If I sample three points uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine?

More generally, for $k<n$ if I sample $k$ points uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine?

An asymptotic form for large $n$ can be deduced in the case $k=2$ from Lemma 4 of Appendix A of Laarhoven's Sieving for shortest vectors in lattices using angular locality-sensitive hashing, but my attempts to generalise to $k=3$ do not match empirical distributions for $n=10$.

If I sample three points independently, uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine?

More generally, for $k<n$ if I sample $k$ points independently uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine?

An asymptotic form for large $n$ can be deduced in the case $k=2$ from Lemma 4 of Appendix A of Laarhoven's Sieving for shortest vectors in lattices using angular locality-sensitive hashing, but my attempts to generalise to $k=3$ do not match empirical distributions for $n=10$.