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Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful. Why do we measure angles in radians? Because th

Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful. Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x. In my opinion drilling down too much on conventions misses the point of math.

uoaei 2y ago
Without delving too far into the philosophy of math as concerns existence vs convention... e pops up quite often when taking limits on a surprising number of varied phenomena. It is much more than a mere convention, unless you subscribe to the nihilistic, anti-epistemological notion that all of mathematics is merely convention. It seems to be the center of the conceptual space particularly around questions of relative and absolute scale. It's true you can use any base for computing things but some are more natural than others in that specific parametrizations have natural interpretations especially when it comes to physics (timescales, information-theoretic optimality, etc.).
programjames 2y ago
No, we measure angles in radians so that d^4/dx^4 (sin x) = sin x.
- cperciva 2y ago
  No, we measure angles in radians so that e^(ix) = cos x + i sin x.
  - nine_k 2y ago
    But this does not depend in the unit.
    - deleted 2y ago
      [deleted]
    - JadeNB 2y ago
      > But this does not depend in the unit. It does! e^z, defined as the series \sum_{n = 0}^\infty z^n/n!, can only be a function of a dimensionless number z. sin(z) and cos(z), defined as power series, technically also work this way. And that's OK, because angles are dimensionless: a radian is just C/(2πr), where C is the circumference of a circle of radius r. But it is sometimes convenient to pick your favorite number of radians, like π/180 of them, and call that a degree, and then to say that sin(x degrees) is the same as sin(xπ/180 radians). With this convention, where the left-hand side of e^(ix) = sin(x) + icos(x) is a function of a dimensionless variable, and the right-hand side can be viewed as a function of a dimensioned argument only in the sense written above, it really is the case that the equation written is true, but the equation e^(ix) = sin(x degrees) + icos(x degrees) is false. (On the other hand, you could make the case that e^(ix) is really a function of an angle, where its value is the complex number that lies on the unit circle at that angle. Then you do recover a "dimensioned" version of e^(ix) = sin(x) + i*cos(x) that's valid even if you measure angles in degrees.)
    - kazinator 2y ago
      If we use a degrees version of sin and cos (call them sind and cosd), then we cannot have e on the left side without a conversion factor. (iπx/180) e = cosd x + i sind x ix π/180 -> e = cosd x + i sind x π/180 -> let f = e ix f = cosd x + i sind x Probem is, f doesn't have nice properties like: d x x - f /= f dx There is something uniquely special about the unit circle, and about using the unscaled distance around the unit circle as the measure of the angle.
      - kazinator 2y ago
        Radians have the property that if we step x by some tiny amount δ, then the cos/sin coordinates will move by that same distance around the unit circle: |[cos(x+δ) + i sin(x+δ)] - [cos(x) + i sin(x)]| = δ This is also related to how we can estimate sin(x) = x for small values next to zero, if using radians. In radians, the derivative sin'(x) is cos(x), and cos'(x) is -sin(x). Derviation just shifts the waveform left by ninety degrees. In units other than radians, we get wacky constant terms that change at each step. That's related to how e^x is its own derivative.
nine_k 2y ago
We measure angles in radians because it's an easy way to measure the length of the circle sector in the units of the circle's radius. It feels neat in many cases. A more practical way to measure angles would be in rotations. 0° = 0, 360° = 1, 90° = 0.25, etc. It would remove a ton of 2π and 4π² factors from a lot of equations in physics.
- mb7733 2y ago
  For trigonometry/calculus/physics radians are by far the most practical because they are dimensionless, so no constants appear when differentiating or integrating. (By the way, these constants will involve factors of pi anyway, it's inherent.) For example, try to work out the Taylor series for sin(x) using degrees (or rotations). It's awful.
  - nine_k 2y ago
    I don't see how Taylor series specifically would be affected. Differentiation of sin x and cos x is the same independent of the unit of x, and nothing else is used in the series. Fourier transform would have 4π² instead of 2π under the exponent, no big deal. The Euler's formula gets a factor of 2π under the exponent though. Given its wide application, it adds plenty of noise, of course.
    - setopt 2y ago
      If you define variants sint(x) = sin(2πx) and cost(x) = cos(2πx) that takes x in units of turns instead of radians, then d/dx sint(x) = 2π cost(x) etc. I agree with you that this is completely fine though. I also find it more natural to think of “how many percent of a turn” an angle is than how many “degrees” or “radians” something is, since we use base-10 everywhere else. My workaround is to mostly write everything in terms of sin(2πτ), cos(2πτ), and exp(2πiτ) when I can, where τ measures turns.
    - School-Cotton 2y ago
      > Differentiation of sin x and cos x is the same independent of the unit of x, That’s not true. If the unit is degrees, d/dx sin(x) = pi/180 * cos(x).
akira2501 2y ago
> drilling down too much on conventions misses the point of math. What is "the point of math?"
- __MatrixMan__ 2y ago
  I thought it was about finding conventions that are useful, surprising, or pleasing in some way. Or, given a set of conventions, finding new ways that they are useful, surprising, or pleasing. So I'm curious to know about this other... non-conventional point.
- ithkuil 2y ago
  Well, the point, in math, is a zero dimensional object.
Aardwolf 2y ago
Other chosen conventions are less great, like not having the circle constant be 6.283185... so that we don't need to deal with factors or divisors of 2 all the time when using radians or almost anything else involving pi 90 degrees is one fourth of a circle, it would be so much more intuitive if we'd use "1/4th of something" rather than "1/2 radians" to express this
pif 2y ago
No! We measure angles in radians because it's the simplest way to link the length of an arc to the radius of the circle.
- aidenn0 2y ago
  But we have an inconsistency because pi relates the diameter to the circumference, not the radius to the circumference, which leaves an annoying 2 in the 2pi radians in a circle.

uoaei 2y ago

Without delving too far into the philosophy of math as concerns existence vs convention... e pops up quite often when taking limits on a surprising number of varied phenomena. It is much more than a mere convention, unless you subscribe to the nihilistic, anti-epistemological notion that all of mathematics is merely convention. It seems to be the center of the conceptual space particularly around questions of relative and absolute scale. It's true you can use any base for computing things but some are more natural than others in that specific parametrizations have natural interpretations especially when it comes to physics (timescales, information-theoretic optimality, etc.).

programjames 2y ago

No, we measure angles in radians so that d^4/dx^4 (sin x) = sin x.

cperciva 2y ago
No, we measure angles in radians so that e^(ix) = cos x + i sin x.
- nine_k 2y ago
  But this does not depend in the unit.
  - deleted 2y ago
    [deleted]
  - JadeNB 2y ago
    > But this does not depend in the unit. It does! e^z, defined as the series \sum_{n = 0}^\infty z^n/n!, can only be a function of a dimensionless number z. sin(z) and cos(z), defined as power series, technically also work this way. And that's OK, because angles are dimensionless: a radian is just C/(2πr), where C is the circumference of a circle of radius r. But it is sometimes convenient to pick your favorite number of radians, like π/180 of them, and call that a degree, and then to say that sin(x degrees) is the same as sin(xπ/180 radians). With this convention, where the left-hand side of e^(ix) = sin(x) + icos(x) is a function of a dimensionless variable, and the right-hand side can be viewed as a function of a dimensioned argument only in the sense written above, it really is the case that the equation written is true, but the equation e^(ix) = sin(x degrees) + icos(x degrees) is false. (On the other hand, you could make the case that e^(ix) is really a function of an angle, where its value is the complex number that lies on the unit circle at that angle. Then you do recover a "dimensioned" version of e^(ix) = sin(x) + i*cos(x) that's valid even if you measure angles in degrees.)
  - kazinator 2y ago
    If we use a degrees version of sin and cos (call them sind and cosd), then we cannot have e on the left side without a conversion factor. (iπx/180) e = cosd x + i sind x ix π/180 -> e = cosd x + i sind x π/180 -> let f = e ix f = cosd x + i sind x Probem is, f doesn't have nice properties like: d x x - f /= f dx There is something uniquely special about the unit circle, and about using the unscaled distance around the unit circle as the measure of the angle.
    - kazinator 2y ago
      Radians have the property that if we step x by some tiny amount δ, then the cos/sin coordinates will move by that same distance around the unit circle: |[cos(x+δ) + i sin(x+δ)] - [cos(x) + i sin(x)]| = δ This is also related to how we can estimate sin(x) = x for small values next to zero, if using radians. In radians, the derivative sin'(x) is cos(x), and cos'(x) is -sin(x). Derviation just shifts the waveform left by ninety degrees. In units other than radians, we get wacky constant terms that change at each step. That's related to how e^x is its own derivative.

nine_k 2y ago

We measure angles in radians because it's an easy way to measure the length of the circle sector in the units of the circle's radius. It feels neat in many cases. A more practical way to measure angles would be in rotations. 0° = 0, 360° = 1, 90° = 0.25, etc. It would remove a ton of 2π and 4π² factors from a lot of equations in physics.

mb7733 2y ago
For trigonometry/calculus/physics radians are by far the most practical because they are dimensionless, so no constants appear when differentiating or integrating. (By the way, these constants will involve factors of pi anyway, it's inherent.) For example, try to work out the Taylor series for sin(x) using degrees (or rotations). It's awful.
- nine_k 2y ago
  I don't see how Taylor series specifically would be affected. Differentiation of sin x and cos x is the same independent of the unit of x, and nothing else is used in the series. Fourier transform would have 4π² instead of 2π under the exponent, no big deal. The Euler's formula gets a factor of 2π under the exponent though. Given its wide application, it adds plenty of noise, of course.
  - setopt 2y ago
    If you define variants sint(x) = sin(2πx) and cost(x) = cos(2πx) that takes x in units of turns instead of radians, then d/dx sint(x) = 2π cost(x) etc. I agree with you that this is completely fine though. I also find it more natural to think of “how many percent of a turn” an angle is than how many “degrees” or “radians” something is, since we use base-10 everywhere else. My workaround is to mostly write everything in terms of sin(2πτ), cos(2πτ), and exp(2πiτ) when I can, where τ measures turns.
  - School-Cotton 2y ago
    > Differentiation of sin x and cos x is the same independent of the unit of x, That’s not true. If the unit is degrees, d/dx sin(x) = pi/180 * cos(x).

akira2501 2y ago

> drilling down too much on conventions misses the point of math. What is "the point of math?"

__MatrixMan__ 2y ago
I thought it was about finding conventions that are useful, surprising, or pleasing in some way. Or, given a set of conventions, finding new ways that they are useful, surprising, or pleasing. So I'm curious to know about this other... non-conventional point.
ithkuil 2y ago
Well, the point, in math, is a zero dimensional object.

Aardwolf 2y ago

Other chosen conventions are less great, like not having the circle constant be 6.283185... so that we don't need to deal with factors or divisors of 2 all the time when using radians or almost anything else involving pi 90 degrees is one fourth of a circle, it would be so much more intuitive if we'd use "1/4th of something" rather than "1/2 radians" to express this

pif 2y ago

No! We measure angles in radians because it's the simplest way to link the length of an arc to the radius of the circle.

aidenn0 2y ago
But we have an inconsistency because pi relates the diameter to the circumference, not the radius to the circumference, which leaves an annoying 2 in the 2pi radians in a circle.