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I'm convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for angles. What I want to know is why this is so (or why not).

I understand that using radians is useful in calculus involving trigonometric functions because there are no messy factors like $\pi/180$. I also understand that this is because $\sin(x) / x \rightarrow 1$ as $x \rightarrow 0$ when $x$ is in radians. But why does this mean radians are fundamentally more natural? What is mathematically wrong with these messy factors?

So maybe it's nice and clean to pick a unit which makes $\frac{d}{dx} \sin x = \cos x$. But why not choose to swap it around, by putting the 'nice and clean' bit at the unit of angle measurement itself? Why not define 1 Angle as a full turn, then measure angles as a fraction of this full turn (in a similar way to measuring velocities as a fraction of the speed of light $c = 1$). Sure, you would have messy factors of $2 \pi$ in calculus but what's wrong with this mathematically?

I think part of what I'm looking for is an explanation why the radius is the most important part of a circle. Could you not define another angle unit in a similar way to the radian, but with using the diameter instead of the radius?

Also, if radians are the fundamentally natural unit, does this mean that not only $\pi \,\textrm{rad} = 180 ^\circ$, but also $\pi = 180 ^\circ$, that is $1\,\textrm{rad}=1$?

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Some discussion in the comments on physics.stackexchange.com/q/11500 –  dmckee Aug 6 '12 at 14:52
    
both the Radian and the Cycle are natural units for such cyclical aspects. The difficulty is in choosing the Base unit, and in showing it (rather than hiding it) to every one. At a fundamental level the mathematicians, the physicists and the average person have different perspectives on the world which don't agree [e.g. xkcd.com/435/] –  Philip Oakley May 11 '13 at 20:26
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7 Answers

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Angles are defined as the ratio of arc-length to radius multiplied by some constant $k$ which equals one in the case of radians, $360/2\pi$ for degrees. What you're effectively asking is what's natural about setting $k$ = 1? Again it's tidyness as pointed out in dmckee's alternative answer.

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Consider the Taylor series for the trigonometric function. For instance sine $$ \sin \alpha = \alpha - \frac{\alpha^3}{3!} + \dots = \sum_{n=0}^\infty (-1)^{n}\frac{\alpha^{2n+1}}{(2n+1)!},$$ or cosine $$ \cos \alpha = 1 - \frac{\alpha^2}{2!} + \dots =\sum_{n=0}^\infty (-1)^n \frac{\alpha^{2n}}{(2n)!}.$$

If you were to choose some other unit or angle these very tidy series would pick up some additional factors in every term.

That kind of thing is "unnatural" to mathematicians.

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Or, if Taylor series feel too esoteric for you, just consider the approximation $\sin\alpha\approx\alpha$ for small angles $\alpha$, which only holds if $\alpha$ is measured in radians. (Formally, of course, that approximation simply arises from truncating the Taylor series after the first-order term, so in a sense it's the same thing.) –  Ilmari Karonen Aug 6 '12 at 19:00
    
Nice........... –  Mike Dunlavey Aug 8 '12 at 17:39
    
The formula is 'wrong' in that it already presumes that alpha is in radians, while we all know sin 90 is 1.0 ;-) The trigonometric relationships (e.g. sin(A+B)=s(A).c(B)+s(B).c(A) ) stand no matter what unit of measure is used. The choice of radians is an [in]convenience. The underlying problem is that SI Length is a norm in a 3d space (no name). Here endeth the proof that 1=3. –  Philip Oakley May 11 '13 at 20:36
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People call things "natural" when they simplify formulas.

Example, if there is a spinning wheel, the velocity $v$ of a point on the periphery is intuitively proportional to rotational speed $\omega$ and radius $r$. If the rotational speed is measured in radians per second, then the exact formula and the intuitive one are identical:

$$v = r \omega$$

rather than something ugly like $r\omega(\pi/180)$.

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Most importantly $$ e^{i x} = \cos x + i \sin x$$ only holds (in this form) in radians.

So now you might ask why $e$ is more natural than any other number ;-)

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This is the tidiest answer of them all, I think. –  Floris 15 hours ago
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I think part of what I'm looking for is an explanation why the radius is the most important part of a circle.

The most important part of a circle is the locus of points that comprise it. Without that, you don't have a circle.

Radius is important in the definition of "circle" but the definition of "circle" is not identical with any circle.

The radian is defined as "the ratio between the length of an arc and its radius".

$\theta = s/r$

It is more "natural" than other angular measures for this reason: the angle in radians is the normalized arc length, i.e., the radian measure of angle is the arc length for unit radius.

EDIT: to address the numerous comments Zendmailer has made to other answers.

Zendmailer asks

What I'm asking now is, if they are indeed natural , how does the claim that 1 radian = 1 fit in?

For any angular measure $\alpha$, we have the almost trivial result:

1 $\alpha = 1$

So, the fact that 1 radian = 1 has nothing to do with the question of naturalness.

As I explained in a comment to another answer, the justification for the naturalness of the radian as an angular measure is geometric.

One can construct a circle with a length of string fixed at one end, the center of the circle, and a pencil. Holding the string taunt, the pencil traces out the locus of points that comprise the circle. The radius of the circle is the length of the string.

Having done that, what is the most natural way to measure length along the circle? Lay the string along the circumference. The arc length is precisely 1 radius. The angle subtended by that arc length is a natural measure of angle, the radian.

The angle is the arc length divided by the radius so the radian measure of angle directly gives the arc length as a multiple of the radius.

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Would $1\,\textrm{rad}=1$ be false inside the degrees system? Therefore the truth of $1\,\textrm{rad}=1$ is social, and David Zaslavsky's statement that 'degree' is a fancy name for $\frac{\pi}{180}$ is only true inside the radians system? (sorry for continuing with all these questions. I'm genuinely just trying to understand) –  Zendmailer Aug 8 '12 at 17:49
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@Zendmailer, 1 radian = 1 is true always for the same reason that $\frac{1km}{1km} = 1$ is true always, regardless of social convention, regardless of what system one is in. It's true by inspection. –  Alfred Centauri Aug 8 '12 at 18:28
    
Okay. If I'm not misunderstanding, doesn't this mean Physikslover is incorrect in saying it depends on the system, since 1 radian isn't the same as 1 degree? –  Zendmailer Aug 8 '12 at 18:56
    
@Zendmailer, if we decide to subdivide the circumference of a circle into $n$ units, the arc length of each unit, normalized to the radius, is $\frac{2 \pi}{n}$. Note that this is a dimensionless number. The angle associated with this arc length is 1 angular unit and there are n angular units in a circle . Note that this is true regardless of our choice for $n$. Our choice for $n$ affects the normalized arc length associated with our angular unit but 1 angular unit = 1 always. –  Alfred Centauri Aug 8 '12 at 21:19
    
When you say 1 angular unit = 1 always, this means 1 degree = 1 always ($n=360$), right? And when $n=2\pi$ we have 1 radian = 1 always. Why isn't it valid to compare them and say 1 radian = 1 = 1 degree, if we have used the word 'always'? –  Zendmailer Aug 9 '12 at 8:41
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The reason radian was adopted was that it was easy to relate with the circumference of a circle as 2*Pi if the radius was one unit. There is no such thing as 360 degree(it was a misconception in early times that one year is made up of 360 days so they took it 360). From the present day statistics it shall be 365 1/4 but it doesn't change calculations and results gets adjusted automatically on calculation.

Calculations were easy to manipulate with Pi rather than Degree,minutes,second and they are both interchangeable. So, a comfort became a tradition.

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I seem to remember when it was considered to divide the circle into 400 "degrees", to make navigation a little less error-prone. As it is, pilots have to just "know" that 7 and 25 are opposite ends of a runway, and be really really careful not to mix up 13 and 31. –  Mike Dunlavey Aug 8 '12 at 17:48
    
    
@dmckee: thanks for correcting me with this link, but I am sharing something that I read in a differential calculus book once some time back. Book is "Introduction to Differential Calculus: Systematic Studies with Engineering By Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh" page no-99 at the last footnote !! –  Rorschach Aug 10 '12 at 12:29
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Let me state some background facts which might be related to your questions and I hope they will help you understand the answers posted by others.

1-There is a difference between units and dimensions. Every quantity that carries dimensions must carry units.

2-The opposite to the previous statement is not always true, for example angles have no dimensions at all because by definition they are length/length, but they have units. The unit in this case are used to identify the quantity as an angle.

3-Angles can be measured in degrees and can be measured in radians, just in the same way that distances can be measured in centimeters or in inches. Consequently there must be a conversion factor between the 2 units.

4-Using $\pi$ radians = 180 degrees, you can see that $1 ~rad= 180^\circ/\pi=180^\circ/3.14\simeq 57.3^\circ$. That is to say 1 rad = 57.3 degrees (to put it in a form similar to something like 1 inch = 2.54 cm).

5-By definition $\displaystyle \theta=\frac{s}{r}$ rad, where $s$ is the length of the arc subtended the angle and $r$ is the radius of the circle. Note that the previous expression for the angle gives you the angle in radians. If you want it in degrees then it will look like this, $\displaystyle \theta = \frac{s}{r}\frac{180^\circ}{\pi}$. As you see the expression in radian is much simpler hence natural as pointed out by Mike Dunlavey.

6-If you have a particle that is rotating around a circle of constant radius $r$, then from the equation $\displaystyle \theta=\frac{s}{r}$ rad you can see that we can get $\displaystyle \omega=\frac{v}{r}$ rad per unit time (where, by definition, $\omega = \frac{d\theta}{dt}$ and $v=\frac{ds}{dt}$). Again, as pointed out by Mike, the equation for the angular velocity will have an extra factor of $180^\circ/\pi$ had we wanted the angular velocity be expressed in degrees per unit time instead of rad per unit time.

7-When an angle, expressed in radians or degrees, multiplies a unit of distance say, the surviving unit is that of the distance. For example: given $\omega = 2 rad/s$ and $r=1 cm$, hence $r\omega = 2 cm/s$. This is why in this case you can say 1 rad =1.

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The difficulty in point 2 is that the two lengths are in independent dimensions (as in 3d space). One has just cancelled Lx/Ly and lost information for one's dimensional analysis (this is a Physics question;-). If one did the same with Charge/Temperature it would be a gross error, but we tolerate it for length. Dimensional analysis is newer than the cubit, so the old inconsistency remains. –  Philip Oakley May 11 '13 at 20:52
    
@PhilipOakley If we are dividing 2 lengths nobody care about whether they are along the same direction or not, units are not attached to directions –  Revo May 14 '13 at 0:04
    
Anybody working in optics definitely cares. There are many measurements that have Angle(radians) as an integral part of their value, and it is a very common error, not spotted by dimension checking, for the angle part to be omitted, double counted, or wrongly applied. –  Philip Oakley May 14 '13 at 7:30
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