Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:
theta = arc len / r
Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?
In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.
How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?
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A quantity is dimensionless if it has same magnitudes in different units.
$$1 \text{rad}=\dfrac{1m}{1 m}=\dfrac{1 nm}{1 nm}=\dfrac{1\text{light year}}{1\text{light year}}=1 $$ As you noted The same units get cancelled.
However length is not so.
$$1m=100cm $$
$1\ne 100$ for obvious reasons.
Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.
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As you point out, the radian measurement of an angle is the ratio of the length of an arc the angle intercepts to the length of the radius of said arc. As both arc length and radius are measured with units of length, these units of length cancel when determining how many radians an angle is. This is why radians are dimensionless - there is no "unit" that describes what a radian measures, because it is a ratio of two different quantities with the same unit of measurement.
A measurement in degrees, however, is simply a different ratio; rather, it is the ratio of the arc to 1/360th of the circumference of the circle corresponding to the arc.
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In short:
Angles are dimensionless quantities (e.g. m/m for rad and m²/m² for sr). They have no base units in SI, meaning angles have no fundamental existence (contrary to a length or a time), they are derived from something else. This actually creates problems and since a long time proposals have been made to give angle a true dimension. Such change has many implications on other quantities.
The status of angle units has never been clear for the BIPM, the bureau in charge of the SI system, but as making angle units true base units creates more problems than it solves, there is a status quo.
An angle is a ratio, but there are different ratios
If you look at the definition of (plane) angle measurement units, they are all the ratio of a length to a length, so have dimension of m/m = 1, i.e. they are dimensionless.
1 rad: The angle subtended by an arc of a circle that has the same length as the circle's radius. Ratio 1/1.
1 turn: The angle subtended by the circumference of a circle at its center. Ratio 2$\pi$/1.
1°: 1 turn / 360. Ratio 2$\pi$/360.
1 gon (grad, gradient): 1 turn / 400. Ratio 2$\pi$/400.
The angle unit indicates which ratio is used
Note angle comes from latin angulus, apex/corner, and is the corner made by the intersection of two lines/planes. Angle in science is actually a shortcut for angle measure.
And as a matter of fact drawing an angle is easy, but measuring an angle requires some specific construction, e.g. a circle (the ratio of the arc to the radius is the measure), a square (the angle made by the diagonals is 1/4 of a turn), etc. All computations ultimately lead to the ratio of a length to a length. So an angle measure, regardless of the unit used, has no dimension.
However the unit indicates which ratio was used, an angle of n$\times$arc/radius is not an angle of n$\times$1/360. So we need to explain which reference was used, this is the definition of a unit.
The assumption is when we use radians, we can omit the unit, and the implicit radian unit in math is due to the simplification it allows, e.g. in Euler's formula linking angles, Euler number and complex numbers.
Still angle units, plane (rad = 1m/m) and solid (sr = 1m²/m²), are of a special kind. From a SI standpoint, they have been supplementary units, separate from base units and derived units. This special class of units was removed in 1995:
The Comité International des Poids et Mesures, in 1980, having observed that the ambiguous status of the supplementary units compromises the internal coherence of the SI, has in its Recommendation 1 (CI-1980) interpreted the supplementary units, in the SI, as dimensionless derived units.
Dimensionless radian is also a problem
Angle units are now derived units. However this classification and the fact angles are considered dimensionless is strongly challenged as it creates inconsistencies when applied to the real world.
E.g. torque is a force resulting from rotation. It is currently measured in N m, which is a Joule and doesn't reflect an angular quantity. It would be more meaningful to use J rad$^{-1}$, but this requires radian to be a base unit with a dimension, this would be the 8th base unit of the SI.
You may read: Implications of adopting plane angle as a base quantity in the SI for more details.
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The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.
For example, you quote the arc length as $s = R\theta$. This is obviously no longer homogeneous in dimensions if $\theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $\theta$, and that $s=R$ for $\theta=1\,\mathrm{rad}$, we conclude that the true form of that relation must be $$s = R\frac{\theta}{1\,\mathrm{rad}}.$$ (When asuuming that $\mathrm{1\,rad = 1}$ is dimensionless, it yields the original expression.)
Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$\exp(x) = \sum_{k=0}^\infty\frac{x^k}{k!}.$$
If $x$ has a dimension, this expression does not make a whole lot of sense. To be correct, we would have to write $\sin(\theta/\mathrm{rad})$ when talking about the sine of an angle.
As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.
What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $\omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $\omega$ would be $\mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.
If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.
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Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=\aleph\theta,$$ and that for the area of a sector
$$a=\frac{\aleph\theta r^2}2.$$
The universal constant $\aleph$ is in per-angle-unit, and $\aleph=1 \text{ rad}^{-1}=0.0174533\cdots\text{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=A\sin(\aleph\omega t)$$ where $\omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=\dot e=A\aleph\omega \cos(\aleph\omega t)$$ is in $ms^{-1}$.
Another universal constant worth to know:
$$\Pi=3.141593\cdots\text{ rad}=180\text{ deg}.$$
denotes the aperture angle of a half-circle.
It fufills
$$\aleph\Pi=\pi.$$
Hence the famous Euler formula,
$$e^{i\aleph\Pi}=-1.$$
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.