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.


marked as duplicate by Zev Chonoles, Amzoti, Andrey Rekalo, user67258, Alex Wertheim Jul 12 '13 at 18:56

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    $\begingroup$ 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. $\endgroup$ – Ragib Zaman Jan 30 '12 at 9:57
  • $\begingroup$ @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? $\endgroup$ – Green Noob Jan 30 '12 at 10:03
  • $\begingroup$ Further, how can we extend this logic for angles in degrees? I don't think it is defined as a ratio of two lengths. $\endgroup$ – Green Noob Jan 30 '12 at 10:08
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    $\begingroup$ @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. $\endgroup$ – Marc van Leeuwen Jan 30 '12 at 10:09
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    $\begingroup$ 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 $x\propto v^2/g \times f(\theta)$ $\endgroup$ – Chris Taylor Jan 30 '12 at 10:34

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.


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