For the past day, I have been trying to calculate the arc length of a sphere from a pole to a longitudinal cross-section with a specified circumference.
I have diagrams and equations belo w. However, I am getting answers that seem incorrect. Are equations correct? Any improvements to it?
Computing the distance $d$ using $\theta$ in degrees is...ill-advised. :) Radians are defined so that $d = r_{0}\theta$; mathematics seldom gets simpler than that. Further, if you're using a calculator to test values, there's a non-negligible chance the outputs are coming to you in radians.
That aside, you're essentially there. If you measure $\theta$ in radians, then \begin{gather*} C = 2\pi r_{i},\quad\text{or}\quad r_{i} = C/(2\pi); \tag{1} \\ \sin\theta = r_{i}/r_{0},\quad\text{or}\quad \theta = \arcsin(r_{i}/r_{0}); \tag{2} \\ d = r_{0}\theta = r_{0} \arcsin\bigl[C/(2\pi r_{0})\bigr]. \tag{3} \end{gather*} For example, if $r_{0} = 1$, the equator has $C = 2\pi$, and the formula gives $d = \pi/2$ as expected. If $C = \pi$, then $d = \pi/6$ or $5\pi/6$, again as expected.