Why do we use radians?
March 9, 2008 5:52 PM Subscribe
What's the advantage of using radians instead of degrees? I've never had a satisfactory answer to this. In high school, we always used degrees to measure angles and starting in university we switch to radians. Is there really any benefit to using radians?
radians are the natural unit for trigonometric functions. in some sense, pi has the numerical value it does because sine and cosine go through a complete cycle in 2*pi radians.
The significance of radians
posted by sergeant sandwich at 5:57 PM on March 9, 2008
The significance of radians
posted by sergeant sandwich at 5:57 PM on March 9, 2008
With radians, arclength is simply r(θ), and sin(θ) ~ θ for small θ.
posted by kickingtheground at 6:01 PM on March 9, 2008
posted by kickingtheground at 6:01 PM on March 9, 2008
Elaborating on what sergeant sandwich said, radians are convenient because we have defined trigonometric functions in a certain way, and using these trigonometric functions is convenient. Because of the way that sine and cosine are defined, if you feed in to the sine function an angle measure whose units are degrees, you get out an ugly answer (or at least one that has a factor of pi/180 that you have to keep track of), whereas if you feed the sine function an angle measure with units of radians, you get out (comparatively) nicer answers.
They both, of course, are arbitrary ways to measure angle, but one turns out to me more convenient for, say, calculus applications because it plays nicer with other functions that are useful.
(There is nothing inherent in a circle that says its total angle measure has to be 2*pi radians, although you can make the case that it is inherent in a circle that the ratio of the circumference to the diameter of a circle is constant, and that we might want to give this constant a name (that is, (2 pi r) / (2 r) = pi for any circle), and that it might (or might not) be useful to use this quantity to define angle measure.)
posted by leahwrenn at 6:18 PM on March 9, 2008
They both, of course, are arbitrary ways to measure angle, but one turns out to me more convenient for, say, calculus applications because it plays nicer with other functions that are useful.
(There is nothing inherent in a circle that says its total angle measure has to be 2*pi radians, although you can make the case that it is inherent in a circle that the ratio of the circumference to the diameter of a circle is constant, and that we might want to give this constant a name (that is, (2 pi r) / (2 r) = pi for any circle), and that it might (or might not) be useful to use this quantity to define angle measure.)
posted by leahwrenn at 6:18 PM on March 9, 2008
Related, previously in Ask Metafilter: Maybe pi should really be ~6.28.
posted by Jimbob at 6:34 PM on March 9, 2008 [1 favorite]
posted by Jimbob at 6:34 PM on March 9, 2008 [1 favorite]
Some good answers, so I'll just tack on that radians are just as much a length measurement. On a circle with radius 1, the length of an arc with an angle of x radians is also x.
posted by monkeymadness at 7:08 PM on March 9, 2008
posted by monkeymadness at 7:08 PM on March 9, 2008
See Natural Units.
If you use degrees, you end up having to carry around a lot of conversion factors (to convert angles to lengths). It's fine, and you can do it, but you might as well just pick units where the conversion factor is 1. If you read through the articles in the thread Jimbob links to, you'll see that people screwed up and actually picked 2. Oh well. Carrying around factors of 2 is better than factors of 180 or 360.
posted by dsword at 7:26 PM on March 9, 2008
If you use degrees, you end up having to carry around a lot of conversion factors (to convert angles to lengths). It's fine, and you can do it, but you might as well just pick units where the conversion factor is 1. If you read through the articles in the thread Jimbob links to, you'll see that people screwed up and actually picked 2. Oh well. Carrying around factors of 2 is better than factors of 180 or 360.
posted by dsword at 7:26 PM on March 9, 2008
Best answer: Several advantages to radians:
1) Conversion from angular displacement to linear displacement is much easier. For example, if a wheel with radius 1 foot turns 50 radians, the wheel will have traveled 50 feet.
2) The definition is less arbitrary. A radian is just one radius' worth of distance on the edge of the circle, whereas a degree is one one-hundred-eightieth of a full rotation. The only less arbitrary unit I could think of would be rotations (though it lacks some of the other nice features of radians).
3) Trigonometric functions are useful for things that do not involve angles of any sort. When you're using trigonometric functions to deal with waves, degrees and radians don't really have any meaning. However, the same familiar trigonometric function that maps an angle in radians to the length of the opposite side of a unit-hypotenuse triangle (this is the one you get if you hit "sin" on your calculator in radian mode, which is what mathematicians mean when they say "sine") also describes a variety of other useful situations. The one that maps an angle in degrees to the length of the opposite side (this is the one you get if you hit "sin" on your calculator in degree mode, hereafter referred to as "degree-sine") requires a little more tweaking to do the same work.
4) Calculus. The derivative of the sine function is the cosine function. The derivative of the degree-sine function is not the degree-cosine function, but rather a (pi/180) times the degree-cosine function. Be thankful that you don't have to deal with these kind of fudge factors.
5) Complex numbers. If you take a complex number z and write its position on the complex plane in polar coordinates (r,theta) (using radians, of course!), then z is actually equal to r*e^(i*theta). If you want to pull of the same trick with degrees, it'll cost you some more fudge factors.
Advantages to degrees:
1) You're already used to it.
2) It's used a lot for measuring angles (though not as often for doing the many other jobs that trigonometry is useful for).
3) There are a whole number of degrees in a full rotation (but really...180? I mean it does have a lot of factors, but talk about arbitrary. At least they could have gone with metric degrees with 100, 10, or even 1 in a rotation or something.)
On preview, I've duplicated a little ground here, but that's what I get for writing long answers.
posted by ErWenn at 7:31 PM on March 9, 2008 [1 favorite]
1) Conversion from angular displacement to linear displacement is much easier. For example, if a wheel with radius 1 foot turns 50 radians, the wheel will have traveled 50 feet.
2) The definition is less arbitrary. A radian is just one radius' worth of distance on the edge of the circle, whereas a degree is one one-hundred-eightieth of a full rotation. The only less arbitrary unit I could think of would be rotations (though it lacks some of the other nice features of radians).
3) Trigonometric functions are useful for things that do not involve angles of any sort. When you're using trigonometric functions to deal with waves, degrees and radians don't really have any meaning. However, the same familiar trigonometric function that maps an angle in radians to the length of the opposite side of a unit-hypotenuse triangle (this is the one you get if you hit "sin" on your calculator in radian mode, which is what mathematicians mean when they say "sine") also describes a variety of other useful situations. The one that maps an angle in degrees to the length of the opposite side (this is the one you get if you hit "sin" on your calculator in degree mode, hereafter referred to as "degree-sine") requires a little more tweaking to do the same work.
4) Calculus. The derivative of the sine function is the cosine function. The derivative of the degree-sine function is not the degree-cosine function, but rather a (pi/180) times the degree-cosine function. Be thankful that you don't have to deal with these kind of fudge factors.
5) Complex numbers. If you take a complex number z and write its position on the complex plane in polar coordinates (r,theta) (using radians, of course!), then z is actually equal to r*e^(i*theta). If you want to pull of the same trick with degrees, it'll cost you some more fudge factors.
Advantages to degrees:
1) You're already used to it.
2) It's used a lot for measuring angles (though not as often for doing the many other jobs that trigonometry is useful for).
3) There are a whole number of degrees in a full rotation (but really...180? I mean it does have a lot of factors, but talk about arbitrary. At least they could have gone with metric degrees with 100, 10, or even 1 in a rotation or something.)
On preview, I've duplicated a little ground here, but that's what I get for writing long answers.
posted by ErWenn at 7:31 PM on March 9, 2008 [1 favorite]
agggg.... 'radian' is not a unit of measurement in the usual sense: it is dimensionless.
from the wiki"
"Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless."
they go on to explain that if you expand sin(x) = x - (1/3!)x^3 + ... in a Taylor series, if 'x' had units you would be carrying along invisible conversion factors to allow the units of the different powers to match up i.e. x has units of degree, x^3 has units degree^3...
posted by geos at 7:41 PM on March 9, 2008
from the wiki"
"Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless."
they go on to explain that if you expand sin(x) = x - (1/3!)x^3 + ... in a Taylor series, if 'x' had units you would be carrying along invisible conversion factors to allow the units of the different powers to match up i.e. x has units of degree, x^3 has units degree^3...
posted by geos at 7:41 PM on March 9, 2008
(That's Wikipedia you just linked to, geos. There are loads of wikis, and if you say "the wiki" in the context of Metafilter you are rather implying this wiki, which tends not to concern itself with trig.)
posted by mumkin at 8:25 PM on March 9, 2008
posted by mumkin at 8:25 PM on March 9, 2008
Geos, I understand your pain at how many people misunderstand these concepts, but you might be misleading people a bit here. Any unit for measuring angles (including degrees, rotations, minutes, seconds, and radians) is
There is a fine point to be made here, that the standard scale (in most mathematical contexts) is the radian scale. This is why angles measured in radians are not generally marked with a unit while angles measured in degrees generally do get a mark to signify that they are written in a nonstandard scale. But I don't think anyone here has made any mistakes (even tiny pedantic technicalities) that spring from treating radian measure as a dimensioned quantity.
posted by ErWenn at 8:46 PM on March 9, 2008
dimensionlessin the sense used in the Wikipedia article you quoted. The difference between radians and degrees is not a difference between dimensionless and dimensioned (
dimensionfulwould be so much cooler, but it's not a word, unfortunately) units of measure. It's a difference of scaling.
There is a fine point to be made here, that the standard scale (in most mathematical contexts) is the radian scale. This is why angles measured in radians are not generally marked with a unit while angles measured in degrees generally do get a mark to signify that they are written in a nonstandard scale. But I don't think anyone here has made any mistakes (even tiny pedantic technicalities) that spring from treating radian measure as a dimensioned quantity.
posted by ErWenn at 8:46 PM on March 9, 2008
A lot of the equations are simpler with radians. Take, for example, the Fourier Transform (or any frequency-domain math) -- if you used degrees, you'd have a mess of coefficients everywhere. With radians, you tend to have a lot simpler equations, with at most a 2*pi coefficient here and there.
posted by spiderskull at 9:54 PM on March 9, 2008
posted by spiderskull at 9:54 PM on March 9, 2008
Absolutely there's good reasons to use radians; it's just that in normal, everyday life we seldom have any need to.
In college I got all the way to Differential Equations, my final undergraduate math class. Naturally I had plenty of exposure to radians along the way -- I thought I understood but didn't really, until quite recently when I saw a WWII training cartoon called "Position Firing" for teaching airborne gunners how to hit their enemy-fighter targets. Wasn't until then that I fully realized how a radian is the radius of any given circle. A big reason for the cross-hairs in a gun-sight is to give the gunner an easily-eyeballed radian which he can them multiply by two or three to guage where to aim so his shots will reach the spot at the same time as the enemy aircraft.
posted by Rash at 11:29 PM on March 9, 2008 [2 favorites]
In college I got all the way to Differential Equations, my final undergraduate math class. Naturally I had plenty of exposure to radians along the way -- I thought I understood but didn't really, until quite recently when I saw a WWII training cartoon called "Position Firing" for teaching airborne gunners how to hit their enemy-fighter targets. Wasn't until then that I fully realized how a radian is the radius of any given circle. A big reason for the cross-hairs in a gun-sight is to give the gunner an easily-eyeballed radian which he can them multiply by two or three to guage where to aim so his shots will reach the spot at the same time as the enemy aircraft.
posted by Rash at 11:29 PM on March 9, 2008 [2 favorites]
Holy crap! I posted that comment over a month ago, and I just now realized that I said
posted by ErWenn at 7:49 PM on April 14, 2008
a degree is one one-hundred-eightieth of a full rotation! I meant
one three-hundred-sixtiethof course. This is the exact same mistake I made in high school on the AIME, screwing me out of a chance at going to the Math Olympiad. You think I would have learned that lesson by now.
posted by ErWenn at 7:49 PM on April 14, 2008
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posted by mpls2 at 5:56 PM on March 9, 2008