Secret trigonometry

Hidden Trigonometry functions

The versine or versed sine is a trigonometric function found in some of the earliest (Vedic Aryabhatia I) trigonometric tables.
The versine of an angle is 1 minus its cosine.
There are several related functions, most notably the coversine and haversine.
The latter, half a versine, is of particular importance in the haversine formula of navigation
(source: Wikipedia)

vsin(Θ) = 1 - cos(Θ)

cvsin(Θ) = 1 - sin(Θ)

vcos(Θ) = 1 + cos(Θ)

cvcos(Θ) = 1 + sin(Θ)

hvsin(Θ) = (1 - cos(Θ)) / 2

hcvsin(Θ) = (1 - sin(Θ)) / 2

hvcos(Θ) = (1 + cos(Θ)) / 2

hcvcos(Θ) = (1 + sin(Θ)) / 2

Hidden Trigonometry calculus, derivates and integrals

Examples

derivates
---------
d(vsin x) / dx = sin x
d(cvsin x) / dx = -cos x

integrals
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∫ vsin(x) dx = x - sin x + C
∫ cvsin(x) dx = x + cos x + C