[RicoElectrico]

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.

[nine_k]

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]

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]

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.

[hansvm]

> Differentiation of sin x and cos x is the same independent of the unit of x, and nothing else is used in the series.

Implicit in that statement is the use of the series definition of sine, which is a "meaningful" or "natural" definition insofar as it represents some function we care about. I'll address at the end what happens if we assume that definition regardless of its independent plausibility, but first consider:

The semantic meaning of sine, at least from its historical roots and how you might independently uncover it from earlier fields like geometry instead of later fields like differential equations, is that given an angle (in some units, we'll touch on that in a moment) we'd like to know the ratio two sides of a particular triangle associated with that angle inscribed in a circle. Given a choice of units for the angle, the triangle is fixed, and so the result (that ratio of side lengths) is also fixed.

Suppose you want to know how that ratio varies with respect to the angle. You can imagine a change of coordinates `y = cx` and consider the derivative of `sin(x)` vs `sin(y/c)`. The latter will have a numeric value `1/c` times less than the former. E.g., imagine a whole circle represented `1` angle instead of `2pi`. Then converting from our normal radians baseline to that new unit you have `y = (1/2pi)x`, and the derivative of the semantic ratio we're considering with respect to the new measure of angle is multiplicatively `2pi` greater than the original.

Going back to your series definition, suppose we pick that series as the definition of sine, independent of units. The problem that arises is that particular uses of sine do have units, and converting from the problem you care about to your particular from-on-high chosen definition of sine will run into the exact sort of problem our `y = cx` paragraph above touched on. The derivative with respect to the quantity of interest still has an extra `1/c` factor, and the fact that our God-blessed choice of sine is independent of units didn't actually solve anything in the composite problem.

[setopt]

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.

[umanwizard]

> 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).

[uoaei]

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.).

[pif]

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]

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.

[Aardwolf]

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

[programjames]

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

[cperciva]

No, we measure angles in radians so that e^(ix) = cos x + i sin x.

[nine_k]

But this does not depend in the unit.

[JadeNB]

> 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]

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]

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.

[akira2501]

> drilling down too much on conventions misses the point of math.

What is "the point of math?"

[__MatrixMan__]

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]

Well, the point, in math, is a zero dimensional object.