The Flight Path From Singapore

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3.2 The Haversine formula
The haversine formula is an important equation in navigation, calculating the haversine of the central angle between two points on a great circle of a sphere using the latitudes and the longitudes of the two points: sin2 (d2r) = sin2 ( Φ2 - Φ12) + cos (Φ1) cos (Φ2) sin2 ( λ2 - λ12)

where Φ1 and Φ2 denote the latitudes of Singapore and Chicago respectively, and λ1 and λ2 denote the longitudes of Singapore and Chicago respectively. We can rearrange the haversine formula to give us the distance d between the two cities.

3.3 The Haversine function
In order to show how the haversine formula is derived, we must first understand the function of the haversine. The term haversine appears to have originated from “half a versed sine”, or “half a versine”. The haversine function is useful as it can calculate the distance of an arc of a great circle. The flight path from Singapore to Chicago is, in fact, an arc of a great circle.

The versine function is one of many trigonometric functions that were once popular in fields such as navigation, but now are practically obsolete and are no longer taught in schools. Commonly abbreviated as versin, the versine function is versin (α) = 1 – cos (α) where α is the variable in the function. However, this does not seem very useful in the derivation of the haversine formula. We can manipulate the versine function into a more useful substitution using the half-angle identity sin2 (12α) = [1 – cos (α)] as sin2 (12α)
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