Angle <i>is</i> technically dimensionless.<p>Not sure how to explain the &quot;zen&quot; of radians; it&#x27;s kind of the angular equivalent of &quot;e&quot; in natural logarithms. People who first learn logarithms say WTF, what is this &quot;e&quot; thing? why not just log10 everywhere? But when you get into calculus you find that &quot;e&quot; is the only log base that doesn&#x27;t introduce extra baggage; if you take the derivative of a^x the result is ln a * a^x, and if you take the derivative of log_a x = ln x &#x2F; ln a, you get 1&#x2F;(x ln a). In both cases the &quot;ln a&quot; term is 1 only when a = e.<p>For radians, the functions sin wt and cos wt (w = &quot;omega&quot;) have derivatives of w * cos wt and -w * sin wt when wt is measured in radians. Otherwise there is an additional constant of proportionality that depends on the angular &quot;unit&quot;. (the term &quot;unit&quot; in quotes because it&#x27;s still dimensionless but just related to radians by a dimensionless scaling factor, like 180&#x2F;pi for radians &lt;-&gt; degrees or 1&#x2F;(2*pi) for radians &lt;-&gt; revolutions.