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?
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19$\begingroup$ It's just a convention, but there aren't many reasons to use degrees and many to use radians. $\endgroup$– anomalyJul 18, 2017 at 17:32
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6$\begingroup$ 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. $\endgroup$– Eric LippertJul 18, 2017 at 22:51
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2$\begingroup$ None of the answers use a nice unit circle and pi to show how useful and "natural" radians truly are, sadly. When you look at a unit circle and the angles involved, you look at it in radans. Converting to degrees is just an unnecessary step. $\endgroup$– PolygnomeJul 19, 2017 at 6:58
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3$\begingroup$ @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. $\endgroup$– DavorJul 19, 2017 at 11:21
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3$\begingroup$ @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.) $\endgroup$– Eric LippertJul 19, 2017 at 18:46
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6 Answers
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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 $$(\sin \theta)' = \cos\theta$$ $$(\sin \theta)'' = -\sin \theta$$
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13$\begingroup$ @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. $\endgroup$– Alex LiJul 18, 2017 at 22:44
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11$\begingroup$ 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. $\endgroup$– mattdmJul 19, 2017 at 3:43
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6$\begingroup$ 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). $\endgroup$ Jul 19, 2017 at 7:55
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1$\begingroup$ @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. $\endgroup$– Chris HJul 19, 2017 at 8:25
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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.
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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).
answered Jul 18, 2017 at 17:34
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It makes computation of circular arc lengths easier: no conversion factor like $\pi/180$ needed.
It makes the Taylor series for functions like $\sin$ and $\exp$ look cleaner.
But yes, it's a convention.
answered Jul 18, 2017 at 17:31
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The main reason is that the derivatives of sinusoidal functions only work if the angles are measured in radians.
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8$\begingroup$ They're certainly still differentiable, but you get an annoying constant factor. $\endgroup$– anomalyJul 18, 2017 at 20:28
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1$\begingroup$ @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 $\endgroup$ Jul 20, 2017 at 5:01
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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.