begingroup$ I really like this answer because it makes clear that many things in mathematics are defined out of convenience and out of a need to be consistent with previously established definitions and concepts, something which confused me for a long time (I can't remember how many times I asked "why" something was defined a certain way, assuming that it was the expression of some absolute reality) and I think in general takes a while for people starting math later in life to understand. I see the realization of this idea as a major milestone in the development of one's mathematical maturity. $\endgroup$ – jeremy radcliff Jan 8 '18 at 1:46
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Also consider: $$ \lim_{x\to0} \frac{\sin x}{x} = L$$

It can be shown that $L=\bf1$ if you use radians; otherwise $L = {\pi\over180}$. And in calculus this is one of the fundamental limits, along with $(1+y)^{1/y} \underset{y\to0}\longrightarrow e$.

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Because it gives you angle in the sense which you can actually represent on the number line without any conversion?

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It has to do with Euler's formula.

If the argument of a complex number $\phi$ were in degrees, that would have to be stuffed with conversion constants.

You might suspect that $e$ is an arbitrary constant in here; but it isn't. It has special properties unrelated to circles like angles, like $e^x$ being its own derivative.

(Well, when we say unrelated, we have to cross our fingers, because it's all related, of course).

$e$ is a very special exponent base. When we multiply the imaginary number $i$ by some real numbers and raise $e$ to this product, we get various points on the unit circle on the complex plane.

It so happens that how this exponentiation using $e$ works is that the range $[0, 2\pi]$ corresponds to the full circle. But the length of that interval is also just the distance around that circle.

From there we just call the distance 1 "one radian" and say there are two-pi of them around a unit circle. From there, we call that an angle measure and say there are two-pi of them in any circle.

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A visual argument for radians:

Imagine rolling the unit circle along the number line. Radians are the only unit where the roll angle is equal to the distance rolled.

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Others have explained why radians are useful. Let me offer a comparison instead: Radians are to degrees as "plain numbers" (or decimal numbers if you will) are to percentages.

It seems that most people like or are used to speaking in percentages. Numbers range from zero to hundred and integers are sufficient for many practical purposes. It is convenient for the less mathematically inclined, but if you want to calculate, the "plain number" $0.21$ is probably going to be a handier form than $21\%$.

Similarly, degrees often range from zero to 90 (sometimes a little higher) and again integers are enough to describe most angles in everyday life. It has the same kind of comfort than using percentages. However, if you want to calculate something — see other answers for examples — radians are the more convenient choice.

Percentages and degrees are useful for everyday communication, but far less useful for doing mathematics.

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There is no "advantage" of radians over angles, for example. Think of it as an unit of measure. For example, kg vs pounds. The only difference is on the choice one makes, and not in what actually represents.

EDIT:

Do not be misled by the down votes. This is an answer from the perspective of a theoretical physicist.

I strongly encourage you to think in, again, units in physics. There is no advantage in using one units or the other, except perhaps your final numerical results and the application of what you want to give them. The theory does not change at all if you use degrees or radians, the angle is still the same angle viewed with different glasses! I have never used degrees since I started the career (and not from that I am going to say that one is advantageous one over the other, because I can google up examples where there is a wide application of degrees over radians), because I don't need them. Perhaps and engineer would need them more and that would make them advantageous for him.

I want to stress that the theory is not change by either using degrees or radians, and from this point of view the advantage type of feeling you have is not really relevant.

If you continue your studies in science, you will find specially in Physics, that most of the equations are given in free-unit form. Maxwell's equations describe the same phenomena no matter which units you choose (the only difference are a couple of prefactors in the equations, which will give you your chosen units). Newton's equations are still the same, whether you choose to measure forces or masses in cgi or international unit system (or any other system unit).

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  • $\begingroup$ In my opinion, it is more convenient to measure mass in kg than in pounds if you use decimal system. $\endgroup$ – Antoine May 12 '17 at 8:55
  • $\begingroup$ From mathematics' POV, there is surely no difference. However, from human's POV there are differences which many happen to be advantages as well. $\endgroup$ – edmz May 12 '17 at 13:00
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    $\begingroup$ This "no advantage" needs a context such that the reader can understand the value system that gives no advantage. For the casual 'person in the street' it is probably right that using radians has no (or even negative) advantage relative to using the common or garden angular measure of degrees (cf grads, mils, and other units). However to the mathematician (and that's the course the OP is doing), it is most convenient to abstract away any (most) real world units, and measuring arc length (in radius's) is the way to go. This then fits the easy sin(x) ~= x formula. $\endgroup$ – Philip Oakley May 12 '17 at 13:56
  • $\begingroup$ I think it is not true. For examples, using radian is more convenient when doing calculus. $\endgroup$ – Alex Vong May 12 '17 at 15:33
  • $\begingroup$ This is not an advantage of what I like, but what it really is. You can do the entire calculus using degreees and only the units will change. And from this, it depends on the application, not what you would like more. You download votes are misleading. $\endgroup$ – user2820579 May 12 '17 at 16:31

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