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Turns are better than radians

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Re: Turns are better than radians

#141

Post by jacksnipe »

mjw_byrne wrote: post #93
Double_a_92 wrote: post #34

Earlier quoted context omitted.

Why do they need to be at multiple integers of pi, instead of at half turns?

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.

Re: Turns are better than radians

#142

Post by [deleted] »

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…

[deleted]

Re: Turns are better than radians

#143

Post by mjw_byrne »

red_trumpet wrote: post #105
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…

> or the imaginary and real parts of e^ix For that particular application, x is exactly the argument[1] of your complex number, though! [1] https://en.wikipedia.org/wiki/Argument_(complex_analysis)

True, however that doesn't mean x represents an angle. It means you can put a geometric interpretation on an abstract formula. e^ix = cos x + i sin x regardless of what x represents - if you were doing electrical engineering it might be time, for example.

Interpretation is often strongly motivated by what "comes first" in the order in which you learn things, so the whole debate is a bit subjective anyway. Another common example: at school we learn the integral is the area under the curve. At university we learn the area under the curve is the integral. The integral is the "real" thing and the area is just a convenient geometric interpretation, which actually makes no sense for many (most?) integrals. At school we learn it "backwards" purely because it's easier that way, and visual aids are helpful. I think something similar applies to sin and cos.

Re: Turns are better than radians

#144

Post by bluGill »

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…

Trig is generally called per-calculus in US high schools. It is not a required course, but it is one of the courses everyone on the college track is expected to take.

Though most people haven't used any of that since college and so don't know it very well anymore. I smelled BS when I read the blog, but couldn't put my finger on why - the comment you replied to explained what I knew was the case but couldn't remember.

Re: Turns are better than radians

#145

Post by RunSet »

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…

> 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 = "go"; g2 = "great"; i = "it"; i2 = "i"; m = "me"; s = "see"; w = "works"; w2 = "what"; w3 = "wrong"

i w g2 f m; i2 d s w2 c g1 w3.

Re: Turns are better than radians

#147

Post by weinzierl »

genewitch wrote: post #110
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 was a bad student through 8th grade, but managed to get selected for a STEM magnet school. I was supposed to enter 9th grade with Geometry, then algebra II, trig, Calc for the 4 years. But they discovered i'd never passed algebra prior, they put me in algebra, which means i would have finished in trig. Due to a crazy 3.5 years, i never got a high school math education. Calculus makes my eyes glaze over, and all i k…

"[..] and then C never made sense to me"

I had my fair share of higher math, but C never made really sense to me either. It's not us, it's C that's to blame.

Re: Turns are better than radians

#148

Post by doliveira »

Olreich wrote: post #130
doliveira wrote: post #88

Earlier quoted context omitted.

Try to solve the Schrodinger Equation for even an infinite well using long variable names. I'm not talking about using it in code, I'm talking about someone arguing that books and articles should do it as well.

If you go watch math lectures, there's a bunch of "x means Puppy Constant" or, "let's substitute in k for the Real component", or "let's signify by collecting these terms into a variable". My argument wouldn't be to replace ALL the variables with meaningful names, just the ones with a lot of meaning that a reader might not understand. It'd also be great if constants, variables, and functions all got naming convention…

Again, try to solve the Schrodinger Equation for even an infinite well using long variable names.

Re: Turns are better than radians

#149

Post by bluGill »

ttoinou wrote: post #72
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…

Correct, radians are a "fake" unit made up to understand better formulas (the same way we use types in programming languages)

While it is a fake unit, it was made to make the math easy. You could call the origin of everything the place where I'm standing - but good luck calculating a path for the mars rovers to travel if I happen to walk to the bathroom.

Re: Turns are better than radians

#150

Post by scambier »

otikik wrote: post #117

Fun fact: PICO-8 only provides turn-based trigonometric functions https://pico-8.fandom.com/wiki/Sin > PICO-8 uses an input range of 0.0 to 1.0 to represent the angle, a percentage of the unit circle. Some refer to these units as "turns". For instance, 180° or π (3.14159) radians corresponds to 0.5 turns in PICO-8's representation of angles. In fact, for fans of τ (tau), it's just a matter of dropping τ from your exp…

Here are the algorithms for PICO-8's sin(), cos() and atan2(), as implemented by the Picolove emulator: https://github.com/picolove/picolove/blob/4b79a3b7ebd1d1bc6c...

Also, since PICO-8 coordinates have their origin in the top-left corner (y goes down), sin()'s result is also inverted.

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