Are Angles Dimensionless?
Date: 08/31/2003 at 12:30:32 From: Christopher Subject: Dimensional inconsistencies In the equation arc length = r*theta the numbers seem to work out fine, but the dimensions seem to be completely incorrect. In any circle there is a specific angle where the radius is equal to the arc length. This angle is defined as one radian and is approximately equal to 57.29 degrees. From this it can then be said that for every length of arc that is equal to the radius there is 1 radian of angle subtended. For any sector the angle will be so many lots of 1 radian, and thus the arc length will be the same number of times of the radius. From this you can then write: arc length / angle (in radians) = radius / 1 radian It can now be said that: arc length = angle (in radians) * radius / 1 radian This is not the same as the equation arc length = r * angle (in radians) Furthermore the r*theta equation seems to imply that the angle is dimensionless (has no units!). This is obviously incorrect, as it seems to imply that an angle can be measured with numbers. You have to have some way of measuring an angle just like you do with mass, or with length. Units are simply a standard measure of some parameter, which tell you what you are dealing with, whether it is length, mass, angle or whatever. The numbers just tell you how many of them you have. How you can say that the angle measured has no units goes against everything I know. But I am not arrogant enough to say the whole world is doing it wrong and I am doing it right. Can you explain to me what is going on here?
Date: 08/31/2003 at 23:16:14 From: Doctor Peterson Subject: Re: Dimensional inconsistancies. Hi, Christopher. Angles are indeed dimensionless; the reason is that an angle is measured as the ratio of arc length to radius (meters per meter, say), which cancels out any units. Other angle measurements, such as degrees, are also dimensionless, though they are defined by different ratios, such as the ratio of the arc length to 1/360 of a circle, which results in the need for a multiplier. It is for this reason that we still need to name the unit, radians, rather than leaving it out entirely. Saying "radians" specifies the way in which the angle is measured; but it is still dimensionless and can be ignored as long as you know that radians are the correct method for the calculation being done. Since this is a tricky issue, I did a web search for references to support my opinion. Here are some: Why Use Radians instead of Degrees? http://mathforum.org/library/drmath/view/54181.html All angle measures can be said to be dimensionless. A radian is the ratio of an arc length to a radius, and the ratio of two lengths is dimensionless. A degree is 180/pi radians, and the constant 180/pi is dimensionless, so a degree is also dimensionless. Trig functions are also by nature dimensionless. They can be defined as ratios of sides of a right triangle. Again, the ratio of two lengths is a dimensionless quantity (a "pure number"). Glossary of Mathematics http://westview.tdsb.on.ca/Mathematics/R.html Radian Measure: A dimensionless unit of angle measure (as opposed to degree which has a dimension). An angle of one radian (1 rad) causes the ratio of the length of the arc of the circle's circumference to the radius of the circle to be one. Since this is a ratio the radian is not a unit and it is dimensionless. We have two different angle measures (pi radians = 180 degrees) since it is very troublesome to integrate or differentiate trig functions whose arguments are in degrees. One radian approximately equals 57.3 degrees. One of the real values of radians is that it is a non-unit "unit" (since it is the ratio of an arc length to the radius of a circle). That is, in any calculation using an angle measure, using radians does not introduce a new unit into the process to muddy the situation. If you are putting an angle into a trig function, it doesn't matter whether you use radians or degrees, since, for example, sin 30 deg. = sin pi/6 rad. = sin pi/6 (no unit). IF you are using an angle value in any other calculation, however, using degrees often produces non-sense. For example, evaluate the expression x^2-2x-4 for x being 30 deg. or pi/6 radians (or, therefore, just plain pi/6 with no unit). With degrees, the result is non-sense -- you get 900 square degrees - 60 degrees - 4. With radians, the non-unit "unit", the result is a pure, unitless number. Evaluating Taylor polynomials in order to get approx. values of the trig functions for any angle, then, requires radians to make any sense whatsoever. Calculators use similar methods to produce trig values, so if you ask your calculator for sin 30 degrees, it internally converts the angle to radians, uses that value in an algebraic expression that it has stored in a memory location that your "sin" key accesses, then produces the appropriate UNITLESS value for sin 30 degrees. Likewise in any other formula - in math, science, engineering, architecture, or whatever - if you have to use an angle in most any way other than in a trig function, radians are required and degees produce non- sense. I disagree with this as to degrees; I would say the degree is still dimensionless, but is a different unit (scaling). But the radian is generally assumed, so that the degree requires conversion. For example, the Taylor series for the sine, as usually given, assumes that the sine function takes an angle in radians; but one could write a Taylor series for the sine of an angle in degrees, which would simply have different coefficients. There is no inherent difference, contrary to what this author says; it is just a rescaling. This is why we do include the radian as a unit in calculations, as a reminder of the correct scaling; but it can be ignored where necessary. Introduction to the Aerodynamics of Flight http://history.nasa.gov/SP-367/appendb.htm The measure of the central angle of a circle is defined as the ratio of the subtended arc of the circle divided by the radius, that is, a ratio of two lengths. Thus, this measure is dimensionless but is assigned a special name of radians. Additionally one may express the angle in degrees by noting that an angle of 1 radian equals about 57.3°. The fact that both radian measure and degree measure are dimensionless means that the numerical value of an angle does not change from one system of units to another. I'm not sure what that last sentence means, but otherwise I agree. The Three Classes of SI Units and the SI Prefixes http://physics.nist.gov/Pubs/SP811/sec04.html The SI supplementary units are now interpreted as so-called dimensionless derived units (see Sec. 7.14) for which the CGPM allows the freedom of using or not using them in expressions for SI derived units. (Footnote 3) Thus the radian and steradian are not given in a separate table but have been included in Table 3a together with other derived units with special names and symbols (see Sec. 4.2.1). This interpretation of the supplementary units implies that plane angle and solid angle are considered derived quantities of dimension one (so-called dimensionless quantities - see Sec. 7.14), each of which has the unit one, symbol 1, as its coherent SI unit. However, in practice, when one expresses the values of derived quantities involving plane angle or solid angle, it often aids understanding if the special names (or symbols) "radian" (rad) or "steradian" (sr) are used in place of the number 1. For example, although values of the derived quantity angular velocity (plane angle divided by time) may be expressed in the unit s^-1, such values are usually expressed in the unit rad/s. This is "official"! The radian is optional, and can be replaced by 1. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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