Tom writes some good stuff, but in general I disagree with his final points about needing two different sets of functions, with a special one for mathematicians....
However from a pragmatic point I had already proposed a set of options for such deserving cases 
and actually ended up with four special cases (that is, four options for use / non-use of angle unit enforcement).
So while I disagree with Tom wanting two variants, I (hypocritically) allow four, but only as options.
Philip
Stuart,
Yes the editor interface is diabolical. I think SI will have an Angle 'dimension' before PTC manages to correct it (at this rate )
I personally disagree that Angle is defined as a ratio of two lengths. A right angle is well defined without any lengths..... The choice of the unit of angle is more of the problem than is discussed. That is, it is because every body appears to assume that the radian defines angle, rather than the the causality being other way around, we have Angular measure as the clear independent dimension, and then we argue about what base unit to use, just like imperial vs metric. In this case we have that the metricators would like to use radians, but all the imperialists still want to use degrees and right angles (90 degrees) as their base unit. And what a mess it causes, with every calculator having different defaults and different methods of changing between them.
Everyone knows sin(30)=1/2 !!! even I can draw that equilateral triangle 
There is a potential problem with lengths of arc. The problem being you can't measure them with a ruler !! but no one mentions that in the formulae, which Is why I proposed, for mathcad the arc(r,theta) function. That is, there is no "transformation constant", rather there is a "transformation function" (using your phrase for 'k')
the y = r.sin(theta); theta in angle, r and y in units of length, has no problems at all in my way of thinking. When theta arrives at the sin function, it has a number and a unit quantity. The number is divided by the unity quantity, measured in an appropriate scale (e.g. fraction of a right angle), and then numerically calculated by say the CORDIC algorithm to produce a numeric value for sin(theta), which is then multiplied into r to get the length y.
I've deliberately avoided both the classic sin(x)=x-x^3/3!+.. formula and the use of radians, just to show that they aren't necessary.
Another diversion is that most pictures of the sin() triangle I would argue as figuratively "wrong", in that the common divisor should be along the x-axis, that is the Hypotenuse of the constructed triangle (yes the right triangle is constructed after the angle : see causality) is along the x-axis. At this point you get to choose how to do the sin(2.theta) vs sin(theta/2), because one can construct the semicircle over the angle to get all sorts of angular doubling formulae..
Have fun.
Philip
Philip Oakley wrote:
I personally disagree that Angle is defined as a ratio of two lengths. A right angle is well defined without any lengths..... The choice of the unit of angle is more of the problem than is discussed. That is, it is because every body appears to assume that the radian defines angle, rather than the the causality being other way around, we have Angular measure as the clear independent dimension, and then we argue about what base unit to use, just like imperial vs metric. In this case we have that the metricators would like to use radians, but all the imperialists still want to use degrees and right angles (90 degrees) as their base unit. And what a mess it causes, with every calculator having different defaults and different methods of changing between them.
I don't have a problem with that; my proposal handles it easily.
There is a potential problem with lengths of arc. The problem being you can't measure them with a ruler !! but no one mentions that in the formulae, which Is why I proposed, for mathcad the arc(r,theta) function. That is, there is no "transformation constant", rather there is a "transformation function" (using your phrase for 'k')
Well, you can measure them with a ruler ... if you cheat and use a bendy ruler sideways 
However, the problem I have with most other proposals (sorry) is that they end up doing something 'special' to handle different circumstances, thus reducing the consistency and presenting potential problems with the theoretical underpinning. My proposal, AFAICT, applies consistently across all dimensions. If you want to define an arc function, fine - I often find it useful to define functions for things that can be done otherwise. In this instance, I believe that one should be able to simply write c = k . r . θ.
the y = r.sin(theta); theta in angle, r and y in units of length, has no problems at all in my way of thinking. When theta arrives at the sin function, it has a number and a unit quantity. The number is divided by the unity quantity, measured in an appropriate scale (e.g. fraction of a right angle), and then numerically calculated by say the CORDIC algorithm to produce a numeric value for sin(theta), which is then multiplied into r to get the length y.
Once again, I wouldn't treat angle any differently to any other dimension. sin(θ), sin(x) and sin(t), where θ=angle, x=length and t=time, are all handled exactly the same, giving results of angle, length and time respectively. My proposal avoids treating angle as special and doesn't require division by anything to get a dimensionless number. y = r . sin(θ) gets rewritten as y = k . r . sin(θ) as above and gives a dimensionally consistent answer, et sim for y = k . a . sin(t), except that k now has dimension 1/time (or, as stated, it might be conceptually more accurate to let k = length/time and a = length/length).
I've deliberately avoided both the classic sin(x)=x-x^3/3!+.. formula and the use of radians, just to show that they aren't necessary.
Tom and I have addressed this - the proper derivation of such formulae shows that there is a conversion constant of appropriate dimension for each terms that makes the expression dimensionally consistent - think of the standard expression for distance covered under constant acceleration, s = s0 + v0.t +1/2 a. t2, where s0=length, v0=length/time and a=length/time2; this gives additional justification for allowing sin(t) without any attempt to provide spurious conversions from time to angle.
I'd also be interested to see if anyone has given a coherent argument against the addition of an angle unit.
You have no idea what you just asked!
We have had many heated discussions about units, including angular units. I'll refrain from calling them arguments, although a lot could accurately be described that way. I'll leave you to decide if they are coherent! Here's a nice juicy one to start you off:
http://communities.ptc.com/message/52193#52193
Once you have plowed through the 370 posts in that thread (no, that number was not a typo) you can find lots more material by searching these forums for combinations of "angle" "unit","Hz", "radian".
I'd also be interested to see if anyone has given a coherent argument against the addition of an angle unit. Any takers out there?
Jean would have had you for asking that!!!!! 
Mike
Philip Oakley wrote:
Yep, the Neper is a ratio measure, so I was being a bit naughty, but then I did hint that in the updated title..
I disagree that one can pass a unit/dimenion through the equation to the outside. One can't take the sin(3.ft) and say it is sin(3).ft Even sin(3) isn't actually well defined (except when in a cliche of mathematicians..). The modern convention has become to assume that it should be in radians, but that isn't necessary. It must simply be an angle.
Stuart_Bruff wrote:
I kind of disagree with this. An amplitude is not measured in nepers; specifically, a neper is the ratio of two field levels - as you've mentioned parenthetically, it is a ratio. As you say, if theta is an angle, then sigma must be an angle for dimensional consistency. The 'correct' expansion, I hypothesize, is:exp(sigma+j.theta) = exp({sigma}[angle]+j.{theta}[angle])
= exp({sigma}+j.{theta})[angle]
= (exp({sigma}).exp(j.{theta})[angle]
where {} & [] indicates the numerical value & unit of a quantity, respectively.
The above allows replacement of angle by any arbitrary quantity and does not require the neper. Any quantity conversions should be handled explicitly.
As stated, I see no theoretical reason to disallow sin(3.ft) - it merely needs care in defining exactly what the expression means. I'm not sure what you mean by sin(3) not being well defined; AFAICT, as a non-quantified number, it is well defined and evaluates according to the standard expansion. Tom has drawn the distinction between the 'mathematical' function and the 'physical' one, but I believe that this doesn't really address the core of the issue, which is that there should be a consistent mathematical structure that allows continuous flow of numerical quantities through equations - after all, nature manages it 🙂
Stuart
Hi Stuart,
Stuart_Bruff wrote:
As stated, I see no theoretical reason to disallow sin(3.ft) - it merely needs care in defining exactly what the expression means. I'm not sure what you mean by sin(3) not being well defined;
AFAICT, as a non-quantified number, it is well defined and evaluates according to the standard expansion.
Tom has drawn the distinction between the 'mathematical' function and the 'physical' one, but I believe that this doesn't really address the core of the issue, which is
that there should be a consistent mathematical structure that allows continuous flow of numerical quantities through equations - after all, nature manages it 🙂
Stuart
I've split your response into what I think are the four relevant parts, so we can be clear about each.
1. It is my contention that "sin" is a mechanism for taking in an angle and from it determining the ratio now attributed to it. In this(my) context, it must have an angle given to it. It has a non-linear response, that is if the angle is doubled, the response isn't twice the previous value.
2. The problem (I see) is the association of that "standard expression" (which is OK as a stand alone mathematical expression) with my "sin of angle" context above(#1). [Note that the standard expression is non-linear]
3. I understood that Tom had the two separate function, which he treated as distinct and separate. This side stepped the difficulties - one foot in each camp.
4. "Nature abhors mathematics, simply getting on with it"
rather mathematics is our abstraction of the situation. For the units& dimensions arguments to work as we systematists desire, we need the summed dimensional components to have a common dimensional power (Buckingham's Pi theorem). This makes the "standard expression" unsuitable for the purpose of evaluating sin(X), rather the standard expression needs to take x=X/rad (I used big and little X for the distinction between with and without the Angle dimension)
Philip
Philip Oakley wrote:
Hi Stuart,
Stuart_Bruff wrote:
As stated, I see no theoretical reason to disallow sin(3.ft) - it merely needs care in defining exactly what the expression means. I'm not sure what you mean by sin(3) not being well defined;
AFAICT, as a non-quantified number, it is well defined and evaluates according to the standard expansion.
Tom has drawn the distinction between the 'mathematical' function and the 'physical' one, but I believe that this doesn't really address the core of the issue, which is
that there should be a consistent mathematical structure that allows continuous flow of numerical quantities through equations - after all, nature manages it 🙂
Stuart
I've split your response into what I think are the four relevant parts, so we can be clear about each.
1. It is my contention that "sin" is a mechanism for taking in an angle and from it determining the ratio now attributed to it. In this(my) context, it must have an angle given to it. It has a non-linear response, that is if the angle is doubled, the response isn't twice the previous value.
2. The problem (I see) is the association of that "standard expression" (which is OK as a stand alone mathematical expression) with my "sin of angle" context above(#1). [Note that the standard expression is non-linear]
3. I understood that Tom had the two separate function, which he treated as distinct and separate. This side stepped the difficulties - one foot in each camp.
4. "Nature abhors mathematics, simply getting on with it"
Philip
Unfortunately, I don't believe that state and religion (physics and mathematics) can be separated so readily. As I've stated somewhere else, I think that the 'necessity' to have distinct 'physical' and 'mathematical' functions to deal with units, is more an indication that something is rotten in Blegdamsvej 17,
2100 Copenhagen than a reflection of the 'reality' (at least, that's my interpretation).
Consider the following (see attached mcad file).
Take a pair of variables, x & y say, in any old units you like, so I'll go for dimension length with unit the metre. Find a blackboard (if you can), and draw a line y = a.x, where a is a constant. Most people, I think, would be prepared to admit that a was either dimensionless or had units m/m. This is interesting, but a little bit predictable. Let's make it slightly more interesting by plotting y = a.x - b.x3/6. Clearly, to be dimensionally consistent, b must have units of 1/m2 (or m/m3). Now, I go into a wild frenzy of series extension, plotting y = a.x - b.x3/6 + c.x5/120 (c having units of 1/m4 (or m/m5)) and drafting in the infinity of myselves in the parallel universes to create an infinite dimensionally-consistent series. As it happens, I set each constant to have numerical value 1; a few seconds cogitation reveals that the resultant curve looks suspiciously like a sine, and a little mathematical prestidigitation allows me to rearrange the final equation as y = sin({x}) m (where {x} is the numerical value of the quantity x). Further reflection, indicates that as my choice of unit, [x], was arbitrary, I could have chosen [x] = V, s, or even rad.
But here's an interesting thing, if [x] was rad then, using your's and Tom's choice, I have sin({x}) rad = sin({x} rad) => sin(x).
However, given that the choice of unit is arbitrary, I have to ask what's so special about the radian that it is allowed as a unit in sin(x) whereas the metre, second, volt, etc aren't? And the answer,AFAICT, is nothing. The nature of our universe is such that physically rotating an object through 360 degrees in 2D results in a physically superposition of that object with itself. This might imply that the radian is special because it's effective domain is 2.pi. However, the total angle is important in our 3D universe, eg if you start turning round on one spot, the fluid in your ears doesn't stop when you pass through 0 degrees again and one complete turn of a winch doesn't unwrap the rope. This makes it physically meaningful to talk about angles in excess of 2.pi.
The apparent problem with attaching a meaning to sin(3 m) is dealt with by noting that the bald statement implies a cyclic period of 2.pi m in the same way that sin(3 rad) implies a period of 2.pi rad. A more general solution for any periodic function allows the user to specify the periodicity given any set of units. Indeed, many of us are introduced to sin (x) where [x]=degree long before we meet the radian. See the attached worksheet for examples, which includes sin (x degrees). Note (changing notation slightly) that this means a rewrite of the standard equations to obtain a 'correct' dimensional understanding, eg, from y=r.sin(theta) to be y=k.r.s where k=1/rad and s=(rad/deg).sin({theta} deg), in a similar way to the redefinition of the arc equation.
I don't see any need to have a foot in each camp if it means bifurcating my body or pretending that the other half doesn't really exist. Mathematics is the handmaiden of physics and if it won't let physics handle units in a consistent fashion that doesn't involve invoking mysterious 'angles' where none exist, then it should be sentenced to bread and water rations for a week.
Message was edited by: Stuart_Bruff SB: The terms x3 and x5 are supposed to be the respective powers of x and show up as such in the editor; however, I can't see them as superscripts when looking at the thread.
