Turns are better than radians
81–90 of 494 posts
Re: Turns are better than radians
#82The major motivation for radians is arc length parametrization, really. Meaning that in a circle of radius 1 unit (in whatever measurement unit you've chosen), an arc formed by a k-rad angle measures k units. There is an intentional coincidence of angles and arc measurements.
Re: Turns are better than radians
#83>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 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.
Excuse me? Have you done any computation in Physics? Have a look at the pendulum equation, for a start...
Re: Turns are better than radians
#84y’’ = -y Radians are God’s chosen angular unit. If you want to do mathematics, you have to use radians.
What subfield of mathematics? Surely if you do differential equations, your trigonometric functions will eat radians. But e.g. for geometry, where you do basic arithmetic operations on angles, turns are a bit more convenient than radians. (Radians are not that inconvenient if you denote 6.28 somehow, but then again, why not just use turns.)
Re: Turns are better than radians
#85Earlier 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…
Or just defining the result of division by zero as zero "for safety": https://www.hillelwayne.com/post/divide-by-zero/ It boggles the mind, truly!
Are you claiming the author is incorrect that x/0 = 0 is mathematically sound?
Re: Turns are better than radians
#86>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…
That is the way to do the math, but not the way to write the code.
That said, I would like for my compiler to combine any multiplications involved down to one factor for input to the fastest sin/cos operations the machine has. And, to treat resulting multipliers close enough to 1, 1/2, and 1/4 as exact, and then skip the multiplication entirely.
But the second part is a hard thing to ask of a compiler.
Re: Turns are better than radians
#87>But math never decreed that sine and cosine have to take radian arguments!
To define trigonometric functions you need angles. Which can be measured in degrees or radians.
If angles, degrees or radians wouldn't matter, those functions would be some ordinary real functions and not called "trigonometric".
Re: Turns are better than radians
#88Earlier 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…
Whats wrong with long variable names?
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.
Re: Turns are better than radians
#89Earlier 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…
> The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series. The addition theorems for trigonometric functions can easily be shown by the multiplication theorem for Taylor series (and adding two Taylor series). This proof would be more convoluted if the Taylor series were not so easy. Also, because of the simplicity of their Taylor series, one immediate…
Those are useful symbolically, but not computationally.
Re: Turns are better than radians
#90Earlier 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…
That is the way to do the math, but not the way to write the code. That said, I would like for my compiler to combine any multiplications involved down to one factor for input to the fastest sin/cos operations the machine has. And , to treat resulting multipliers close enough to 1, 1/2, and 1/4 as exact, and then skip the multiplication entirely. But the second part is a hard thing to ask of a compiler.
Yeah, seems to me that languages should allow way more semantic expression than most do today.
I wish I had done CS, those kinds of compiler optimization sounds so fun. I'd love to work on that