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I was trying out Dimensional Analysis on a few equations and realized that angles have no dimension. Otherwise equations such as $s=r\theta$ are not dimensionally consistent.

Further, why don't trigonometric ratios have any dimension?

PS: I couldn't find any appropriate tag for this question. Could someone re tag as appropriate? Thanks.

MJD
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Green Noob
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    The answer to both your questions is the same - trig ratios and radian measure are both dimensionless because they are defined as the ratio of two lengths, which have the same units so they cancel. – Ragib Zaman Jan 30 '12 at 09:57
  • @RagibZaman In that case does it mean that Dimensional Analysis cannot be applied to equations which involve ratios of 2 quantities with the same unit? – Green Noob Jan 30 '12 at 10:03
  • Further, how can we extend this logic for angles in degrees? I don't think it is defined as a ratio of two lengths. – Green Noob Jan 30 '12 at 10:08
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    @GreenNoob: No it does not mean Dimensional Analysis cannot be applied, it just means such ratios have an empty dimension. If they are equated or compared to an expression with a non-empty dimension, then there is an error, but if they are equated or compared to another such ratio or an explicit number, then no error is detected. – Marc van Leeuwen Jan 30 '12 at 10:09
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    When doing dimensional analysis on a problem which has an angle as a parameter, you generally find that the solution can involve an arbitrary function of the angle, as in e.g. the problem of how far a ball travels under a gravitational field $g$ if thrown with velocity $v$ at angle $\theta$ (the dimensional analysis solution is $x\propto v^2/g \times f(\theta)$ – Chris Taylor Jan 30 '12 at 10:34
  • More precisely, recall the definition of the radian: you are in essence dividing the length of a circle's arc by the length of its radius. The two quantities you're dividing have the same dimensions, and thus... – J. M. ain't a mathematician Jan 30 '12 at 14:18
  • For dimensional analysis purposes, however, it's fine to treat the radian as $\frac{\text{meter}}{\text{meter}}$, $\frac{\text{cm}}{\text{cm}}$, or whatever length unit is found convenient in the application. – J. M. ain't a mathematician Jan 30 '12 at 14:19
  • It's possible to treat angle with dimensions, but then you would have to recast the dimensional analysis of all things, and some quantities could be split as a result. In Leo Young's book on EM units, solid angle appears as a specific dimension. The definition of a radian as $\frac{metre}{metre}$ supposes that the constant of circumference to radius is dimensionless, whereas in $C = k R \theta$, $k$ would have units of $degree^{-1}$. A similar dimensioned constant appears in the CODATA (1000 mol/kg), where the previous was purely a numeric ratio. – wendy.krieger Jul 14 '13 at 07:19

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Compare to a ratio of weights, it is weights that you compare. You get no units for the ratio, right ? I mean the result is independent of choice of units. But with angles your ratio is with lengths ! not angles ! so don't be surprized that you get "units". Any partition of an angle is named with "units" The funny thing is that unit transformation with angles obeys the same laws as with any other unit.

Engel John
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