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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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