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Asked this before. What is the geometrical significance of the relation

$$ \frac{ \sin A}{ \sin a}=\frac{ \sin B}{ \sin b}=\frac{ \sin C}{ \sin c}= ? $$

in a spherical triangle?

How can we see it as some ratio of lengths, as some angle, a solid angle, or a cross ratio, integral curvature ? or whatever. It is found also in the century old book copy by George S. Carr, A Synopsis of Elementary Results in pure Mathematics, page 894, used by S. Ramanujan.

When the circumradius $r$ is small compared to sphere diameter, the relation reduces to $1/{2R}$.

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    there is nothing special about this. The thing that is new, compared with planar trigonometry, is the two laws of cosines. Because, you see, on the sphere, similar triangles are congruent. – 2017-01-08
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    Actually two laws of cosines is just small stuff. The big thing is the presence of dual triangles. Given any spherical triangle, there is a second or "dual" spherical triangle, such that the sides of either triangle are supplementary to the angles of the other. This is the real difference from planar geometry, and it causes many laws such as the Law of Cosines to come in pairs. – 2017-01-09
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    Thanks, there is such a tag as geometric- interpretation. – 2017-01-09

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