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This is actually a technical question about what kind of mathematical practices and notation lead to a more energy-efficient society, and the expertise required to authoritatively answer this question for posterity's sake lies within this question's target audience. Thinking and computation in general (both human and non-human) are an energy and time-intensive process. Time and physical energy are scarce commodities in terms of the survival of an individual in a hyper-competitive society, and in terms of the survival of the entire society overall because the amount of energy a society can use for its survival is finite due to entropy and other factors. So this a technical question, but if anyone believes that this question can reach a more authoritative and influential group of respondents elsewhere please say so in the comments and I'll delete this and post it there instead. Also feel free to edit this post.

As many of you know, $\tau$ (tau) is defined as the length of the circumference of a circle of radius 1, and $π$ (pi) is defined as half of that circumference. Replacing radians with units of tau would, for example, lead to writing $\sin(90°) = \sin( π/2)$ as just $\sin(\tau/4)$ instead. Sin of a quarter of a circle. Easy. For three-quarters of a circle we just write $\sin(\tau*3/4)$ instead of $\sin( 2π*3/4) = \sin( π*3/2) = \sin(3*90°) = \sin(270°)$. So much precious time and effort was wasted in my high school math classes over this needless overcomplication. For a mathematicians, scientists, and engineers the difference in notation is trivial, but for first-time students its the difference between intellectual abuse and conceptual clarity.

EDIT: My original question also asked if we should replace radians with units of $\tau$ also. It turns out that that makes the derivatives of the trigonometric functions messy, because if $x$ is in units of $\tau$ instead of radians then the corresponding function for sine is $sin(x*\tau)$ and its derivative, in units of tau, is $\frac{d}{dx}sin(x\tau) = \tau*cos(x*\tau)$...I think. So radians are better and that part of my question has been definitively answered. Thank you.

Ariel Hernandez
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  • History is rife with many things that seem to have been the wrong way to do things. $\pi$ is a usable constant, and while degrees may have limited utility for mathematics, they are widely used elsewhere and unlikely to be supplanted... Also, $\pi$ has many definitions, notable simple-terms versions include "the ratio of a circle's diameter to its circumference" as well as "the area of a unit circle." – abiessu Nov 06 '21 at 23:31
  • Unless we all become participators. – Ariel Hernandez Nov 06 '21 at 23:32
  • But therein lies the problem... "We all" make up a small community which is a subset of all persons involved with mathematics, which is a further subset of all persons engaging with the various topics here. There is little change on these lines that is likely. – abiessu Nov 06 '21 at 23:36
  • I am invested in the outcome and therefore biased to think and act differently. Thank you for your feedback. – Ariel Hernandez Nov 06 '21 at 23:42
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    The Math Educators StackExchange might be a better place for this question. – Blue Nov 07 '21 at 00:05
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    Does this answer your question? Why is $\pi$ = 3.14... instead of 6.28...? Also, 1, 2, 3 and more. – dxiv Nov 07 '21 at 01:12
  • $\tau$ does not change radians. Radians are radians. $\tau$ changes how we talk about radians. ${90}^{\circ}$ does not change for $\frac{\pi}{2}$ radians to $\frac{\tau}{4}$ tradians or whatever. These quantities are equal. If $\tau$ changed radians, it would be a reason not to use it, since derivatives of trigonometric functions would become more complicated. – Thomas Anton Nov 07 '21 at 01:19
  • @ArielHernandez: This is like debating the relative merits of percentages and fractions. Everything's a trade-off. For instance, representation in terms of $\tau$ (which I kinda like) may make the names of angles more intuitive, but it makes the arithmetic of angles a bit more of a pain: eg, "$30^\circ+60^\circ=90^\circ$" vs "$\frac1{12}\tau+\frac16\tau=\frac14\tau$", or comparing $90.0001^\circ$ vs $1.5703$ radians to a right angle. – Blue Nov 07 '21 at 02:28
  • @Blue: Good point, but if you already happen to be in base 10 then you can also write $0.08333\tau + 0.16666\tau = 0.25000\tau$ ...But still not easier than adding with degrees. – Ariel Hernandez Nov 07 '21 at 06:20
  • By the way, sometimes angular measure does use a unit called a turn, a cycle, a revolution, and sometimes other names, which I think is essentially what you proposed. For example, "revolutions per minute". Presumably it's a good idea for some applications. Just perhaps not for calculus, as you've discovered. – David K Nov 07 '21 at 21:25

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Degrees are a useful convention that is not going away. Radians make calculus results very elegant. The derivative of $\sin x$ is $\cos x$ when $x$ is in radians. When $x$ is not in radians, it is something else. Radians and degrees are both important, and two systems of angle measurement are quite enough already.

D_S
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  • There are already more than two systems of angle measurement. See https://en.wikipedia.org/wiki/Gradian#:~:text='angle')%2C%20grad%2C,%CF%80200%20of%20a%20radian. – Gerry Myerson Nov 07 '21 at 02:30
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    Never heard of it, and glad I didn't have to learn it. Looks like it never caught on – D_S Nov 07 '21 at 02:33
  • I think that if you modified the sin function to accept units of tau instead of radians, that you would still get cos(x), because $\frac{d}{d(x\alpha)}sin(x\alpha) = \frac{1}{\alpha}\frac{d}{d(x)}sin(x\alpha) = \frac{1}{\alpha}\alpha*cos(x\alpha) = cos(x\alpha) $. – Ariel Hernandez Nov 07 '21 at 06:01
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    Actually... No I'm wrong about that, I think, because if $x$ is in units of tau then the corresponding function is $sin(\frac{x}{\tau})$ and $\frac{d}{d(x)}sin(\frac{x}{\tau} )= \frac{1}{\tau}*cos(\frac{x}{\tau}) $. – Ariel Hernandez Nov 07 '21 at 07:08