An Alternative Introduction to Trigonometry
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- Опубліковано 24 тра 2024
- This video is an alternative introduction to trigonometry that uses oscillations, not triangles, as its starting point. In my experience, thinking of trigonometry in terms of oscillations makes much more sense than using triangles. Furthermore, it shows us that one of our fundamental mathematical constants is not what you think it is...
Happy τ day! Here's a link to the τ manifesto: tauday.com/tau-manifesto
To actually calculate the sine and the cosine, you need some calculus. Here is how we do this. In the video, we said that the magnitude of the spring's acceleration is equal to the magnitude of its displacement, and the acceleration points in the opposite direction of the displacement. This leads to the differential equation y'' = -y. Then the sine is the solution to this equation with the initial conditions y(0) = 0 and y'(0) = 1. The cosine is the solution to this equation with the initial conditions y(0) = 1 and y'(0) = 0. From this description, there are a few options for how to solve this equation:
1. You can use generic numerical analysis techniques. Most of the simple differential equation algorithms aren't that accurate though.
2. From y'' = -y, we in general have y^(n) = -y^(n - 2). We can use this to determine the value of the (co)sine and all of its derivatives at 0, and then use this to find the Taylor series around zero. For example, sin(0) = 0, sin'(0) = 1, sin''(0) = 0, sin'''(0) = -1, sin^(4)(0) = 0, sin^(5)(0) = 1, ... This leads to the Taylor series sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ... Around zero, this Taylor series can be pretty accurate with only a few terms, and then because the sine and cosine are periodic we can add or subtract multiples of τ to get the input close enough to zero for the accuracy to be high.
3. If you want to learn more about this subject, you can look up various algorithms such as CORDIC.
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Sections
00:00 Introduction
00:59 Oscillations
01:37 Basics of Oscillations
03:55 Spring Simulation
07:46 Cosine and Sine
10:59 Circles
12:20 Radians
13:34 Triangles
14:33 The (obvious) Twist
16:38 τ＞π
Honestly, the unit circle is all we need as a start point here.
Yeah agreed. Unit circle, then right triangles, then non right triangle trig.
Happy Tau Day!
I somehow want this comment to get to 628 likes, and then stay there. :D
_“... thinking of trigonometry in terms of triangles and angles is only telling one side of the story...”_
So I guess we'll need to explore the other two sides? 😁
why did i laugh
Because it’s funny
I am wholly appreciative of your videos. I love you. This exact niche of "Applied Mathematics" in Physics / Engineering is severely lacking in schools and media, that's why thank you for being part of this niche. Thank you for your efforts.
Really like this approach, especially making tau the fundamental entity.
Also appreciate your Geometric Algebra series. That's a mind blower which, to me, corrects some other bad choices.
20 minutes ago I hadn’t heard about tau. Now I want to burn pi at the stake. Thanks for the insight!
The tau v. pi "debate" is more entertaining than probably any sports rivalry, atleast imo.
It's not much of a debate. π is a historical accident.
This is the best introduction to trigonometry I've ever seen. The alternative method starting with oscillations is genius, and perhaps more relevant than the standard way using triangles first. And of course, I love just how naturally τ shows up everywhere.
Already liked because I know it'll be good
2
same
After a bunch of videos on geometric algebra, I sure am excited for a change of pace!
Woah, this is SUCH an underrated channel! As a fellow educational UA-camr, I understand how much work must have gone into this- amazing job!! Liked and subscribed :)
I love these types of videos! Not necessarily for overturning conventional approaches, but rather to tackle the topic from a different direction. You mention this is only one side of the story, and indeed some of the ideas here are better shown from the triangles perspective (for example I think the existence of sine AND cosine is better justified using triangles in circles), while others are better shown in this oscillation perspective (for example I think it shows perfectly the relevance of tau and pi in simple harmonic motion, instead of showing them as magic constants or results of memorized definitions). By exploring more sides of the story, we're better able to identify what does and doesn't make sense to teach, resulting in more autonomous expressions of ideas.
A call to action for the new generation of students. Join the Tau movement today!
Thank you and happy tau day!
Great use of Tau instead of Pi!
This was amazing thank you very much
I woke up early just to watch this. Greetings from Europe!
for dirac's belt trick logic, it'd be more reasonable to use 4pi as the "important constant" so really it's arbitrary which one should be used more
Isn’t that usually thought of as “two full rotations”? So it would be more reasonable to just call it 2tau
@@jakobr_ but it's actually one full rotation in this context, we just think of 360° or 2π or tau when defined in one of the standard ways as a full rotation when it's not a full rotation in other cases as it doesn't get back to the initial state. And the choice of which context to start in is arbitrary so choosing tau, π, or 4π or any other constant is just as arbitrary. The math will all work out fine and logically arise from your original context beautifully.
Time to check out the Geometric Algebra notation developed by David Hestenes. See ROTORS and SPINORS.
Also thank you for starting trig like this!
TAU GANG
Awesome!!
Oh hey it's tau day!
Pi probably came around because it was easier for ancient people to measure pipes’ diameters than radii. (No guessimations of where the center is would be needed.)
The old ways are sometimes not ideal. Electrons should have a charge of +3, not -1. Astrology?!? List goes on...
When you think about it, an even more natural measurement of angle is the full turn: that way, pretty much all important angles are rational.
And that's kind of what we're doing when we use degrees. AIUI, historically degrees (and minutes and seconds) weren't specifically measurements of angle, but a way of expressing fractions, that mostly died out once decimal point notation was developed.
The issue with this idea is that radians are what you have to use when doing advanced mathematics. But we can get the best of both worlds by giving the amount of radians of one full turn a name, and then we can express angles as fractions of that value. Hm, I wonder what name that should be...
I disagree as Oscillations as a good introduction to trigonometry. I'm guessing you work with waveforms a lot, so you probably see waveforms as the primary use for sin and cos, but I find it's a lot easier (and in my corner of the world, more useful) to approach it from the geometric perspective. The explaination "They're X and Y coordinates on a circle" does a pretty good job of explaining sin and cos, as well as the use for tan, and why atan2 is a thing (if anyone happens to go that route). Also Tau is nice, but it's pretty hard to measure a radius with a set of calipers.
Valid point, but I have a different take on his video. While his video is interesting and informative, I get the feeling that beginners might struggle with this, but for a different reason. My issue with it, is that if you don't already know calculus, the idea of measuring speed at a single moment in time seems like mathematical witchcraft.
Up until the point when you learn what derivatives are, the formula for speed is just taught as a finite change in distance (delta x) over a finite nonzero time interval (delta t). If you taught the person about derivatives (and vectors) first (totally doable), then it would make more intuitive sense how the instantaneous speed idea relates to oscillations and how that relates to trig. That. Is my biggest critique. Other than that, I like the video.
@Quantum Passport All without looking at a triangle? Then you have no context for the Pythagorean Theorem. Here is an out of nowhere equation, play around with it.
@@alexandertownsend3291cos(alpha)^2+sin(alpha)^2=1; x^2+y^2=1. you can downscale 3:4:5 triangle to 0.6:0.8:1, obviously 0.36+0.64=1. you can find angle/phase for which cos(alpha)=0.6 you may try to find fractions such 3^2/5^2+4^2/5^2=1 for random numbers (such as 3,4,5) and you can simplify them to a^2+b^2=c^2. Now you can forget triangle. sin and cos can be written as taylor series which sum together to e^kx where k is, by some accident, i*pi.
I mostly agree, but I will nitpick: I don't think I've ever used calipers in a math class. Or a physics class (which was my major). Or my job doing GIS work for environmental scientists. In fact, they all gave me the numbers, and I didn't measure a single thing myself; the last time I was given a diameter I was probably 14.
Hmm, and iirc, lathe cross slide dials might measure diameter or radius or both depending on the model. Regardless, you're thinking about changes in radius when you decide on a depth of cut. So, if you're in manufacturing, you just have to be bouncing between the two measures anyway.
Yes! A trig explanation using tau, uploaded on 6/28.
I've not been a fan of tau, when I first heard about it, but now you convinced me.
Now I follow two "mathematical" cults. The other one being the seximal/niftimal number system.
@@porky1118 one of us, one of us! ;) ;) ;)
How do we know that the period of circle occultation is the same period of the spring (with initial velocity being 0 and the acceleration being equal to the position)?
For the movement along the circle, if you compare the velocity vector and the position vector, the velocity vector is just rotated by 90°. The acceleration vector, being the derivative of the velocity vector, is rotated by 90° again, so 180° compared to the position vector. Thus, the acceleration is just minus the position. This is also true component-wise, so y''=-y, which is exactly the spring equation. Therefore, the y-component moves exactly like the spring.
If you'll notice, I never actually specified the rate of oscillation of the circle. We could make it be whatever we wish. Because the sine and the cosine work for this, using them is the simplest choice, so the simplest choice for the rate of oscillation of the circle is to make it the same as the sine and the cosine.
The reason PI and PI/2 is used instead of Tau is for the same reason the Discrete Cosine Transform (DCT) was abandoned for the Fast Fourier Transform (FFT) -- computational performance. The Fast Fourier Transform exploits the periodic nature of all sinusoidals, only calculating the half- or quarter-period, and then inferring the remainder of the wave by employing symmetry-based transformations along one of its axis. All of this would not be possible without a prior knowledge of Tau/2 (ie. PI)
Thank you very much, I like your videos very much, especially the one about the geometric product.
Ps
6.28 my birthday, called from myself the 2π day....but much more are 6 and 28 the first and second "vollkommene Zahlen"...lg from vienna!
I will only accept tau if I get to eat two pies on tau day
That's what I do every Tau Day! 😁
use telescope to look at Tau ceti
I question the use of scalars rather than multivectors and the geometric product. It makes sense to give an orientation to the orbital path (counter-clockwise(CCW)) but how would one justify using a bivector of two radial vectors pointing outward and using the scalar product? Does a vector HAVE to be a straight oriented line segment? Why can't it be curved? Does the sum of the angles of a triangle ALWAYS have to be tau/2 radians? But then, why would the vector along the x-axis point outward rather than inward to represent attraction to the center? I'm just curious about establishing a consistent conceptual and notational foundation that can be used in higher dimensions. For example, Buckminster Fuller once described a circle as a planar triangle with 3 angles of pi radians.
Good idea; I've used 2*pi so many times while studying EE that I just see it now as a single symbol.
Let's start a petition to remove misconceptions from education, starting with:
1) pi
2) conventional current flow
3) your comment here...
I like the idea that math notation would be much better if people wrote functions at the right of their arguments, i.e. something like x.f instead of f(x). I have doubts if it is actually a good idea, but at least diagram chasing would be slightly easier.
Thx Mani
am a lifetime fan of sudgy
Like Euler found value of PI as 6.26 im his calculation before the original nomenclature of PI says that Euler basically used Tau for his methods.
This is false. To start with, Euler never used the symbol π to denote a constant. He used it to denote angular variables, just as we do with θ today. In certain circumstances, he fixed its value to be some specific quantity. Sometimes it was equal to 3.14..., other times it was equal to 6.28..., and yet other times, he would let it be equal to things like 1.57... or 12.56... (these correspond today to π/2 and 4·π, respectively). Also, the modern nomenclature for π follows the nomenclature used by the Ancient Greeks. So, no, Euler's work does not precede the modern nomenclature.
There is no τ in my calculator. So, sorry, happy 2π day for everybody!
I've been fully onboard with the whole tau thing even before I heard of the tau manifesto. However, I have one disagreement -- the symbol. I don't like using yet another Greek letter to represent a number that is SO important to math and science. It needs something more universal and fundamental, something not bound to language (i.e. "tau for turn" in English).... something more like the ten basic Arabic numeral digits we use. My proposal is ⊙ because of the symbology of its importance to circles/oscillations; it's also the symbol for the Sun, which is very appropriate, as 6.28... is basically as important as the Sun (not to mention all the orbit references invoked by that).
There's already another number that's incredibly important in math and we just call it e. I don't think we need to come up with a new symbol, and to be honest I think a new symbol is a bad idea because it will be harder for people to know how to type/pronounce it. Also the tau manifesto already has an entire section on this: tauday.com/tau-manifesto#sec-conflict_and_resistance
@@sudgylacmoe I actually think e should have a better symbol too because it's so easily (no pun intended?) confused with a constant or variable. I didn't bring it up because it was out of the scope, but my suggestion for that would've been Ↄ. Something similar to the shape of an exponential curve (for which it's most famous for), which is kind of like a crescent on its side, but also symbolically speaking of crescent, this also is the closest one-stroke symbol to the alchemical symbol for the moon. As for how to pronounce say these characters? Simple. 'Sun' and 'moon'. Translate it to whatever language you'd like. Both are one symbol, and no more than 1-2 strokes to write.
Where I can read more about it? Other than tau manisfest
Just everyone stay away from string theory and you'll probably be fine.
Not sure that it makes sense to use calculus concepts as an intro to trigonometry. In Algebra II, the last thing most students learn is how to divide complex numbers, factor polynomials using complex numbers, and deal with real-valued exponentials and logarithms. If you want to move on to trigonometry from there, I feel like introducing functions like (-1)^x is a good place to start. It literally draws the unit circle in the complex plane. (-1)^(1/4) is the special 45-45-90 triangle and the solution can be found by finding √i = a+bi, which is a really interesting exercise any algebra 2 student should manage. The 30-60-90 triangle is (-1)^(1/6) = ³√i = a+bi. From there you can define sine and cosine using (-1)^(x/180°) = cos(x) + i*sin(x), which is just Euler's Formula written a bit differently so that you don't need complex exponents and can use degrees rather than radians.
*In algebra II, the last thing most students learn is how to divide complex numbers, factor polynomials using complex numbers, and deal with real-valued exponentials and logarithms.*
Perhaps where you live, but not outside that region. At least in Western countries (including Latin America), exponential and logarithmic function are introduced in pre-calculus, or calculus I, not algebra II, and even then, they are taught extremely poorly as concepts. In fact, case in point,...
*I feel like introducing functions like (-1)^x is a good place to start.*
This is problematic, because exponentiation, strictly speaking, is ill-defined when the exponent is not an integer (sometimes, it is ill-defined for negative integers). To be rigorous, exponentiation for natural exponents is defined by x^(m + n) = x^m·x^n for all x, m, n. To extend this to the integers, we must restrict x to be the invertible elements of the structure one is working in. Extending this to the rational numbers (let alone further), is problematic, because the equation alone is not sufficient for even well-defining something like 4^(1/2), and there are no natural additional restrictions that one can add that hold in the general case. The concept of exponential functions is related to that of exponentiation, but strictly speaking, they are different concepts. An exponential function on an algebraic structure A is defined as a nonzero function E on A such that E(x + y) = E(x)·E(y). If A is the set of real numbers, then we typically add another restriction: E must be continuous. But this restriction requires using concepts from calculus. Also, the issue is that while it makes sense to talk about the base of E, there may multiple functions with the same base, so it is incoherent to talk about "the exponential function base b," and some numbers also cannot be the base of an exponential function. As such, using the traditional exponent notation makes no sense. Instead, we use the exp notation to denote the natural exponential function (which is easy to define naturally) and then we can prove that all other exponential functions can be expressed in terms of this one. This generalizes to the complex numbers very simply. Meanwhile, logarithms do not generalize well to structures beyond the real numbers, because logarithms depend very intrinsically in how the rea numbers themselves are defined.
Anyway, the point is that, given that the concept of something like (-1)^x is not even coherently defined, it does not make sense to suggest that one should teach the concept. One of the big problems plaguing modern mathematics education is that we teach a lot of incoherent concepts to children that they, later on, have to unlearn, which is an extremely difficult task, and which creates a lot of confusion, because what they learned as laypeople does not match at all what actual mathematics and mathematics students know about the concept. Trying to solve this problem by.... teaching more incoherent concepts... that does not make sense.
*From there, you can define (-1)^(x/180°) = cos(x) + i·sin(x)*
No, why? This is terrible. It is so unnatural, and it just makes it seem like we are supposed to stick to degrees, which is a terrible idea, objectively speaking. Besides, part of the problem with how we teach trigonometry is employing this widely spread misconception that degrees and radians are "units for measuring angles, and radians just happen to be more natural," which is conceptually inacurate, and it leads to students making many, many mistakes. For example, in metrology (science), it is well-understood that non-polynomial functions like exp(x), ln(x), and cos(x), have inputs which are dimensionless, and thus, have no units. It is meaningful to talk about exp(L/5 meters), where L is some length, or to talk about sin(5 Hz·t), where t is a variable in units of time. This is because the inputs are dimensionless, but it is meaningless to talk about ln(5 kg), for example. Yet, having something like cos(15°) is valid, even though we are treating degrees as a unit of measurement? That is nonsense. Besides, since angles are dimensionless anyway, by definition, it does not make sense for them to have units of measurement in the same way that dimensionful quantities do. Also, treating degrees and radians as units of measurements gives the wrong understanding for how trigonometric functions work, and also the wrong understanding behind why radians are "more natural than" degrees. In the first place, one needs to discuss and understand what it really means to say that radians are more natural that degrees, given their dimensionlessness. No such discussions are ever had in the classroom, and this is part of the problem. What exactly is an angle? How do we know two angles "have the same measure"? How do we know if they "have different measure"? What does it even mean for them to have a measure at all, when the way we "measure" them is dimensionless? These are the questions that a student needs answered to properly learn the concepts and actually understand them, rather than for them to memorize vocabulary and symbols without understanding why.
@@angelmendez-rivera351 Jesus dude, edit your comments into a presentable form appropriate to the medium. I wrote a fairly short comment and you responded with an opinion article.
Harmonic motion indeed gives nice intuitions about trig, though I like to define them for myself via exponentiation right now. It raises questions why do that at all, but the answers aren’t too far away, just a little algebra and analysis, and suddenly you see that you move uniformly on a circle, or you have something to do with parameterizing the group of plane rotations (when we show SO(2) ≅ U(1)). “The right way” is to parameterize SO(2) in a good manner (only one of the parameterizations gives us radians, and we need to require that the speed is 1 and it seems mathematically a mouthful), as is with hyperbolic trig, when we do SO(1, 1), Minkowski plane, and “parabolic trig” “cosp” t = 1, “sinp” t = t (SO(1, 0, 1), related to galileian mechanics). But through exponentiation in an appropriate algebra (complex numbers, split-complex numbers, dual numbers), we get the requirement of unit speed automatically. The speed is unit in the corresponding geometry; the usual trig goes with euclidean plane, hyperbolic trig comes with the Minkowski “relativity plane”, and parabolic goes with the degenerate galileian geometry of dual numbers. Also we can of course require that the speed of incrementing the area sweeped by our radius vector is unit. That even can be done using Clifford algebras of those planes, so maybe you’ll make a video about that (or already have? I haven’t watched them yet); and exponentiation is there too, of course, with the way to represent SO(...). So it’s all tied; and all that exponentiation is right from Lie algebras, taking a “speed of rotation” (represented by bivectors in Cl. algebras) to a rotation which will be made with this constant speed after an unit of time. Someone should implement all those connections in a single video plz.
Also we can even unify those three algebras (or corresponding Clifford algebras of course, but that won’t look new): we just take, formally, ℝ ⊕ J ℝ where J is a new quantity and J² is some real number. When that number is negative, we have an algebra isomorphic to complex numbers, when it’s positive then that’s split complex numbers, and when it’s zero this is our degenerate case of dual numbers and parabolic trig. (Which no one seems to talk about anywhere at all. Well, probably because it’s degenerately simple, but still it does exist and it does even relate to galileian mechanics. Which is a good approximation we still use left and right.) But it seems a great pain to prove results right in this unified setting without going to the three cases of different concrete geometries. I wasn’t even able to show several technical lemmas one would need. But it’s pretty easy to show exponentiation indeed gives an unit speed and an unit area sweep.
Ah, also you’ll need to define conjugation (formally changing J to −J which we see leaves J² the same, so that’s a pretty natural operation in our algebra) and also you’ll need to borrow a good norm from somewhere, as z z* using this conjugation isn’t a norm squared in general (ow). And we need norm to define power series (and do exp through one) or derivative (and do exp through a diff. eq.).
If I could invest in UA-cam channels, this would be the one I put my money on!
Good news: You can invest in it! www.patreon.com/sudgylacmoe
@@sudgylacmoe ahahah that's not what I meant but sure!
I was thinking this graph this same day and this pops up
τ gang🤘
Where were you before my maths exams
this is insane
Was about to make a petty comment how there is tau propaganda inserted but in the end you still referred to pi so it's a fair treatment. I see its utility (including the 1/2 c x^2 form that shows up from integration) but maybe future generations that learn C = Tau r first without seeing pi can make the change 😀
Beginnings of trigonometry are with circles and chords not triangles. Name trigonometry comes from XVI century.
Omg this feels like indoctrination :O even more after just having watched the tau manifest.
(Because it begins without defining anything, btw I'm not criticizing that, the video is pedagogically oriented to do the intuition first)
While normally I would agree that tau is generally better than pi, there is a subject where I find pi is a better choice, and it's one that I know mainly from this channel. While tau is the circumference of the unit circle, pi is the area of the unit circle. Now, I will be bringing up hyperbolic trigonometry, so this isn't exactly an entry level explanation. In hyperbolic trigonometry, since the arc length doesn't have a simple expression, angles are instead represented as twice the area enclosed by a region.
When doing geometric algebra, or even just quaternion algebra, the standard way to rotate an object is to apply a rotor twice, which rotates by twice the angle given. But what if you didn't give it the arc length, but instead the area enclosed by it? So in this case, a full rotation would be pi rather than tau, and applying a rotor generated with the value pi will give exactly that rotation, sweeping out a region of area pi. This also works for hyperbolic transformations as given earlier, as well as simple translations! For some unit 2-blade B (I'm not certain about arbitrary bivectors), the rotor e^xB will generate a transformation that sweeps out sector with an area of exactly x when applied to the vector (1, 0, ...).
Edit: remember that bivectors _are_ areas. So a bivector B is an oriented plane segment with an area of |B|. e^B then represents a rotation that sweeps out a representation of B, in the plane B̂ and with an area of |B|.
I would say this is a place where the factor of one half should be there. The reason for that half is because to apply a rotor to something you need to apply it on both sides, making you have to make the rotor do only half of its full job. Furthermore, in two dimensions, you don't need to apply it on both sides anyway so in this simple case you wouldn't have the factor of two at all. The Tau Manifesto also covers the formula for circle area (at tauday.com/tau-manifesto#sec-circular_area), where it shows that the factor of one half in the area formula arises naturally.
I will admit that there's one area that π seems better suited: Fourier series. In Fourier series we want half of a full period for various reasons. However, given that τ is better than or the same as π in practically every other circumstance, I still use τ in Fourier series simply for consistency.
The point of tau is not only that it makes expressions simpler by eliminating hidden factors, it's that it more accurately represents the underlying mathematic structures of circles and oscillations. As pointed out in the Manifesto, the area of the unit circle only becomes exactly pi out of coincidence, because when expressing area as a function of the radius, which is a one-dimensional measure, we are implicitly integrating, which introduces a 1/2 factor that cancels out the 2 in 2pi.
I don't understand what you're proposing in the second paragraph, but sounds like you're mixing concepts in an unsound way, by mixing areas and angles.
@@andremeIIo Hyperbolic angles are _defined_ in terms of areas, and the same relation works both for a vertical line and a circle. (The shapes generated from B² = 0 and B² = -1 respectively.)
@@andremeIIo It may be true that τ more accurately reflects the mathematical structure of circles and oscillations if you are not looking at it from a foundational perspective and are only worried about circles in Euclidean space, specifically. The Tau Manifesto fails to account for the fact that (a) in terms of more fundamental mathematics where π may show up (prior to even having sufficient sttucture for oscillations to be a concept), τ is not more natural than π, (b) all of Euclidean circle mathematics emerges more naturally from the mathematics of Euclidean conic sections, where the naturalness of τ is diminished in relation to ellipses (c) π can be defined purely analytically, and without needing to resort to geometry, for example. The same can be said about the naturalness of radians, actually. The difference is, radians as a standalone concept always remain natural when taking those things into consideration.
@@angelmendez-rivera351 since you took the time to write that long and structured reply, I'll politely point out nothing you said makes any sense. You are either trolling or drunk, or simply didn't read the manifesto at all and are just talking out of your ass. If you trying to upset me, congrats, you made my eyes roll. If you are being genuine... All I can say is to read the manifesto again (and maybe study math from better sources, because that's a lot of misconceptions there).
That segway into tau propaganda was truly seamless
Just change the unit of angle measure to "turns", so that 1 turn= 1 tau.
Then try to find the diameter or radius in terms of turns and circumference.
Use an enscribed hexagon
Homework for you:
For T="turn units", let Circumference C=1 [T] . r =1[L] , where L="length units".
So, r= 1[L/T]
Work out radius r in length units [L] using an enscribed hexagon.
prnt dot sc slash Z16h_ySQed61
@@PasajeroDelToro This is particularly unnatural. If L denotes the dimension for measurements units of length, then, in the formula C = τ·R, where C denotes the the circumference and R denotes the radius, then both C and R ought to have dimensions of L, since both represents lengths of a 1-dimensional set in the Euclidean space of dimension 2 or higher. Yes, the circumference of a circle can be traced by a radial line segment being rotated by exactly 1 turn, hence there is a relationship between the rotation parameter (which we call an angle) and the circumference. However, in metrology, the angle parameter is best conceptualized as being dimensionless, while circumference is best conceptualized as having dimension L. The angle parameter is best conceptualized as a point, a 0-dimensional object, in the circle.
A distinct point is that, although both circumference and radius have the same naive dimension L, additional information distinguishing the two can be included by distinguishing between circumferential lengths and radial lengths. This is an extension to dimensional analysis.
@@angelmendez-rivera351 "However, in metrology, the angle parameter is best conceptualized as being dimensionless, while circumference is best conceptualized as having dimension L. The angle parameter is best conceptualized as a point, a 0-dimensional object, in the circle."
What about hyperbolic angles where magnitude is the area of hyperbolic sector, can we measure angles in units of lenght squared?
My question was why teach us degrees at all
Degrees are good for giving us whole number measurements of small angles, which can be very convenient in practice. Degrees split the circle up into 360 slices, and 360 is a really nice number to take fractions of. One fifth of a circle is 72 degrees, a whole number. One third is 120 degrees. One twelfth is 30, and so on.
If you want to add angles, it’s really easy if both angles are whole number multiples of the same unit, as opposed to radians, where it isn’t at all obvious that tau/6 plus tau/3 equals tau/2 without some extra multiplication. But in degrees, 60 + 120 = 180.
Yeah but how can Spherical Trigonometry be understood in terms of oscillations? How is Spherical Trigonometry covered and applied? Why don't you tackle that?
I came here to learn trigonometry and NOT to be BRAINWASHED into a FALSE BELIEF that tau is better than pi!
You will never convert me...
(This is a joke)
Since tau is just 2 pi, i don't see what's gained by trying to change a convention at this point.
Thank you for this!
And hmm, this appears _not_ to be a part of the Summer of Math Exposition? Cool. :) (Note: it'd be cool if it was, too; just interesting.)
nc
13:03 while i like radians, there was an arbitrary decision. you could have instead chosen that the inputs be the number of times you went around the circle. the reason radians are natural is because of the following fact: d^4 sin(x angle units)/dx^4 = sin(x angle units) * [1 angle unit / 1 radian]^4. This is a fact that can only be discovered via calculus as far as i am aware, but most definitely it is an arbitrary decision in the face of other options
cos and sin are real and imaginary parts of x |-> exp(ix)
Maybe it's time to look at the geometric product of two vectors in Geometric Algebra.
Speaking of oscillations, I recommend pulling back on the high pass filter. Really you could get away with setting it to 90hz. If you are worried about plosives and you are also using something LIKE adobe audition (any single audio file editor like audacity will work) just look for the obvious low frequency bursts and selectively filter.
I'm actually not worried about plosives (I deal with those with my mic positioning). Without the high pass filter (which I have set at 100 hz) there's a low-frequency hum that gets in the way of some of the other things I do. I'm surprised you even noticed anything, other than the hum (which is already barely audible) I can't audibly tell the difference between with and without the high pass filter.
@@sudgylacmoe ah ok, it's the microphone that has the sound I am hearing. Have you tried a notch filter at the frequency of the hum?
؟؟
Gods this is painful. Either use energy arguments or solve Newton’s second law to convince people it’s a sine wave. Math people suck at explaining physics.
This is an interesting video and it would work well as a review or maybe end of course summary, I don't think it really works as an actual intro to trig for someone learning it for the first time. That's because it crams too much information into a single lesson. IMO, an actual intro lesson for beginners should focus more on a single concept and go into more detail, explaining the same concept in a few different ways and giving more concrete examples.
I respectfully disagree. I believe that introductions should give a rough overview of an entire topic without going into too much detail.
As an example of where I think this worked well is my Swift Introduction to Geometric Algebra. It moved very fast, and glossed over a lot of details. However, despite not really enabling anyone to do anything, it has exploded in popularity. Why? Because it shows people why we care and gives them the motivation to look for more. Imagine if I made my introduction entirely about the details of how multiplication in geometric algebra works, without ever talking about the geometric applications of it. Nobody would care about it, and the video would never have gotten popular.
Years ago I read betterexplained.com/calculus/preface/ by Kalid Azad and I think it is a good summary of my views on the matter. Ever since I read it I've always gone through textbooks twice: once skimming it just to get an overview of the topic, and then another time going through it in detail. Doing this has helped me out tremendously in my learning journey.
I think making this big of a deal about tau vs. pi could be taken as the definition of the expression making a hen out of a feather. That making the switch from pi to tau would really have anything other than a negligible impact is wishful thinking; there are so many things wrong with math education today, and having pi used instead of tau is NOT one of them. Being a tau hardliner feels to me like something that can make teachers go "yeah of course, this is the thing that has been holding my teaching back all these years. After I make this switch to tau, everything will make so much sense to my students!" and then they implement tau instead of pi in their teaching, while not truly reflecting about what is actually wrong with their teaching.
If you think that tau is legitimate, then you automatically think that pi is just as legitimate, and vice versa. And the thing is, pi is the one out of the two that has been grandfathered into the system whether you'd like it or not. It is not better or worse than tau. If you say that tau is better than pi because it gives a nicer periodicity when working with oscillations, then I can counter by pointing out that the angle sum of a triangle is pi and that triangles are the most fundamental shapes in geometry. And, of course, when teaching mathematics we definitely begin with basic geometry way before we ever get into oscillations or any kind of periodic processes...
I would say that we can't know it won't work for certain until we try it. And some people have tried it and said it makes things better. tauday.com/a-tau-testimonial is an example, and another example is myself. In making the animations for my videos (especially this one), I have to use τ/π a lot, and using τ has made things so much simpler.
@@sudgylacmoe Well, that can be said about almost any 'solution' to any kind of problem. Even if we can't know for CERTAIN, we can still be pretty sure about a lot of things. One of these things is pi vs. tau question. Using either of the two instead of the other is really a minute change in perspectives at most, and for a change to really have a major effect, the change needs to be way more substantial than this.
One of the major problems with the testimonials you're providing is that there is a serious risk of bias in the sample; nobody who started out using pi and who thinks pi makes things easier than they would think tau is would ever give a testimony to the virtues of pi if not pressed on the matter (since they've probably never even looked into the pi vs. tau question). But since tau is not standard those who prefer it will be way more vocal about it.
@timpani112 Your counter can be countered by noting that the sum of the exterior angles of *all* polygons (not just triangles) is tau radians.
@@focoma I assume that you're only talking about convex polygons? Otherwise the statement makes very little sense. And all of this is of course completely missing my point by a mile. I don't see a major difference in utility between pi and tau, and therefore I don't see the point in trying to start what I see as a completely meaningless revolution for the sake of what exactly? Aesthetics?! If we focus on the non-issue of pi vs. tau, then we may be blinded to the actual issues that are plaguing mathematical education worldwide, and even worse we may end up thinking that we have made a meaningful difference to mathematics when all we did was to replace pi with tau.
@@sudgylacmoe I agree with timpanii. The education surrounding mathematics is abysmal all over the world, but the π vs τ situation contributes to less than 0.001% of those problems. Even if we limit this discussion specifically to learning and teaching Euclidean geometry, the education is plagued with far greater problems than the understanding of π vs τ. Leonhard Euler actually used the symbol π to accomodate an entire family of values, rather than a single fixed constant. When you look at the foundational mathematics, which I think are actually far more important to teach, π and τ emerge as special cases of a more foundational and basic mathematical structure, and in analysis, π emerges more naturally from the definitions and concepts than τ. This is also true in the context of engineering as well, and even physics, and I say this being a physicist.
This is not to say that the things being stated by the Tau Manifesto are wrong. Rather, the point is that the Tau Manifesto actually just suffers from the same problem ordinary mathematics education does: it tells everything from only side of the story, and one which is fundamentally not very different from how regular education does it, aside from this one single change in notation. It tries to provide a solution to a non-problem, and as a result, it does not actually solve much at all. Now, that is not to say that τ is not more useful than π in some contexts. It definitely is, but the point is that this one change in definition does not require an entire movement, especially since it is not universally useful.
tau = 2π making it better than π is the same way as saying American should switch from imperial to metric but they prefer not to
As an American, I agree that we should switch from imperial to metric
Metric being better than imperial is however a stronger case than Tau being better than pi
@@MrNikeNicke Pi is the only concept that relates to the diameter. Everything else in math deals with the radius, which is reflected with Tau.
The manifesto book has lots of examples worth considering.
@@rlf4160 I'm not saying Tau isn't better than pi, just that the difference is much less than between metric and imperial
@@rlf4160 Pi would be fine for angles too if you switched to "diametrians". However, that's a PITA conceptually, so you wouldn't want to.
nothing alternative is here
This is a really good video! However- I smell tau propaganda in here…
String theory's 4D is a "hypersphere".
This is a contradiction. There is no 4D "hypersphere".
2D circle ✅
2D sphere 🚫 contra.
3D circle 🚫 contra.
3D sphere ✅
4D circle ✅ (see quaternion)
4D sphere 🚫 contra.
5D circle 🚫 contra.
5D sphere ✅ (see hypersphere)
Here's my 1D-9D:
3 sets of 3 dimensions;
1D, 2D, 3D are spatial
4D, 5D, 6D are temporal
7D, 8D, 9D are spectral
1D, 4D, 7D line/length/continuous
2D, 5D, 8D width/breadth/emission
3D, 6D, 9D height/depth/absorption
So 4D = temporal length of spatial 3D (current reality)
And 5D = temporal length/breadth of spatial 3D (space-time)
The sheer volume of nonsense in one comment is staggering
@@jakobr_ there's no such thing as a 4D "hypersphere" or "space-time". That is 5D. This is not a debate lol.
@@ready1fire1aim1 > Posts some crazy bullshit.
> Cannot explain.
> Do not debate, please. 😰 uwu
pi>>>tau
16:46 NO! We should NOT use 𝜏 instead of ℼ!
ℼ Is not wrong - its definition is wrong!
We should simply redefine ℼ = 6.28... 😊😍🥰
The letter 𝜏 is just so ugly - like a legless ℼ. 🤢🤮🤮🤮
丆 is better than 丌, got it.