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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.

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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.2012-01-30
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    @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?2012-01-30
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    Further, how can we extend this logic for angles in degrees? I don't think it is defined as a *ratio* of two lengths.2012-01-30
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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.2012-01-30
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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)$2012-01-30
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    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...2012-01-30
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    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.2012-01-30
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    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.2013-07-14

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