Angle, like any other quantity that is fed to a pure mathematical function (like the sine or cosine function), <i>must</i> be dimensionless [0]; and indeed it is.  &#x27;Degree&#x27; (as in 30 degrees) is not really a unit, but the number π&#x2F;180, representing π units of arc along a circle of radius 180 units.  &#x27;Radian&#x27; is also not a unit, but rather the number 1, representing 1 unit of arc along a circle of radius 1 unit.<p>EDIT:  [0]  For example, one way of defining sin(x) is as the sum of the power series `x - x^3&#x2F;6 + x^5&#x2F;120 - …`; this cannot be dimensionally consistent unless x is dimensionless.  Another is to define y(x) = sin(x) to be the solution of `d^2y&#x2F;dx^2 = -y` satisfying certain initial conditions; but the units of `y` can only be equal to those of its second derivative if the argument of `y` is dimensionless.<p>EDIT 2:  One easy way to fix the potential dimensional inconsistency is to replace `d^2y&#x2F;dx^2 = -y` by `d^2y&#x2F;dx^2 = -k^2 y`, where `k` is just a conversion factor for dimensional consistency (so that `[k] = [x]^{-1}`).  Then a solution is `y = sin(k x)`; and, once again, the argument of the sine function is dimensionless.