= Spherical law of sines
{title2=$\sin A/\sin a=\sin B/\sin b=\sin C/\sin c$}
For a nondegenerate convex minor <spherical triangle>, with angular sides $a,b,c$ opposite interior <angles> $A,B,C$, the <spherical law of sines> states
$$
\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}.
$$
The <sines> of sides refer to central <angles>, so a physical side <length> $\ell_a$ on a <sphere> of <radius> $R$ must enter as $\sin(\ell_a/R)$. All six <angles> lie in $(0,\pi)$, and the denominators are positive. The reciprocal form, $\sin a/\sin A=\sin b/\sin B=\sin c/\sin C$, is equivalent. This is Tatum's sine formula, equation 3.5.1, in https://www.astro.uvic.ca/~tatum/celmechs/celm3.pdf[Celestial Mechanics, section 3.5: spherical sine and cosine formulas].
A short proof explains the <symmetry> of the <spherical law of sines>. Let $u,v,w$ be the vertex <unit vectors> at $A,B,C$. At $u$, the outgoing unit <tangent vectors> towards $v,w$ are $t_B=(v-\cos c\,u)/\sin c$ and $t_C=(w-\cos b\,u)/\sin b$. Their <angle> is $A$, so the magnitude of their <cross product> is $\sin A$. Since both <tangent vectors> are <orthogonal> to $u$, the <scalar triple product> obeys
$$
|u\cdot(v\times w)|=\sin b\sin c\sin A=\sin c\sin a\sin B=\sin a\sin b\sin C.
$$
Dividing by $\sin a\sin b\sin c$ proves the <spherical law of sines>. Its beauty is that the same three-dimensional <scalar triple product> can be calculated at any vertex, producing all three ratios at once.
For the <spherical triangle> worked out under the <spherical law of cosines>, $A=90^\circ$, $b=c=60^\circ$ and $\cos a=1/4$. Hence $\sin a=\sqrt{15}/4$, and the <spherical law of sines> gives
$$
\sin B=\frac{\sin b\sin A}{\sin a}=\frac{\sqrt3/2}{\sqrt{15}/4}=\frac2{\sqrt5}.
$$
The <inverse sine> gives $B\approx63.434949^\circ$, but the supplementary <angle> $116.565051^\circ$ has the same <sine>. The <spherical law of sines> alone therefore leaves two candidates for this <angle>. The <spherical law of cosines>, using the known three sides, gives $\cos B=1/\sqrt5>0$ and selects $63.434949^\circ$; likewise $C=B$. When only two sides and an opposite <angle> are given, two distinct valid <spherical triangles> can remain. A <cosine> relation resolves an <inverse sine> branch only when enough additional data are known; it does not make every ambiguous data set unique.
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