Say I start with the cone shown in the left in the diagram below. I can find the angle a formed by the wall of the cone as 2*arcsin(R/S).
If I cut the cone open and flatten the wall into a 2D surface, it forms a segment of a circle having radius S and an arc length equal to the circumference of the cone, 2πR. The segment angle b can be found as the fraction of the circle comparing the segment arc to the circle circumference, which reduces to 360*R/S.
If I want to calculate the actual angles, I need a trig table for the cone. If I'm only interested in the relationship between a and b, I could calculate the angles and compare them. However, it intuitively seems like there might be some simple ratio or relationship between the two angles.
That's my question. Is there a simple ratio or relationship between the cone angle and the angle of its flattened surface?
I'll do this first with all angles in radians, since the relationships are simpler that way. Therefore from your formula for $b,$ but using $2\pi$ radians rather than $360$ degrees, we have $$ b = \frac{2\pi R}{S}. \tag1$$
Therefore, after dividing both sides of Equation $(1)$ by $2\pi$, $$ \frac RS = \frac{b}{2\pi}. \tag2$$
You also found that $$ a = 2 \arcsin\left(\frac RS \right). \tag3$$
Now use Equation $(2)$ to substitute for $\frac RS$ in Equation $(3)$: $$ a = 2 \arcsin\left(\frac{b}{2\pi} \right) = 2 \arcsin\left(\frac{b}{360^\circ} \right). \tag4$$
The version with division by $360^\circ$ is in case you insist on measuring $b$ in degrees and don't want to convert it to radians.
To get $b$ in terms of $a$ we just undo all the things we had to do to $b$ in order to get $a$, starting with the multiplication by $2.$ That is, divide by $2,$ take the sine, and multiply by $2\pi$:
$$ b = 2\pi \sin\left(\frac a2 \right) = 360^\circ \times \sin\left(\frac a2 \right). $$