Why Radian Measure Makes Life Easier In Mathematics And Physics

The two most commonly used measures for angles are degrees and radians. There are 360 degrees in a full circle (a right angle is 90 degrees), and $2\pi$ radians in a full circle (there are $\pi/2$ radians in a right angle), so there are about 57 degrees in a radian.

Students typically learn about degrees before they learn about radians, which brings up the question: Why learn about radians if degrees are good enough for measuring angles? There are two reasons, but both are grounded in the fact that the radian is a unitless measure, which I’ll explain in the following paragraph.

The radian is defined to be (see the diagram) the ratio of the length of the arc of a circle (indicated by $s$ in the diagram) to the length of the radius of the circle (indicated by $r$ in the diagram), where each length is measured in the same unit. Therefore, when you divide $s$ by $r$ to get the radian measure of the angle, the units for the two lengths cancel, and you end up with a measure that has no units:

$\theta = \dfrac{s}{r}$

When $s$ is the entire circumference of the circle, the corresponding angle is that of the entire circle. Since the circumference of a circle is $s = 2\pi r$, the angle of a full circle is $\dfrac{2\pi r}{r} = 2\pi$.

Unless there is a good reason, one is free to use any measure whatsoever for angles, such as degrees, radians, or one of the less common ones. There are two good reasons to use radians:

1. If you are working with the derivative of a trigonometric function, then it is preferable to use radian measure for angles, because then derivative formulas (and limit formulas) are easier. For example, using radians, the derivative formulas for sine and cosine are:

$\dfrac{{\rm d}}{{\rm d}x} \sin x = \cos x$

and

$\dfrac{{\rm d}}{{\rm d}x} \cos x = -\sin x$

However, if degrees are used, the derivative formulas for sine and cosine are:

$\dfrac{{\rm d}}{{\rm d}x} \sin x = \left ( \dfrac{\pi}{180} \right ) \cos x$

and

$\dfrac{{\rm d}}{{\rm d}x} \cos x = - \left ( \dfrac{\pi}{180} \right ) \sin x$

The latter pair of formulas are a pain because of the additional factors, so why wouldn’t we use the simpler first pair of formulas? It’s the laziness principle that is used so often in mathematics … we always do the easiest thing. Therefore, we use radians whenever we are dealing with derivatives of trigonometric functions. (The derivatives of the other trigonometric functions are similarly simplified if radians are used … work them out, if you wish!)

2. In describing rotational motion, one rearranges the definition of radian measure to relate the linear displacement to the angular displacement:

$s = r \theta$

Differentiating both sides of this relation with respect to time (and assuming a constant radius), one obtains a relation between the linear velocity $v$ and the angular velocity $\omega$ for motion in a circle:

$v = r \omega$

If $\theta$ is measured in radians, then $\omega$ is measured in radians per second, and then the speed comes out in a natural unit; for example, if $r$ is measured in metres, then $v$ comes out in m/s. However, if $\theta$ were measured in degrees, then the units for $v$ would be m·degrees/s, which is an unnatural unit, and hard to understand.

Differentiating the relation from the previous display with respect to time once more, we get a relation between the linear acceleration and angular acceleration for an object moving in a circle:

$a = r \alpha$

A similar comment about units is also relevant here: Measuring $\alpha$ in radians/s2 means that if $r$ is measured in metres, then $a$ comes out in natural and easily understood units, m/s2. But if $\alpha$ is measured in degrees/s2, or in revolutions/s2, then the units for $a$ come out an awkward mess.

Finally, there is something arbitrary about degree measure (why 360? why not 100, or 1000, or some other round number?) that makes it aesthetically not pleasing. On the other hand, there is something aesthetically pleasing about the definition of radian measure, as it is a simple ratio. This makes it a natural measure, wouldn’t you say?

Which leads me to think about the use of the word “natural” in mathematics … it would be nice to discuss this another time.

p.s.: By the way, did you notice that the factors in the formulas for the derivatives of sine and cosine in degree measure are the same as the factor that is used to convert degrees to radians? This makes sense if you think about the conversion from degrees to radians as a scale change (i.e., a change of variable) and then use the chain rule for differentiation.

Update, 20 September 2012: Ted Burke notes in a comment to this post that Euler’s formula

$e^{i\theta} = \cos \theta + i\sin \theta$

is valid only when the angle $\theta$ is measured in radians. If the angle is measured in degrees, Euler’s formula would have to be written in the considerably uglier form

$e^{i\pi \theta / 180} = \cos \theta + i\sin \theta$

This is another situation in which radian measure is more convenient than degree measure.

Update, 28 February 2016: Parker Harris contributed a very interesting comment, which leads me to doubt that “unitless” is a good term for the radian. Perhaps “ignorable unit” is a better term, as discussed in my response to Parker’s comment.

Thanks very much to Parker for his thought-provoking comment!

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. https://qedinsight.wordpress.com Further resources, and online tutoring, can be found at my other site http://www.qedinfinity.com
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26 Responses to Why Radian Measure Makes Life Easier In Mathematics And Physics

1. Richard Hayes says:

This a very nice post and came up when I posed the question to my colleagues (the type of basic question which you sometimes have to be old(er) to pose) ‘why radians?. I started thinking about the answer when I got lazy and did a google instead. I am teaching a class on ac circuits and phasors and realized that I wasnt sure (had forgotten?) why I had to use the radian setting on the calculator to plot the sine wave, i.e. why the sine formula is x(t) = sin( 2 PI f t)

• Thanks, Richard! The word “phasor” brings back some fond memories. All the best in your teaching!

2. batchloaf says:

Thanks, a very good explanation! One other thing that I was reminded of while reading your post is that radian angles make Euler’s formula particularly elegant.

• Thanks, Ted! And it’s a good point about Euler’s formula, which I’ll add to the post. All the best!

3. Jennifer says:

I agree that students would benefit from learning radian measure earlier. Think of how beautiful the interior angle sums for polynomials are in radians! Triangle: pi, Quadrilateral: 2pi, Pentagon: 3pi… My precal kids got so flustered that I ended up making radian scale protractors for them! You can check them out on my website, proradian.net.

4. Jennifer says:

Reblogged this on jensilver.

5. tomcircle says:

Reblogged this on Singapore Maths Tuition and commented:
Why radian is preferred in Math? Very good explanation here…

6. Skye says:

Very nice explanation! So radians are used because it’s more beautiful!

7. kyledelaney1988 says:

I’m still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I’m discovering that I’ve been wrong all along! I’m guessing that when you differentiate sine, the step that only works for radians is when you replace sin(dx) with just dx, because as dx approaches 0 then sin(dx) equals dx because sin(0) equals 0. But isn’t the same true for degrees? As dx approaches 0 degrees then sin(dx degrees) still approaches 0. But I’ve come to the understanding that sin(dx degrees) approaches 0 almost 60 times slower, so if sin(dx radians) can be replaced with dx then sin(dx degrees) would have to be replaced with pi/180 times dx degrees.

But the question remains of why it works perfectly for radians. How do we know that we can replace sin(dx) with just dx without any kind of conversion applied like we need for degrees? It’s not good enough to just say that we can see that sin(dx) approaches dx as sin gets very small. Mathematically we can see that sin(.00001) is pretty darn close to .00001 when we’re using radians. But let’s say we had a unit of measurement “sixths” where there are 6 of them in a full circle, pretty close to radians. It would also look like sin(dx sixths) approaches dx when it gets very small, but we know we’d have to replace sin(dx sixths) with pi/6 dx sixths when differentiating. So how do we know that radians work out so magically, and why do they?

• These are good questions.

One answer to why radians work out so nicely is that you just do the calculation (of the derivative of the sine function) in arbitrary angle units, and then see which one comes out the simplest. As you suggest, you could use “sixths” or any other angle units. When you try this, you find that the derivative formula comes out simplest for angles measured in radians. This is motivation to use radian measure in calculus, but if you wished you could use any other angle measure you wish, provided that you accepted the specific derivative formula for the sign function that comes with that angle measure.

You can find the derivation of the derivative of the sine function in many places; search for a derivation involving a limit calculation. If you go through the details carefully, you’ll note that the limit of sine theta divided by theta is 1 only when theta is measured in radians. For other angle measures, this limit is not 1, and so the derivative formula must be multiplied by this factor.

OK, that’s one answer. Another answer is geometrical. Think of the slope of a line, which is defined as rise divided by run. If you change the units for the run, you will get a different value for the slope.

A derivative can be interpreted as the slope of the tangent line to the graph of the function in question. Using different angle measures amounts to changing the units of the run, which changes the slope. The units for the slope are (rise units)/(run units). Changing the angle measure changes the units of the run, but not the rise.

Does this make sense?

• kyledelaney1988 says:

Yes, I understand all that about slopes, but that’s not really what my question was about. I understand why it would be different depending on which units you use. I just don’t understand how to mathematically prove that radians are those perfect units. I guess I need to look further into proving limits.

• OK, I get your question now. There is a geometric argument that you might find compelling:

Draw a circle centred at the origin. Now draw a ray from the origin angled off into the first quadrant somewhere, intersecting the circle. The purpose of doing this is to convince ourselves that the limit of sin (dx) divided by dx as dx approaches 0 is 1.

Drop a vertical line segment from the point of intersection of the ray and the circle down to the x-axis. This produces a right triangle, formed from the vertical line segment (of length Y), the line segment along the x-axis (of length X) and the segment along the radius of the circle (of length R). Label the angle as dx. Then in the right triangle, sin (dx) = Y/R.

Note that the arc of the circle subtended by the angle is not very different in length from Y; furthermore, the approximation gets better and better as the angle gets smaller. Thus, in the limit as dx approaches 0, Y can be replaced by the arc of the circle subtended by the angle, which we can call S. Thus, as dx approaches 0, sin (dx) gets closer and closer to S/R.

But, if we measure the angle in radians, then S/R = dx, an exact relation that is the definition of radian measure. Thus, in the limit as dx approaches 0, sin (dx) approaches dx, and so sin (dx) divided by dx approaches 1.

If we used some other angle measure, then we would have to modify S/R = dx by some factor, and the same factor would appear in the derivative formula.

• kyledelaney1988 says:

Okay, I get it now, thank you. And I also found a detailed proof here: http://oregonstate.edu/instruct/mth251/cq/Stage4/Lesson/sinProof.html

8. Hello Mr. Bushell,

If you can point to an error at a specific location in the post, then I’m happy to hear it. Otherwise take your verbal abuse elsewhere.

9. Parker says:

If you have a moment, I’d love some help understanding why radians have a better claim at being unitless than degrees. Here’s what I’m thinking now: To calculate and angle in radians, you divide s/r, which is, say, feet/feet, which cancel out. Suppose I want to calculate he same angle in Parkers, a unit of angle I’ll define as a full rotation. I’d divide s (feet) by 2*pi* (r (also feet)), getting again a ratio, so Parkers are unitless as well? Just as radians are defined as the ratio of arclength to radius, Parkers are defines as ratio of arclength to circumference. To calculate degrees I’d divide s by 2pi and then divide that by 360. Again the units I use are going to be feet/feet, aren’t they? I can’t quite make sense of exactly why but this seems related to the relationship between force, distance, and energy, and power. Energy can have units length*force, for example ft*lb. Power, or energy per second, can have units ft*lb/s. However a more common unit of power is the horsepower, equal to 550 ft*lb/s. The 550 is arbitrary and inconvenient, but I don’t think it changes anything fundamental. What do you think?

• I think you’ve made a very important point, which leads me to conclude that “unitless” doesn’t capture what is special about radian measure. That is, I don’t think “unitless” is the right word, and I’ll have to cast for a better word.

Here’s the difference between the radian and the Parker (or degree, or any other angle measure): In the formula $v = r\omega$, if you measure $\omega$ in radians per second, and if $r$ is measured in metres, then multiplying the two values gives you a speed that is in metres per second. On the other hand, if you measure $\omega$ in Parkers per second, and if $r$ is measured in metres, then multiplying the two values does not give you the correct speed in metres per second; you’ll have to do some kind of conversion to convert from metres.Parkers/second to m/s.

So the key difference is that when you multiply (or divide) some combination of units by a radian, you can just ignore the radian in the final unit, as if it were not even there. You can’t do this with degrees, Parkers, or any other angle unit. This property is what I tried to capture with the term “unitless,” but it’s not clear that this is the best word. But it’s hard for me to summarize this property of radians with a single word.

Maybe “ignorable unit” would be an acceptable term, because that captures the fact that in a product or quotient of units, if the radian is included, it can be safely ignored without introducing error, whereas this is not the case for the degree, the Parker, etc.

Thank-you for a very good comment!

• kyledelaney1988 says:

I’m glad to hear you say all that. I’ve encountered many mathematicians who maintain that angles measured in radians are indeed unitless and thus that angles themselves are dimensionless, i.e. not a dimension that is distinct from ratios or spatial dimensions on a graph.

10. Songwhe Huh says:

The natural constant ( e ) is the key to understand the mystery of radians.
e^x function is very special because it is invariant under differentiation with x.
For the e^ix, with geometrical explanation on complex plane, it is the point on a circle with radius 1 and the arc distance from the reference point (1+i0) is x. ( it can be proved by invariant property)
So, now we can define angle as the arc length of unit circle from (1,0) without length unit.
Then we can interpret the Euler formula as the definition of sine and cosine with radian unit.
With that definition and simple complex algebra, you can intuitively see differentiation of sin(x) is cos(x).

11. Hardik Ghoshal says:

Thus makes it easier to study complex number arguments.

12. Ted says:

Radians are great for hand calculations, but when it comes to computing there are some advantages to using cycles or degrees:

1. Trigonometric libraries are slowed down when calculating large numbers, where cycles could make it trivial to perform the calculation: simply drop the whole number, and do the calculation on the whole number part.

2. 360 divides nicely by factors of 10, 9, 8. This means the degree measure for any of those angles can be stored exactly as a finite binary number.

As for Euler’s formula, cos t + j sin t = e^jt is kind of nice, but when you start doing Fourier transforms, you see e^(j*2*pi…) terms, and the reason the 2*pi is there is because you’re using radians instead of cycles. Use cycles, and the term goes away.The Fourier transform (along with its variants like cosine transform) is widely used in engineering and science for everything from jpeg image compression to radio frequency modulation. A silly thing that slows down trig functions in FFT calculation is the common use of radians for cosine and sine in standard libraries where it’s completely unnecessary.

13. Mark Girard says:

Comment:
When illustrating to students the connection between the unit circle and its map into a sine or cosine function, one could say that the distance of the wave line y=sinx is the radian value of x.

— So, the distance from the start of the sine function to the end of its first iteration at 2pi (360) is 2pi. It would seem so, but do not have any formal proof for it.

14. lockedmoon says:

Do you know of a logical decomposition of radian vs degree vs arc distance?
There is no SI unit of “degree” or even “angle”.
Length along the arc is clearly a unit of distance. So is length of the radius. So one divided by the other is a simple number. This defines the angle.
But you can have an angle without a distance. “Angle” could be a unit of rotation. Any ideas why we don’t have an SI unit of rotation?

• Rodney says:

You can also define angle by the area of the sector by saying by using r^2 as a unit of measure. To find the angle you can just divide the area swept out by the angle you are measuring by r^2. You can use radius/2 as a unit which is half radians. It is totally arbitrary. I can divide the circle into 4 equal pieces and call each piece some made up name pegrees or something and use it as a measure of angle and 90 degrees would now be equal to 1 pegrees. Totally arbitrary. The whole point is that radians just give you cleaner looking mathematical results that is where the term naturalness comes from in regards to radians…And also the whole unit arc length is just to justify the naturalness of radians.. That is it. Dont get hung up on that.

15. Rodney says:

From what I understand every arbitrary unit you use to measure angle requires the set of real numbers. It is just that you get different domains depending on the arbitrary unit you use to measure angle. And the important point is that they are all just as valid as the units of radians. However, radians just give you nicer looking mathematical results but it doesn’t mean the results you get from other arbitrary units are not valid. You will always have that pi/X factor where X depends on the arbitrary unit you use. To Songwhe Huh the choice of units dictates the form of the derivative of the sine and cosine not e. From your argument you assumed radians already for the euler formula and pretty much proved what you already assumed.