The arclength formula, θ = S/R is simliar in format to the trigonometric formula tan θ = opposite / adjacent, where S= opposite, and r = adjacent (although the trigonometric formula is for right angled triangles, it seems to me they show some similarities).
This uses tan, whereas the arclength formula does not.
Why is this the case?
On
The spirit of the arc-length formula is the same as that of Thales' theorem. When the radius is $1 $, the arc-length is the angle, by definition of an angle. When the radius is $ r $, you have scaled your picture by a factor of $ r $, and every length of every curve is thereby multiplied by a factor of $ r $ in the process, meaning that the arc-length is now $ r\theta $.
As for the tangent - well, it measures something completely different. But it is a definition rather than a formula that you can prove.
The answer is very simple: $\tan\theta = S/R$ would only apply to a right triangle! This is a sector, which has a curved edge and is not a triangle.
For a wonderful counterexample, consider the angle $\pi$, which corresponds to half a circle. On the unit circle the arc length is $(1)(\pi)=\pi$, but how could this possibly relate to the tangent function?