Editing Law of sines (section)

=== Geometric proof ===
Consider a unit sphere with:
<math display="block">OA = OB = OC = 1</math>

Construct point <math>D</math> and point <math>E</math> such that <math>\angle ADO = \angle AEO = 90^\circ</math>

Construct point <math>A'</math> such that <math>\angle A'DO = \angle A'EO = 90^\circ</math>

It can therefore be seen that <math>\angle ADA' = B</math> and <math>\angle AEA' = C</math>

Notice that <math>A'</math> is the projection of <math>A</math> on plane <math>OBC</math>. Therefore <math>\angle AA'D = \angle AA'E = 90^\circ</math>

By basic trigonometry, we have:
<math display="block">\begin{align}
  AD &= \sin c \\
  AE &= \sin b
\end{align}</math>

But <math>AA' = AD \sin B = AE \sin C </math>

Combining them we have:
<math display="block">\begin{align}
\sin c \sin B &= \sin b \sin C \\
\Rightarrow \frac{\sin B}{\sin b} &=\frac{\sin C}{\sin c}
\end{align}</math>

By applying similar reasoning, we obtain the spherical law of sines:
<math display="block">\frac{\sin A}{\sin a} =\frac{\sin B}{\sin b} =\frac{\sin C}{\sin c} </math>

{{see also|Spherical trigonometry|Spherical law of cosines|Half-side formula}}