Talk:2748: Radians Are Cursed
how do transcript 172.70.127.37 19:23, 10 March 2023 (UTC)
https://en.wikipedia.org/wiki/Square_degree may be of some help with this one. 162.158.166.124 19:44, 10 March 2023 (UTC)
The comic isn't actually correct. A radian is not equal to the length of a circle's radius; it is equal to the length of the radius, multiplied by 2π, divided by the perimeter, which is why it has no units, while the length does. In other words, radian/2pi=length of radius/length of perimeter. 172.70.46.84 19:51, 10 March 2023 (UTC)
As suggested by the above Wikipedia link, square degrees are in fact often used in astronomical contexts. Also, it's quite standard to say that radian=1; see for example SI derived unit. An angle is the ratio between the arc length and the radius, and we just optionally append "radian" for clarity. So 1 = 57.3 degrees is correct; Randall simply used the wrong argument to obtain it. Aseyhe (talk) 20:57, 10 March 2023 (UTC)
- I always understood radian to be the name of the unit, so by definition 1 radian=1. Barmar (talk) 21:17, 10 March 2023 (UTC)
- It is a shame that astronomers don't use the proper unit for such things: the steradian. It is literally there for describing the 3D equivalent of angle. Oh well... --172.69.79.137 04:16, 11 March 2023 (UTC)
- It is a shame that astronomers don't use the proper for length, preferring ad-hoc units based on the solar system. But if you use a different ad-hoc unit based on the properties of the solar system they throw a hissy fit.172.70.38.150 06:51, 12 March 2023 (UTC)
- Indeed, what is the "proper [distance unit?] for length"? Light-year, based on Earth's orbital period. AU, based upon Earth's orbital radius. (Kilo)metre, based (approximately, and quartered) upon Earth's circumpolar circumference. Parsec, based upon Earth's orbital radius and a notionally arbitrary subdivision of angle. (Which can be avoided by mathematically more pure "paradians"???) Planck-lengths, might be not solar-/geo-centric but creates horribly huge numbers even at the human scale. ;) 172.70.86.128 16:07, 12 March 2023 (UTC)
- It is a shame that astronomers don't use the proper for length, preferring ad-hoc units based on the solar system. But if you use a different ad-hoc unit based on the properties of the solar system they throw a hissy fit.172.70.38.150 06:51, 12 March 2023 (UTC)
Someone fix the vandalism, how do you upload images? --Purah126 (talk) 03:06, 11 March 2023 (UTC)
- I'm doing it but that user needs to be blocked.
- To revert images, scroll down and click the revert link next to the last good version.
- And do not feed the trolls. ~ Megan she/her talk/contribs 03:10, 11 March 2023 (UTC)
On reading this I vividly remembered a maths teacher once asking our class "What's 10% of a straight line?", and the looks of disgust and bewilderment when he said the answer was 18 degrees. 172.70.86.147 08:31, 11 March 2023 (UTC)
- I just hope that was Celsius degrees (or Kelvin), rather than Fahrenheit(/Rankine). ;) 172.71.242.190 10:51, 11 March 2023 (UTC)
- If you use Kelvin with degrees you have already lost...172.68.51.178 13:29, 11 March 2023 (UTC)
So the volume of the sky is 4/3 π r³ = 7,092,429 cubic degrees
I remember in the quantum mechanics class we figured that if \hbar is defined to be h/2π, then we might as well introduce the notation \pibar as an alternative for 1/2. Captain Nemo (talk) 11:08, 12 March 2023 (UTC)
The logic is fine once you recall the formula s = r x theta. The arc length subtended by an angle is equal to the radius times the angle. On the unit circle, the radius is 1 (no unit). Therefore, the subtended arc length of 1 radian is s = 1 x 1 radian = 1 radian. 172.71.22.117 21:45, 12 March 2023 (UTC)
- "...the radius is 1 (no unit)." There's definitely a unit. It's whatever the unit the unit circle is reflecting (even if that's mathematical Unity). And in the case of dimensional analysis, it's a particular dimension that you'd need to account for, and the difference between this radians thing and the degrees thing is only the inclusion of dimensionless pi-based constant of conversion. Doesn't change the understanding of the issue, but I believe that some explanations/comments aren't then conveying it onwards accurately. 172.69.79.184 22:15, 12 March 2023 (UTC)
- I mean, I'm sorry, but respectfully, you are wrong. The unit circle is *by definition* a circle of radius 1. There is no unit attached to that. 172.71.82.41 01:55, 13 March 2023 (UTC)
There is actually some dispute about whether angles should be measured using units. I can't find it now, but there was an article by someone arguing that the current SI definition of the radian as 1 rad = 1 m / 1 m was flawed. He felt that units of angle should have a dimension, A, and rewrote several formulae slightly to accommodate this. But more often today, the radian is considered dimensionless with a value of exactly 1, making it not actually a "unit" so much as a hint telling how the angle was measured. In this definition, an angle has a measure of x (radians) iff the circular arc it intercepts as a central angle has an arclength of x times the circle's radius. Under this definition, the following become mathematically correct:
- rad = 1
- ° = π/180
- Radius of unit circle = 1 = (180/π)(π/180) = (180/π)° = 57.29577...°
- (1°)² = π²/32400
There is really nothing mysterious about it. Here, we are just defining the radian and degree as real numbers. This is how we treat them in Calculus. For instance, d/dx sin(2x rad) = 2 cos(2x rad), not (2 rad) cos(2x rad) as the chain rule implies. This is because 2 rad = 2. This also helps explain why Phil Plait's bizarre dimensional analysis actually does work. In particular, the last equation above would normally be written with "rad" on the right-hand side, giving a conversion between square degrees and square radians. Using the fact that the area of a sphere is 4πr², we see that the area of the unit sphere must be 4π square radians, and thus 4π * (32400/π²) * (1°)² = (129600/π)°² = 41252.961...°². Note that a "square radian" is also equal to a "steradian" by definition, which is the solid angle that subtends 1/(4π) of the surface of the sphere. 172.70.127.38 02:56, 13 March 2023 (UTC)