What are the benefits of radians?

What are the benefits of radians?

Would you be able to get the same benefits of radians if they were replaced with "turns" (e.g. a 1/2 turn would be pi radians)?

"Turns" seem more intuitive desu.

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nothing it's pretentious bullshit
0-360 degrees are more understandable

What if we could device a system such that common fractions such as half a turn, one third of a turn, one fourth of a turn, etc. could be described without the use of fractions or repeating decimals?

Apparently radians makes things easier.

Sadly the only example I can think of is arc lengths. To find an arc length with radians all you have to do is:

[math]\theta r[/math]

Where [math]\theta[/math] is in radians and [math]r[/math] is the radius.

Can anyone else think of some examples where radians are better?

if you really don't know then you don't belong here. go read a calculus book.

The approximation sin(x) = x works only when the units are in radians. The lives and careers of millions of engineers and physicists depends on this.

Radians are the best system for calculus

It works for degrees too.

But any other more accurate approximations don't work, since sin(x°)≠cos(x°)

tau for turn would be better, yes

tauday.com/tau-manifesto

just like a system with twelve digits instead of ten would be better. Thanks, frenchies

first of all, kys retard
secondly: that's completely different from what op proposed, tau is irrational

>tau
oh shut the fuck up

No it doesn't.

literally people who cannot understand the number 2

this thread already exists and tau apologists are still getting BTFO left and right in there

If not for radians, we wouldn't have [math]e^{i\pi} + 1 = 0[/math].
But seriously, radians are just better. More intuitive, for one: just look on the unit circle, go counter-clockwise on the edge for [math]x[/math] units, and the angle you will have traveled will be [math]x[/math] radians!

1 tau literally equals 1 turn, as OP specified.

d/dx sin(x) = cos(x) only works in radius
exp(i x) = cos(x) + i sin(x) only works in radians
The power series for sin and cos only work in radians
Basically everything only works in radians

Degrees are retarded.

Radians and a unit circle approach are literally the only way that trig functions and their power series work. By extension, forget about polar coordinates in 2 space, and polar/cylindrical/spherical in 3 space without radians. Also, if you can't do radians you generally can't do proper linear transformations within the same vector space, which is only a little bit completely essential to 3d graphics and physics engines.

It's only non-intuitive because we still waste time teaching kids degrees. I honestly don't know why they still bother.
Just teach radians from early age. It fucking makes me mad that I learned degrees at all and that it's my default thinking mode when looking at geometry.

There are useful applications for degrees in physics, but yeah, for the most part you are correct.

>There are useful applications for degrees in physics

Name one.

If you are doing lower level stuff and you are doing 2d trajectory calculations. You figure out the vertical and horizontal components the object hits the ground with, but need the angle, that sounds like a good time to use arctan with the degrees.

Calculus

And for the most part, if you need to approximate something using a calculator, it is generally easier to get a good approximation that makes sense with degrees. I'm more of a radians fan myself, but if you are putting into the calculator, and getting a number out, 83.2° means more to me than 1.452 radians. Radians are great if you can do it as a coefficient and pi, but if you don't really have that option, radians are a pretty rough choice for most people.