The small spherical triangle

When a spherical triangle becomes smaller and smaller on a sphere of fixed radius, its angles remain finite but its sides tend to zero length. The triangle, in fact, approximates more and more to a plane triangle and we would expect that the formulas of spherical trigonometry would degenerate into the well-known formulas of plane trigonometry. This indeed is the case.

Figure 1. flight route from Prestwick to Gander

As an example take the cosine formula, namely cos a = cos b cos c + sin b sin c cos A.
If θ is any angle, in circular measure,

........(1)

If θ is sufficiently small, we may write
cos θ = 1 − θ²/2 sinθ = θ.
Now let the spherical triangle ABC shrink so that the cosine formula may be written as

or

We can neglect b2c2/4 since it is of smaller order than a2/2, etc.
Hence,
a² = b² + c² − 2bc cos A
the well-known formula of plane trigonometry.