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Re: Haversine- how to derive it?
From: Robert Bernecky
Date: 2015 May 17, 07:01 -0700

I would write it as "haversine of angle 45° has the numerical value 0.1464466"

hav(45°)= 0.1464466

to calculate hav(x) you can use either

$\frac{1-\cos x}{2}$ or $\sin^2\left(\frac{x}{2} \right )$

the latter expression means first halve the angle e.g. 45/2 = 22.5 deg. , then find the sin(22.5°) = 0.3826834

finally square this to get 0.1464466 which matches what you get using the first expression (1- cos x)/2

You will also want to know how to find the "inverse haversine": given that haversine of some angle is 0.1464466 , what angle is that?

hav(x)= 0.1464466

as Frank Reed posted, you can find x using

cos x= 1- 2*0.1464466

cos x = 0.707106781

now use a calculator to find the inverse cosine of 0.707106781

x = cos^-1 (0.70710678) = 45°

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