Angles are absolutely more dimensionless than lengths are. For an easy check you can't add quantities where the dimension differs, which means it doesn't make sense to add a length to its cube. On the other hand it does make sense to add an angle to its cube - this is a necessary component of computing sin(angle) by the power series sin(angle) = angle - (angle^3)/6 + ...
Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless.
The reason why this is possible is that the arguments of such nonlinear functions are either explicitly or implicitly not the physical quantities, but their numeric values, i.e. the ratios between those quantities and their units, which are dimensionless.
In the case of the nonlinear sinus function, what is usually written as sin(x) is just one member of a family of functions where the arguments are angles implicitly divided by units of plane angle:
sin(x) is the sinus function with the angle implicitly divided by 1 radian
sin(x * Pi/2) is the sinus function with the angle implicitly divided by 1 right angle
sin(x * Pi*2) is the sinus function with the angle implicitly divided by 1 cycle a.k.a. turn
sin(x * Pi/180) is the sinus function with the angle implicitly divided by 1 sexagesimal degree
It is very sad that the logical thinking about angles of most people has been perverted by what they have been taught in school, which is just a bunch of nonsense copied again and again from one textbook to another.
> sin(x) is the sinus function with the angle implicitly divided by 1 radian
This to me sounds like the most natural explanation. For example, in a sibling comment someone mentioned that "you can calculate e^(-t)", but I disagree: in physics it's always e^(-t / T), where T is some time constant, so that the argument of the exponential is dimensionless. Same applies to sin(x): usually we write something like sin(2pi f t), where the units of f and t cancel out, and the 2pi is there to cancel out the invisible implicit 1 radian. sin(ft) would be wrong, at t = 1 / f you wouldn't have advanced by a full cycle.
Units are not the same as dimensions, something can have a dimension of 1 (which is what we usually mean by "dimensionless") and still have different units, just as something can have a dimension of length but still be measured in meters or feet.
As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you should use radians. Otherwise you use something else.
Actually the dimensions are by definition the same as the fundamental units, i.e. the units that are chosen freely, independently of all other units.
A dimensional formula of a quantity just writes its unit as a function of the fundamental units.
In any equality of physical quantities, in the two sides not only the dimensionless numeric values must be equal, but also the units must be equal, which is usually expressed by saying that the dimensions must be the same, and it is verified by writing in both sides the dimensional formulae, i.e. the units of both sides as functions of the fundamental units.
A dimensionless quantity is a ratio of two quantities that are measured by the same unit, so that the units simplify during the division.
There may be different but related dimensionless quantities, which are differentiated by different definitions of those quantities, but a dimensionless quantity cannot have different units.
This is just meaningless mumbo-jumbo that has been sadly introduced in the documents of the International System of Units, in 1995, after a shameful vote of the delegates, who have voted automatically, without thinking or discussing, a vote equivalent with establishing by vote that 2 + 2 = 5.
Sure, but that's orthogonal to the "angles don't really have units" assertion and the "it does make sense to add an angle to its cube" assertion, which are the ones I'm responding to.
As another example for the second assertion, you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed, etc, which comes up all the time. But that doesn't mean `dimensionless + seconds + seconds^2` implies seconds are dimensionless any more than sin's series with `angle + angle^3` implies that angles are dimensionless.
you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed
The argument of exp does have to be dimensionless, exactly because adding seconds to seconds squared doesn't work. If t has units of time, there has to be another factor with units of inverse time, for example continuous compound interest is exp(rate*time).
If you say "angles have units" I agree with you - obviously you can measure them in degrees or radians or whatever you want. I was responding to the claim
> Angles aren't dimensionless any more than lengths are dimensionless
They are dimensionless, but they still have units. The concepts are orthogonal.
As for the question about adding an angle to its cube, I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.
> They are dimensionless, but they still have units. The concepts are orthogonal.
The concepts are not orthogonal, they are incompatible.
A dimensionless quantity (which angles are not) is by definition the ratio of two quantities that are measured with the same unit.
When you compute the ratio by division, the two identical units disappear from the result, therefore the result is indeed dimensionless.
There is no way to choose a unit for a dimensionless quantity in the usual sense.
At most you could define a new different dimensionless quantity, as the ratio of two dimensionless quantities, i.e. as a ratio of ratios, but because it needs a different definition this should better be viewed as a different quantity, not as the same quantity with a different unit.
Ok, you've replied to quite a lot of my comments, with some fairly confusingly worded responses. Some of your claims appear to be incompatible as far as I read them.
For example you claim that angles are not dimensionless, but that dimensionless quantities are formed by the ratio of two quantities with the same unit. Since the angle subtended by an arc in a circle is the ratio of the arc length and radius, it would seem that these two claims contradict each other.
I do agree that without some aditional work the power series argument I wrote above does seem to be wrong.
No, even if you have reproduced a definition of the plane angle that is encountered in many textbooks "the angle subtended by an arc in a circle is the ratio of the arc length and radius", this definition is very incorrect.
This very wrong definition forced upon many students is the root of all misconceptions about plane angles.
The reason why this definition is wrong is because some words are missing from it and after they are added it becomes obvious that its meaning is different from what many teachers claim.
First there is no relationship whatsoever between the magnitude of a radius and the magnitude of an angle. What that definition intended to say was:
"the angle subtended by an arc in a circle is the ratio of the arc length and of the length of an arc whose length is equal to the radius".
By definition, a radian is defined as the angle subtended by an arc whose length is equal to the radius.
To explain how plane angles are really defined would take more space, but the only thing that matters is that the characteristic property of plane angles is that the ratio between two plane angles subtended by two arcs of a circle is equal to the ratio of the lengths of the two arcs.
Introducing this characteristic property of the angles in the so-called definition from above reduces it into the sentence "the angle is measured in radians". This is either a trivially true sentence when the angle is indeed measured in radians, or it is a trivially false sentence when the angle is measured e.g. in degrees. It certainly is not a definition.
If that had been the definition of plane angle, that would have meant that the plane angle was discovered only in the second half of the 19th century, together with the radian, while in reality plane angles have been used and measured with various units for millennia.
To measure plane angles, it is necessary to first choose an arbitrary angle as the unit angle, for instance an angle of one degree.
Then you can measure any other angle by measuring both the length of the corresponding arc and the length of the arc corresponding to the chosen unit angle. Then the two lengths are divided, giving the numeric value of the measure of the angle.
I would argue that dimensionful quantities are never used as arguments to power series, however you are correct that this does not imply that angles are dimensionless, since (like we do with other quantities, we can divide by whatever unit we like to get something dimensionless). I withdraw that argument.
A better argument that angles are dimensionless is that dimensionless quantities are formed by the ratio of two quantities with the same dimension, and the angle subtended by an arc in a circle is given by the ratio of the arc length and the radius.
