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Turns are better than radians

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Re: Turns are better than radians

#291

Post by mech422 »

mathieuh wrote: post #195
js2 wrote: post #183

Earlier quoted context omitted.

I'm a 50 yo programmer. I have a CS degree. I don't even remember my college calculus much less my high school trig. I just haven't had cause to use it in my career, not as a sysadmin, not as a programmer. My son is taking calc 3 and I knew I happened to have my calc 3 notes from the mid-90s, so I pulled them out of the filing cabinet and my very carefully taken notes, my proofs, my hand drawn graphs, it was all gibb…

I'm 27, educated in the UK, all I remember about trigonometry is SOHCAHTOA.

I'm a 50+ year old American, of british decent... I never managed to remember the 'american' mnemomic, but my dad taught me one the used to use in England around WWII: Percy has a bald head, poor boy

Perpendicular/hypotanuse = Sin

Base / Hypotanuse: CoSin

Perpendicular / Base: Tan

edit: try to fix the HN god awful formatting

Re: Turns are better than radians

#292

Post by i_no_can_eat »

adrian_b wrote: post #182
i_no_can_eat wrote: post #83

Earlier quoted context omitted.

> there are no important applications for those series. Excuse me? Have you done any computation in Physics? Have a look at the pendulum equation, for a start...

Only in school exercises you can solve a differential equation by expanding a sine function into a Taylor series. In practical physics computations, the solution of differential equations requires numerical methods that do not use the Taylor series of specific functions, even if the theory used for developing the algorithms may use the Taylor series development of arbitrary functions. For accurate prediction, the sim…

> Only in school exercises you can solve a differential equation by expanding a sine function into a Taylor series.

> In practical physics computations, the solution of differential equations requires numerical methods that do not use the Taylor series of specific functions, even if the theory used for developing the algorithms may use the Taylor series development of arbitrary functions.

I'm sorry, but you have no idea what you're talking about. Series expansions is one of the most widely used techniques in Physics. Obviously some equations require full blown numerical methods to be solved, but one can do a whole lot with analytical techniques by doing series expansions and using perturbation theory.

Saying that this is only used "in school exercises" shows that you're completely out of touch with reality.

Re: Turns are better than radians

#293

Post by shadowgovt »

jefftk wrote: post #122
nurbl wrote: post #13

I agree that this makes sense for the kind of situations that the article talks about. If you only need to express common angles like 90 degrees, 45 and so on, radians are just messy (though in physics, you get used to it). But in other cases, radians are useful. For example consider the case of small deviations from a direction. If you give it in radians, let's say three mrad (milliradians), it's very easy to estima…

To elaborate a bit: that works because sin(x) is very close to x for small x, but only when x is measured in radians.

That's because of the equality relationship between 2π radians and the length of the unit circle perimeter. If one is working with a sine taking in turns, one can just adjust by saying sin(q) ≈ 2π * q for small q.

Re: Turns are better than radians

#294

Post by schindlabua »

cjbgkagh wrote: post #272
js2 wrote: post #183

Earlier quoted context omitted.

I'm a 50 yo programmer. I have a CS degree. I don't even remember my college calculus much less my high school trig. I just haven't had cause to use it in my career, not as a sysadmin, not as a programmer. My son is taking calc 3 and I knew I happened to have my calc 3 notes from the mid-90s, so I pulled them out of the filing cabinet and my very carefully taken notes, my proofs, my hand drawn graphs, it was all gibb…

I’m the opposite, Im 15 years into my career of applied research which for me is like an extension of university. I tend to lean on Mathematica to do my calculus though. I think high school curriculum was optimized to expose a lot of people to things they won’t need on the off chance that a few will end up as researchers of some sort. It would be more efficient to identify such people earlier and split them off. I th…

I think the point of high school is to make kids' brains do work, and what you are learning is secondary.

People love to hate on their school curriculum and all the useless knowledge they had to acquire but I'm positive it makes you a smarter person overall, and the body of high school knowledge makes learning more specialized knowledge easier (even if that's baking bread or whatever)

(People also love to talk about how little they remember from school, yes the brain is a muscle and you stopped working out, congratulations.)

Re: Turns are better than radians

#295

Post by adrian_b »

q-big wrote: post #78
adrian_b wrote: post #63

Earlier quoted context omitted.

The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series. There is only one consequence of those series that matters in practice, which is that when the angles are expressed in radians, for very small angles the angle, its sinus and its tangent are approximately equal. While this relationship between small angles, sinuses and tangents looks like an arg…

> The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series. The addition theorems for trigonometric functions can easily be shown by the multiplication theorem for Taylor series (and adding two Taylor series). This proof would be more convoluted if the Taylor series were not so easy. Also, because of the simplicity of their Taylor series, one immediate…

The proof of any property of the trigonometric functions is trivial when the sine and the cosine are defined as the odd and even parts of the exponential function of an imaginary argument, and the proof uses the properties of exponentiation.

Any proof that uses the expansion in the Taylor series is a serious overkill.

Moreover, those proofs become even a little simpler when the right angle is used as the angle unit, instead of the radian.

In this case, the sine and the cosine can be defined as the odd and even parts of the function i ^ x.

Re: Turns are better than radians

#296

Post by nkrisc »

WastingMyTime89 wrote: post #67
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

The writer don’t seem to realise that radian is not an arbitrary unit but a dimensionless one which is defined so that 1rad is actually just 1. Reading the submission and the comments here, I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA. While I’m slightly envious you might not have to suffer developing powers of cosine and sine but that would explai…

Use it or lose it. Most people have no reason to need knowledge of trigonometry, so even if they’re taught it they quickly forget it.

I never really learned trigonometry until I started doing game programming in my spare time when suddenly that knowledge and linear algebra became necessary to understand. They only way I learned it was by needing to know it.

In fact, I regularly forget knowledge I don’t need to know. The stuff I do need to know remains fresh in my mind.

Re: Turns are better than radians

#297

Post by p_j_w »

mcv wrote: post #221
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

> > But math never decreed that sine and cosine have to take radian arguments! > Ummm, actually it did. No, it didn't. Some specific uses looking better with radians does not mean you have to use radians always. When I first learned sine and cosine, we used degrees, and that worked fine. Later we switched to radians, but there's no reason why you shouldn't use turns, and the article gives a very good argument why in…

>Some specific uses looking better with radians does not mean you have to use radians always.

It's not just some specific use cases, it's the majority of cases if you look across all of math and science. Switching to turns would be stupid, especially once you start doing differentiation and integration. The fact that we use radians almost across the board isn't some accident.

Re: Turns are better than radians

#298

Post by oddthink »

vrc wrote: post #133
shultays wrote: post #33

This is like complaining about g in F = g * m, in gravitional force formula, and redefining gravitional constant g as 1 and multiplying "gram" by 1/9.8. Sure it simplifies things for you but you are breaking everything else that used g constant Or we can redefine seconds perhaps and multiply it sqrt(1/9.8)

See Gaussian-CGS v. SI units [0]. It really confused me going from a CGS physics class to an EE electrodynamics course! [0] https://en.wikipedia.org/wiki/Gaussian_units

Oh yes. This is definitely a pet peeve of mine. CGS is so much nicer. I did E&M from Jackson before he converted it to MKS, and I still can't keep all those epsilon_0 and mu_0's straight. (Not that it comes up all that much.)

Re: Turns are better than radians

#299

Post by NohatCoder »

doliveira wrote: post #52
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

sin(x) ~~ x only in radians, so honestly that's reason enough. Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighte…

What really bothers me is that mathematicians seemingly never distinguish between doing and presenting mathematics.

You can do your own scribbles with single letters, so do I, it works fine.

But when you present maths in a scientific article, maths book, Wikipedia article or similar, your convenience as a writer should be secondary. Your task is to present information to someone who does not already know the subject. Presenting an equation as six different Greek letters mashed together means that the equation itself convey almost no information. You need a wall of text to make sense of it anyway.

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