# Why do we use radians for polar coordinates rather than degrees?

Is there a specific reason why radians must be used when converting from Cartesian to Polar coordinates, is it just a convention or does it not matter at all?

• qedinsight.wordpress.com/2011/03/14/… – Quantaliinuxite Jul 18 '17 at 17:31
• It's just a convention, but there aren't many reasons to use degrees and many to use radians. – anomaly Jul 18 '17 at 17:32
• Are you familiar with Euler's Formula, and its relationship to polar coordinates on the complex plane? That makes it very clear why radians are a natural measure of an angle for a complex number. – Eric Lippert Jul 18 '17 at 22:51
• @Polygnome - to be honest that makes no sense. You look at it "in radians" if you are used to radians. There is nothing intrinsic about it. – Davor Jul 19 '17 at 11:21
• @Davor: No, radians are the intrinsic way to measure angles. Imagine a slice of pie. How many radians is the angle it makes? The ratio of the length of the crust to the length of the radius. It's not a question of convention; it is the most concise definition possible. (Also, one radian is a nice size for a piece of pie, which makes it very convenient.) – Eric Lippert Jul 19 '17 at 18:46

Radians provide a nice association between the area of a section of a circle and the size of the angle. i.e. $$A = \frac 12 r^2 \theta.$$ They also behave nicely with arc lengths. $$C = r\theta$$

And so if you have something like the rotating wheel, and you want to know the distance a belt attached to that wheel has moved, then rotation in radians avoids a conversion factor. This makes radians handy for problems in physics and engineering.

But the bigger bonus of this association between arc length and radians comes up when $$\theta$$ is small. For small $$\theta, \sin\theta \approx \theta.$$ And that is really important when you get to calculus. Once you start calculus you will wonder why you ever thought degrees were easier.

We haven't really talked about polar coordinates. And, honestly, there is no reason you cannot use degrees when you work with polar coordinates. And plenty of "real-world" applications use degrees and polar coordinates. Problems in navigation, for example, are almost always done in degrees, and are effectively problems in polar coordinates.

Your teachers want you to use radians to be more comfortable with them when the time comes that degrees become inefficient.

If I were to ask you "What is a degree?", could you say anything other than "There are 360 degrees in one revolution"? And then I would ask you "Why 360? Why not 100? Why not 17? Is there anything natural about 360?"

On the other hand, radians are a natural measure of an angle. Given an angle, form a circle (of any radius you like) centered at the vertex of the angle. The radian measure of the angle is exactly the number of times the radius length fits on the arc of the angle. It's independent of whatever radius you chose.

When angles are measured in radians, you also have nice relationships between trig functions like $$(\sin \theta)' = \cos\theta$$ $$(\sin \theta)'' = -\sin \theta$$

• and nice series development – G Cab Jul 18 '17 at 18:20
• @JohnR.Strohm 360 was chosen because it is very divisble, a 'superior highly composite number'. When you're talking about right angles you can say 90 degrees, which is a nice integer. Equilateral triangles have 60 degrees, again a nice integer. As opposed to something like "43.4 degrees" is a right angle. – Alex Li Jul 18 '17 at 22:44
• 2, 3, 4, 5, 6, 8, 9, 10, 12, 15... and it is exactly the same as the number of days the year! I mean, give or take a few feast days which don't really count. – mattdm Jul 19 '17 at 3:43
• The Babylonians used a base-60 counting system, so 360 is a nice round '$60' (and '6' is "round" like "2" or "5" in base 10). – Martin Bonner supports Monica Jul 19 '17 at 7:55 • @mattdm but not 16, and dividing a circle into 16 is quite common e.g. in navigation, where even 32 divisions were used. Halves, of course, aren't too bad to work with. – Chris H Jul 19 '17 at 8:25 The radian has a geometric meaning, it is the ratio between the arc's length and the circunference's radius. Other measures of angle have arbitrary scales, so numerical values in degrees or gradians have no special meaning. Maybe because of their more 'natural' definition, by using radians, we find ellegant derivatives and power series for the trigonometric functions (exponentials too, because of Euler's formula). 1. It makes computation of circular arc lengths easier: no conversion factor like$\pi/180$needed. 2. It makes the Taylor series for functions like$\sin$and$\exp$look cleaner. But yes, it's a convention. The main reason is that the derivatives of sinusoidal functions only work if the angles are measured in radians. • They're certainly still differentiable, but you get an annoying constant factor. – anomaly Jul 18 '17 at 20:28 • @anomaly I didn't say they weren't differentiable. I said if you use radians, statements like (sin x)' = cos x don't hold, which is clearly true – Lanier Freeman Jul 20 '17 at 5:01 As many others commented, radians are the natural unit. Degrees exist only for historical reasons and are not an extremely practical unit (not mentioning the minutes/seconds issue). One of their only assets is that$360$is divisible by$2,3,4,5,6,8,9,10$and others. IMO, we are lacking a truly practical unit, which does not involve the constant$2\pi$: revolutions, i.e. a full turn corresponding to$1\$ unit.