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I needed a refresher on trig last year for some robotics simulations I was working on. The trig lesson was extremely helpful: http://betterexplained.com/article

by hugs 11y ago

I needed a refresher on trig last year for some robotics simulations I was working on. The trig lesson was extremely helpful: http://betterexplained.com/articles/intuitive-trigonometry/ http://betterexplained.com/articles/intuitive-trigonometry/ And fwiw, I finally learned an intuitive understanding of radians by reading the Tau Manifesto (not on BetterExplained). It would be awesome if BetterExplained used tau instead of pi in the lesson on radians, but that's a minor nitpick for a very helpful set of lessons.

jeffwass 11y ago
Out of curiosity do you get confused when people talk about 360 degrees, or 90 degree angles, or turning around 180 degrees, etc? Most people I know intuitively understand degrees and could easily understand radians the same. Why should BetterExplained use tau? Tau is a fairly niche idea that will only confuse way more than explain. That's a bit like saying BetterExplained should be written in Esperanto. Personally while I agree that historically pi would have been better off if defined as 6.28... instead of 3.14... (And even told my students this long before tau manifesto came out) I don't feel that a factor of 2 warrants using up a whole other Greek letter. (I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles).
- jerf 11y ago
  The anecdata that tau is helpful to learners is, IMHO, pretty strong at this point. Facts beat theory. And since anecdata is all we have, well, it's all we can use to talk about. I think it's way, way too easy for people on the terminal end of education to forget just how easy it is to get tripped up by very, very little things. Especially if they were themselves "good at math" and really aren't bothered by extra factors of 2 flying around. This is not the normal experience.
  - jeffwass 11y ago
    When someone says 90 degree angle, does that confuse you?
    - jerf 11y ago
      As it happens, I've tutored people in the middle of high school geometry. Radians confuse them. Full stop. If there is something that can make it easier to understand, while at the same time sacrificing absolutely nothing of mathematical consequence, it's a good thing. What confuses me is not the relevant issue. I've got $BOATLOADS of higher ed. I'm not the interesting case.
      - jeffwass 11y ago
        Ok, we're far away from my original point which was only on using tau in one website. My actual only personal argument against Tao as Hartl proposed it is the use of a standard Greek letter. Really would have preferred he picked something that was more "backwards compatible", exactly as physicists did with hbar. If you're for reducing confusion, do you think adding a new fundamental constant to the body of mathematics that is a Greek letter already used for countless other variables in history, will this cause confusion between old and new mathematical science texts and papers? Asking students of science for any texts and papers they read whether tau is 2pi or some other variable, and keeping track across them, seems more confusing to me than just consistently using pi and extra factors of 2.
        hugs 11y ago
        You have a valid complaint about reusing a letter that already had meaning in other contexts. He does address that in the manifesto, though.
    - hugs 11y ago
      "90 degrees" is not confusing to me in the same way that measurements in feet and miles is not confusing. I was raised in a system that emphasized degrees, feet, and miles (USA). But in engineering, metric is the standard/default system of measurement. And in software libraries (e.g. JavaScript's Math built-in lib) or many calculators, the default is radians, not degrees. So even though degrees is more natural for me, it was worthwhile to get a deeper understanding of radians. The Tau Manifesto got me there. (90 degrees is 1/4 turn around a circle, which happens to be expressed as 1/4*tau -- and I didn't even have to look that up!)
  - coldtea 11y ago
    >Facts beat theory. Not necessarily. Facts (in the sense of "anecdata") are simple observations. The only make any sense against or in favor of something when we combine them with a theory why it is so. OK, students faring better on exams when they are taught with tau is an argument in favor of tau only if we theorize (assume) that it was the tau/pi thing that made the difference. Whereas it could just be that the second group of students was just better naturally (e.g. a good school vs a mediocre one), or just a chance outcome, due to the small sample we picked, or the teacher going for tau was also better at explaining and would have fared just as good if he also was the one to teach the pi group.
    - jerf 11y ago
      It is one of the great errors of the "SCIENCE!" attitude that it convinces people that if we do not have peer-reviewed papers, we must necessarily claim full and total ignorance and berate anybody who attempts to do otherwise. I'm not obligated to stop using my brain just because we have no SCIENCE! papers to work with. In fact, you can't. You have no choice. You must do your best with the data you have, because we do not have "science" for all the questions we encounter every day. Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching.
      - jkire 11y ago
        > It is one of the great errors of the "SCIENCE!" attitude that it convinces people that if we do not have peer-reviewed papers, we must necessarily claim full and total ignorance and berate anybody who attempts to do otherwise. In fact, I think quite the opposite. Its one of the great errors that people assume peer-reviewed papers are always correct, a lot of the time they're not. We should be sceptical of people's claims when not supported by evidence. That doesn't mean we should ignore said claims, just that we shouldn't blindly accept them. This is the point where we should go and get some evidence, try to get a few teachers to use tau from the beginning and see if that helps. Though I do agree that we shouldn't be scared of doing things just because we don't categorically know that it is right. > Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching. Or thinking about the same concept in different ways help you to understand the concept better. Its certainly something I believe has helped me a lot previously.
  - mrcsparker 11y ago
    I just read the Tao Manifesto and - while I was skeptical at first - I have to agree that it is much easier. Note that this is just my opinion, but I have kids and this gives me an alternative viewpoint when they have questions. I wonder what my more math oriented friends will think about Tau
    - jeffwass 11y ago
      Of course the right constant historically should have been 6.28... And not 3.14... Nobody's disputing that (or at least I'm not). The question is whether you improve things by abandoning a universal standard with hundreds of years of support, momentum, and ginormous corpus of math/science literature all to save a 'simple' factor of two. My primary gripe with Tau is that its symbolic representation is not "backwards compatible". Ie, Hartl proposed to use yet another Greek letter which is widely used elsewhere in all sorts of formulae and constants, instead of something that would cause far less ambiguity. For example, a new symbol, as physicists did by introducing hbar instead of h (Planck constant in radians instead of cycles).
- shmageggy 11y ago
  Because tau is more intuitive and this is all about intuition. If you were baking a pie (pi?) and someone told you how much of an ingredient to add per half pie, that would be less intuitive. And no, I do not get confused by concepts like 90 degrees, but this is about inspiring intuition, not the ability to do calculations without error.
- jameshart 11y ago
  You can't just baldly assert that radians are just as easy to understand as degrees - clearly they require more effort to understand, from an intuitive point of view. It seems obvious to me why degrees are easier to handle than radians: 180 is an easier number to understand than pi. People don't instinctively understand why on earth they would use a measurement unit where '1' of the unit has no obvious purpose. What can you do with a 1 radian angle? It's a bit less than the angle in an equilateral triangle. It's too big for usefully measuring things - most angles you come across will be somewhere between 0 and 3 radians, which is just weird. What's that, you don't use radians like that? You use 'fractions of pi' radians? But my calculator can only approximate pi to 9 decimal places, so how does that help me? It's incredibly easy for people to intuit that a ninety degree angle is the same as a 30 degree angle plus a 60 degree angle. But it takes deeper intuition about fractions to figure out that a pi/2 radian angle is the same as a pi/6 radian angle plus a pi/3 radian angle. People are not used to units of measure that are typically denominated with rational fractions of irrational numbers.
  - jeffwass 11y ago
    Every argument on radians you presented applies equally well to Tau as Pi. My original point is that IMHO, mainstream radian educational materials should NOT use the lesser-known and niche 'tau' in lieu of the globally understood 'pi' (other than a possible footnote or sidebar). Ie - it can cause more confusion to the student, especially when connecting it back to the greater scope of their studies.
    - jameshart 11y ago
      I wasn't addressing your tau/pi issue, but rather your assertion that if you understand '90 degrees' that you should be able to understand radians. That seems like a case of 'well I find it simple, why doesn't everyone else?'
- plaguuuuuu 11y ago
  >(I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles). I suggest using pi, but with the number 2 before it. /s
ebbv 11y ago
No they should not use Tau, because the point of education is to give you knowledge that you can build on later. Nothing else uses Tau, so teaching you Tau does not prepare you for further education. I'm not even gonna touch the Tau vs. Pi debate because it's like a sports team rivalry, and not a rational discussion.
- TuringTest 11y ago
  > Nothing else uses Tau, so teaching you Tau does not prepare you for further education. Nitpicking, but that is a self-defeating argument. If everybody used Tau, there would be no problem in teaching it. Not that I disagree that doing the bootstrapping its usage and reaching there would be a problem, but all by itself is not a reason to not try changing it.
kalid 11y ago
Hey, thanks for the feedback! Glad to hear the trig article helped (one of my favorite aha moments -- all the trig functions, including stuff like the exotic cosecant, can be drawn together.) I'm a fan of Tau as an interesting simplifying concept but don't think mentions of pi should necessary be removed. My intuition is that radians are "distance traveled" as you go along a circle. Pi is the "neutral to max to neutral" distance and Tau is the full cycle (neutral, max, neutral, min, neutral). Ideally we just think about "distance" and context determines whether we want the journey to be "there and back" or just "there".
jgamman 11y ago
i (mostly) agree. i think thinking about a fraction of a full circle is the easiest way to work with trig. having said that, the mental overload on your average high school student flipping between pi and tau is too much. my personal compromise is to never cancel out the '2' and always work with 2pi. it makes some equations look a bit odd from the textbook but whacks me in the forehead to remember to think about circles...

by hugs 11y ago

jeffwass 11y ago

Out of curiosity do you get confused when people talk about 360 degrees, or 90 degree angles, or turning around 180 degrees, etc? Most people I know intuitively understand degrees and could easily understand radians the same. Why should BetterExplained use tau? Tau is a fairly niche idea that will only confuse way more than explain. That's a bit like saying BetterExplained should be written in Esperanto. Personally while I agree that historically pi would have been better off if defined as 6.28... instead of 3.14... (And even told my students this long before tau manifesto came out) I don't feel that a factor of 2 warrants using up a whole other Greek letter. (I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles).

jerf 11y ago
The anecdata that tau is helpful to learners is, IMHO, pretty strong at this point. Facts beat theory. And since anecdata is all we have, well, it's all we can use to talk about. I think it's way, way too easy for people on the terminal end of education to forget just how easy it is to get tripped up by very, very little things. Especially if they were themselves "good at math" and really aren't bothered by extra factors of 2 flying around. This is not the normal experience.
- jeffwass 11y ago
  When someone says 90 degree angle, does that confuse you?
  - jerf 11y ago
    As it happens, I've tutored people in the middle of high school geometry. Radians confuse them. Full stop. If there is something that can make it easier to understand, while at the same time sacrificing absolutely nothing of mathematical consequence, it's a good thing. What confuses me is not the relevant issue. I've got $BOATLOADS of higher ed. I'm not the interesting case.
    - jeffwass 11y ago
      Ok, we're far away from my original point which was only on using tau in one website. My actual only personal argument against Tao as Hartl proposed it is the use of a standard Greek letter. Really would have preferred he picked something that was more "backwards compatible", exactly as physicists did with hbar. If you're for reducing confusion, do you think adding a new fundamental constant to the body of mathematics that is a Greek letter already used for countless other variables in history, will this cause confusion between old and new mathematical science texts and papers? Asking students of science for any texts and papers they read whether tau is 2pi or some other variable, and keeping track across them, seems more confusing to me than just consistently using pi and extra factors of 2.
      - hugs 11y ago
        You have a valid complaint about reusing a letter that already had meaning in other contexts. He does address that in the manifesto, though.
  - hugs 11y ago
    "90 degrees" is not confusing to me in the same way that measurements in feet and miles is not confusing. I was raised in a system that emphasized degrees, feet, and miles (USA). But in engineering, metric is the standard/default system of measurement. And in software libraries (e.g. JavaScript's Math built-in lib) or many calculators, the default is radians, not degrees. So even though degrees is more natural for me, it was worthwhile to get a deeper understanding of radians. The Tau Manifesto got me there. (90 degrees is 1/4 turn around a circle, which happens to be expressed as 1/4*tau -- and I didn't even have to look that up!)
- coldtea 11y ago
  >Facts beat theory. Not necessarily. Facts (in the sense of "anecdata") are simple observations. The only make any sense against or in favor of something when we combine them with a theory why it is so. OK, students faring better on exams when they are taught with tau is an argument in favor of tau only if we theorize (assume) that it was the tau/pi thing that made the difference. Whereas it could just be that the second group of students was just better naturally (e.g. a good school vs a mediocre one), or just a chance outcome, due to the small sample we picked, or the teacher going for tau was also better at explaining and would have fared just as good if he also was the one to teach the pi group.
  - jerf 11y ago
    It is one of the great errors of the "SCIENCE!" attitude that it convinces people that if we do not have peer-reviewed papers, we must necessarily claim full and total ignorance and berate anybody who attempts to do otherwise. I'm not obligated to stop using my brain just because we have no SCIENCE! papers to work with. In fact, you can't. You have no choice. You must do your best with the data you have, because we do not have "science" for all the questions we encounter every day. Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching.
    - jkire 11y ago
      > It is one of the great errors of the "SCIENCE!" attitude that it convinces people that if we do not have peer-reviewed papers, we must necessarily claim full and total ignorance and berate anybody who attempts to do otherwise. In fact, I think quite the opposite. Its one of the great errors that people assume peer-reviewed papers are always correct, a lot of the time they're not. We should be sceptical of people's claims when not supported by evidence. That doesn't mean we should ignore said claims, just that we shouldn't blindly accept them. This is the point where we should go and get some evidence, try to get a few teachers to use tau from the beginning and see if that helps. Though I do agree that we shouldn't be scared of doing things just because we don't categorically know that it is right. > Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching. Or thinking about the same concept in different ways help you to understand the concept better. Its certainly something I believe has helped me a lot previously.
- mrcsparker 11y ago
  I just read the Tao Manifesto and - while I was skeptical at first - I have to agree that it is much easier. Note that this is just my opinion, but I have kids and this gives me an alternative viewpoint when they have questions. I wonder what my more math oriented friends will think about Tau
  - jeffwass 11y ago
    Of course the right constant historically should have been 6.28... And not 3.14... Nobody's disputing that (or at least I'm not). The question is whether you improve things by abandoning a universal standard with hundreds of years of support, momentum, and ginormous corpus of math/science literature all to save a 'simple' factor of two. My primary gripe with Tau is that its symbolic representation is not "backwards compatible". Ie, Hartl proposed to use yet another Greek letter which is widely used elsewhere in all sorts of formulae and constants, instead of something that would cause far less ambiguity. For example, a new symbol, as physicists did by introducing hbar instead of h (Planck constant in radians instead of cycles).
shmageggy 11y ago
Because tau is more intuitive and this is all about intuition. If you were baking a pie (pi?) and someone told you how much of an ingredient to add per half pie, that would be less intuitive. And no, I do not get confused by concepts like 90 degrees, but this is about inspiring intuition, not the ability to do calculations without error.
jameshart 11y ago
You can't just baldly assert that radians are just as easy to understand as degrees - clearly they require more effort to understand, from an intuitive point of view. It seems obvious to me why degrees are easier to handle than radians: 180 is an easier number to understand than pi. People don't instinctively understand why on earth they would use a measurement unit where '1' of the unit has no obvious purpose. What can you do with a 1 radian angle? It's a bit less than the angle in an equilateral triangle. It's too big for usefully measuring things - most angles you come across will be somewhere between 0 and 3 radians, which is just weird. What's that, you don't use radians like that? You use 'fractions of pi' radians? But my calculator can only approximate pi to 9 decimal places, so how does that help me? It's incredibly easy for people to intuit that a ninety degree angle is the same as a 30 degree angle plus a 60 degree angle. But it takes deeper intuition about fractions to figure out that a pi/2 radian angle is the same as a pi/6 radian angle plus a pi/3 radian angle. People are not used to units of measure that are typically denominated with rational fractions of irrational numbers.
- jeffwass 11y ago
  Every argument on radians you presented applies equally well to Tau as Pi. My original point is that IMHO, mainstream radian educational materials should NOT use the lesser-known and niche 'tau' in lieu of the globally understood 'pi' (other than a possible footnote or sidebar). Ie - it can cause more confusion to the student, especially when connecting it back to the greater scope of their studies.
  - jameshart 11y ago
    I wasn't addressing your tau/pi issue, but rather your assertion that if you understand '90 degrees' that you should be able to understand radians. That seems like a case of 'well I find it simple, why doesn't everyone else?'
plaguuuuuu 11y ago
>(I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles). I suggest using pi, but with the number 2 before it. /s

ebbv 11y ago

No they should not use Tau, because the point of education is to give you knowledge that you can build on later. Nothing else uses Tau, so teaching you Tau does not prepare you for further education. I'm not even gonna touch the Tau vs. Pi debate because it's like a sports team rivalry, and not a rational discussion.

TuringTest 11y ago
> Nothing else uses Tau, so teaching you Tau does not prepare you for further education. Nitpicking, but that is a self-defeating argument. If everybody used Tau, there would be no problem in teaching it. Not that I disagree that doing the bootstrapping its usage and reaching there would be a problem, but all by itself is not a reason to not try changing it.

kalid 11y ago

Hey, thanks for the feedback! Glad to hear the trig article helped (one of my favorite aha moments -- all the trig functions, including stuff like the exotic cosecant, can be drawn together.) I'm a fan of Tau as an interesting simplifying concept but don't think mentions of pi should necessary be removed. My intuition is that radians are "distance traveled" as you go along a circle. Pi is the "neutral to max to neutral" distance and Tau is the full cycle (neutral, max, neutral, min, neutral). Ideally we just think about "distance" and context determines whether we want the journey to be "there and back" or just "there".

jgamman 11y ago

i (mostly) agree. i think thinking about a fraction of a full circle is the easiest way to work with trig. having said that, the mental overload on your average high school student flipping between pi and tau is too much. my personal compromise is to never cancel out the '2' and always work with 2pi. it makes some equations look a bit odd from the textbook but whacks me in the forehead to remember to think about circles...