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What formula relates rational sides and hypotenuse in spherical geometry?

I obtained an example quite by chance:

$$ \cos \frac{5}{18} \cos \frac{1}{3} = \cos \frac{31}{72} $$

with radius of sphere $72$, sides $(20,24)$ and "hypotenuse" $31.$

.. like we have flat Euclidean sides/hypotenuse geometry using rationals $(m^2-n^2,2mn, m^2+n^2 ).$

Narasimham
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    $\cos\left(\frac5{18}\right)\cos\left(\frac13\right)=0.9087340898$ while $\cos\left(\frac{31}{72}\right)=0.9087340145$ – robjohn Mar 22 '20 at 11:15
  • @robjohn Rational solution is not possible due to the transcendental nature of the relation,right ? – Narasimham Mar 23 '20 at 05:29
  • I don't see an easy proof that there are not any rational solutions, but there probably are not any easily generated families of solutions as there are for Pythagorean triangles. – robjohn Mar 23 '20 at 08:57

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