Haversines

\DeclareMathOperator{\vers}{vers} % versine

These are slight modifications of standard trig functions;
they're a bit useful for spherical trig.
(in particular, for astronomy and navigation;
 EG, you can calculate distances between points using lattitude and
 longitude (using the north (or south) pole, construct a triangle))

A cute note in spherical trig:
a nautical mile is one minute, IE, 1/60th of a degree;
as the earth isn't perfectly spherical, this is ill-defined, but
there's an SI standard.

It's so-called b/c it's pretty close to a mile (about 1.2 miles).

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The \Def{versine} is the compliment of the cosine:
$\vers x := 1 - \cos x$
(abbreviated from ``reversed sine'')
(and you get $= \sin^2 (x/2)$)

The \Def{coversine} is obviously
$\covers := 1 - \sin x$

The \Def{exsecant} is
$\exsec x := \sec x - 1$

The \Def{haversine} (abbreviation for ``half versine'') is
$\hav x := \half \vers x = \half (1 - \cos x)$.
