Talk:Radian

message: Surely, we could just say "A radian is a unit of angular measurement of a circle that is equal to ~57.295779513082320876°". That way, I could read the header, and then know how to apply it in an example. Gehyra Australis (talk) 10:07, 20 January 2022 (UTC)
replies: text: Simple English Wikipedia jumps to that punchline much more quickly, and is a great resource if you're looking for that kind of stuff, particularly when it comes to articles concerning subjects often steeped in rigour or technical detail, like mathematics and science. —JivanP (talk) 19:50, 11 May 2022 (UTC)
thread_title:  Can we not have a simple definition in the Lead for laymen? 

message: I think it would be helpful to insert this text as the third paragraph in the History: 18th and 19th Century section. [The "h" that appears twice at the end should be the lowercase Greek eta.]
Leonhard Euler’s seminal work Theoria Motus Corporum Solidorum seu Rigidorum (Theory of motion of solid or rigid bodies), published in 1765, implicitly adopted the radian as the angle unit for all equations involving rotation. For example, his Definition 6, states (in Latin) that ”angular velocity in gyratory motion is the speed of some point, expressed in the unit of the distance from the axis of rotation”. He therefore defined angular velocity ω as ω = v/r, so that within his equations the angular velocity ω always represented radians per unit time, and the radian was treated as equivalent to the number 1. With an unspecified angle unit, the equation would need to contain a constant term such as ω = v/(hr), where h is described above. Pquincey (talk) 08:39, 5 September 2022 (UTC)
replies: text: Be bold. Dondervogel 2 (talk) 08:47, 5 September 2022 (UTC)
thread_title:  Euler's role in the history of the radian 

message: Here is the key insight: in the context of circles, angles and rotation, units of length are either tangential or radial. Let there be two new units that replace good old \rm m (meters): \rm m_{tan} (tangential meters) and \rm m_{\rm rad} (radial meters). The unit of a \rm rad is the conversion or ratio between these two units.
\rm rad = \frac{\rm m_{tan}}{\rm m_{\rm rad}}
1 = \frac{\rm m_{tan}}{\rm m_{\rm rad}}\frac{1}{\rm rad}
1 = \frac{\rm m_{\rm rad}}{\rm m_{tan}}\rm rad
An angular frequency in this context should be \rm rad/s, not 1/s. A formulation of the relationship of L and \tau is a little more clear I think if add \Deltas or ds like this: \tau \cdot dt = dL. Doesn't really matter for this question, and I'll use s instead of ds or \Delta s.
Take the left side, \tau \cdot dt. The units are (N\cdot \rm m_{\rm rad})\cdot s. Notice the use of radial meters.
Expand N to give (\rm kg\cdot \frac{\rm m_{tan}}{s^2}) \cdot \rm m_{\rm rad}\cdot s and notice the use of tangential meters. Simplify to give: \rm kg \cdot \rm m_{tan} \cdot \rm m_{\rm rad} /s. 
Now multiply this by 1 = \frac{\rm m_{\rm rad}}{\rm m_{tan}}\rm rad, which can also be thought of as "converting" the \rm m_{tan} to \rm m_{\rm rad}:
\rm kg \cdot \rm m_{tan} \cdot \rm m_{\rm rad} \cdot \frac{\rm m_{\rm rad}}{\rm m_{tan}}\rm rad/s
Simplifying, this now gives the units you were looking for, with the m now specified as radial meters:
\rm kg\cdot (m_{rad})^2 \cdot rad/s
Alternatively, you could multiply by 1 = \frac{\rm m_{tan}}{\rm m_{\rm rad}}\frac{1}{\rm rad} instead, and you'd end up with the expression that alfC gives, though those meters are now revealed to be tangential meters:
\rm kg\cdot (\rm m_{tan})^2 / \rm rad/s
This issue must will be addressed clearly and concisely yet. Firstly, thinking of \rm rad as dimensionless is not useful, and thinking of \rm rad = 1 is not useful. In certain frameworks, *technically*, \rm rad = 1 and "\rm rads are dimensionless" are workable, but these statements are somewhat counterproductive for grasping the key insight.
Additional related information is . Voproshatel (talk) 05:46, 5 May 2023 (UTC)
replies: text: This is discussed in the dimensional analysis section, actually the Mohr/Phillips 2015 paper is already cited for its definition of angle in terms of sector area. Regarding your proposal for tangential/radial units (which is WP:OR? I haven't seen anything like it in the literature), I am not sure what it offers - just let \rm m = m_{rad} and then naturally \rm m_{tan} = m \cdot rad, and it is simply the traditional unit system but with an angle dimension and radian base unit. Mathnerd314159 (talk) 15:11, 5 May 2023 (UTC)
thread_title:  Radian is dimension unit because it must indicate the information about a perpendicularity of the radius to the angular values