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The Derivative of Haversed Sine


Versine

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Haversed sine is short for half versed sine and is better known by another shortening: haversine, which in turn is abbreviated to {$\operatorname{haversin}$}. At any rate, it really is half of versed sine.

The versed sine ({$\operatorname{ver}\theta$}) is equal to {$ 1 - \cos\theta $} so ({$\operatorname{haversin}\theta$}) is equal to {$ (1 - \cos\theta)/2 $}

Since

{$$ \operatorname{ver}\theta = 2\sin^2 \left( {\theta \over 2} \right) $$}

{$$ \operatorname{haversin}\theta = \sin^2 \left( {\theta \over 2} \right) $$}

I do not know of a good geometric interpretation of haversin except to point at versine and say it is half of that. Haversine is used in navigation formulas which theoretically could be solved by the spherical law of cosines, but in practice are not because cosines of small angles are so close together that computing with them is perilous.


{$ y = \operatorname{archaversin}(x) $} and {$ y=\operatorname{haversin}(x) $}


Sources:

  1. FooPlot: Online graphing calculator and function plotter

Recommended:

Category: Math Calculus Trigonometry


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August 05, 2017

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