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Turns are better than radians

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Re: Turns are better than radians

#161

Post by mjw_byrne »

jacksnipe wrote: post #141
mjw_byrne wrote: post #93

Earlier quoted context omitted.

The main thing to realise is that sin and cos are not fundamentally tools for doing geometry. The fact that you can use them for working out side lengths of triangles or converting polar to cartesian coordinates is somewhat incidental. It doesn't help that at school our first look at sin and cos is all about adjacent sides and opposite sides in right-angled triangles. It's understandable, because jumping straight int…

Well this isn’t very fair. Yes, triangles have very little to do with the true nature of sin and cos. It’s also true that they are the basic building blocks of cyclicity. But to say they are not geometric tools is dishonest. They instead show us that geometry is deeply connected to many other, sometimes-surprising, areas of mathematics.

"triangles have very little to do with the true nature of sin and cos" <- this is what I mean.

Re: Turns are better than radians

#162

Post by chucksmash »

RunSet wrote: post #145
doliveira wrote: post #52

Earlier quoted context omitted.

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…

> 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. That could never work. If anything the words comprising mathematical texts should be defined once and thereafter truncated to their first letter to reduce cognitive burden and facilitate greater comprehension. c = "could"; d = "don't"; f = "for"; g1 = "g…

Reference Error: s is not defined.

Re: Turns are better than radians

#163

Post by shoxidizer »

moffkalast wrote: post #136
WastingMyTime89 wrote: post #67

Earlier quoted context omitted.

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…

Well it's not exactly surprising, the US is fundamentally built on arbitrary baseless measurement units so getting out of that mindset is probably difficult. A unit that could be inherently defined by math itself and not a farmer looking at their hands and feet? Preposterous!

> A unit that could be inherently defined by math itself and not a farmer looking at their hands and feet?

Where do you think base ten comes from?

Re: Turns are better than radians

#165

Post by happimess »

mihaaly wrote: post #151

This constant need to redefine the known world around a favourite detail amazes me. Are people that bored?! (also I itch hearing the idea of redefining interface - and the world - to fit the implementation detail. how about reimplementing using the [0...0.7854] domain instead of the [0...1] if this is such a huge worry after decades of computing - on slower machines - with the natural radian (arc_length/radius) value…

I think that the author is speaking about their world, in which they regularly encounter a specific use of trigonometric functions which would be simplified (conceptually and computationally) by skipping the conversion from and into radians.

> redefine the known world around a favourite detail

I think that's a good way to think about software optimization. Deep inside nested loops of a game engine (TFA's example code comes from Godot), that's often what you need to do to squeeze some performance characteristic into your hardware.

Re: Turns are better than radians

#166

Post by panda-giddiness »

As already mentioned by others, radians are not arbitrary units for angles; in fact, they are the "natural" "units", so to speak.

By definition, an angle is just the ratio of a circular arc (s) to its radius (r), θ = s/r (as an exercise, imagine how to apply this definition to the angle between two intersecting lines). When the length of the circular arc equals its radius (s = r), the angle subtended is exactly 1 radian; of course, since this is just a ratio, 1 radian is exactly the same as 1 numerically, which is why I put "unit" in quotes earlier -- a radian is not really a unit at all!

A degree, in contrast, equals pi / 180 radians. Of course, since 1 radian = 1, that really just means that 1 deg = pi / 180, similar to how 1%=0.01. Putting this all together, it is perfectly parsable (although not recommended) to say that a $5 burger costs roughly $29000% deg.

Re: Turns are better than radians

#167

Post by bluGill »

the_other wrote: post #48

I'm interested in this from the perspective of learning maths, rather than writing code (for now). I've wondered for a few years now whether teaching angles and trig using turns, rather than degrees or radians, would be better from the very beginning. Degrees are arbitrary and based on the numeric preferences of a dead culture, rather than on what's happening on the page or in 3d space. Radians seem better because th…

Radians makes the most sense just because (as others have pointed out) pi appears all over in math, and in all those places if you are working with a unit of raidans the math becomes a lot easier. In a few cases turns are easier to work with, but teaching them is a dead end to the vast majority of students who will never work in those domains, and even the few students who work in a domain where turns make sense they will still spend time in other areas of math that works much better in radians.

Re: Turns are better than radians

#168

Post by adrian_b »

computerfriend wrote: post #135
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…

> there are no important applications for those series. I cannot believe I just read this.

When have you ever used the Taylor series of sine and cosine for anything (outside school) ?

When you approximate functions by polynomials, including the trigonometric functions, the Taylor series are never used, because they are inefficient (too much computation for a given error). Other kinds of polynomials are used for function approximations.

The Taylor series are a tool used in some symbolic computations, e.g. for symbolic derivation or symbolic integration, but even in that case it is extremely unlikely for the Taylor series of the trigonometric functions to be ever used. What may be used are the derivative formulas for trigonometric functions, in order to expand an input function into its Taylor series.

The Taylor series of arbitrary functions (more precisely, the first few terms) may be used in the conception of various numeric algorithms, but here there are also no opportunities to need the Taylor series of specific functions, like the trigonometric functions.

The Taylor series obviously have uses, but the specific Taylor series for the trigonometric functions do not have practical applications, even if they are interesting in mathematical theory.

Re: Turns are better than radians

#169

Post by gus_massa »

bmacho wrote: post #56
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 sine and cosine that are defined with Taylor series are not the same sine and cosine that are defined for right triangles. The former are R->R functions, while the latter are defined on Angles (Angle is unfortunately not an SI physical dimension yet, but I expect it soon to change), and they don't care about the measurement unit. I have no idea what you mean by radians generalizing for higher dimensions, but not…

In 2D you can measure the solid angles using steradians.

I guess that turns interpreted as parts of whole circles generalize to parts of whole spheres, and you should divide by 4pi instead of 2pi???

Re: Turns are better than radians

#170

Post by rockostrich »

weberer wrote: post #119
WastingMyTime89 wrote: post #67

Earlier quoted context omitted.

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…

>I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA Basic trig is taught in middle school, but exclusively using degrees. Advanced trig is optional in high school if you take the "hard math" track.

> Advanced trig is optional in high school if you take the "hard math" track.

This depends on the state. NYS absolutely requires everyone to learn "advanced trig" in high school.

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