Sine versus and cosine versus

Versed sine (including sinus Versus , cross-sine , Versinus or Versus , abbreviated formulas ) and the cosine versus (also Koversinus or Querkosinus , abbreviated formulas ) are trigonometry rarely used today trigonometric functions . Semiversus (English haversine , abbreviated in formulas ) is half the sine versus. ${\displaystyle \operatorname {vers} }$ ${\displaystyle \operatorname {covers} }$ ${\displaystyle \operatorname {sem} }$

Sine versus

Illustration on the unit circle:
The sine versus forms a radius 1 ( ) together with the cosine , the cosine versus forms a radius 1 ( ) together with the sine . ${\displaystyle CD}$ ${\displaystyle OD}$
${\displaystyle GF}$ ${\displaystyle OF}$

The sine versus is defined as using the cosine or sine function

${\displaystyle \operatorname {vers} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}.}$

It is the difference between the cosine and +1 (shown in green in the illustration opposite).

The sine versus can be extended to the whole complex number level.

Semiversus

The semiversus is half of the sine versus:

${\displaystyle \operatorname {sem} \theta ={\frac {\operatorname {vers} \theta }{2}}=\sin ^{2}{\frac {\theta }{2}}}$

Cosine versus

The cosine versus is shown in the figure opposite in the color cyan and as cvs .

${\displaystyle \operatorname {covers} \theta =1-\sin \theta =\operatorname {vers} \left({\frac {\pi }{2}}-\theta \right).}$

It is the difference of the sine to +1 and also the sine versus the counter-argument (π / 2 - θ )

Related functions

Sometimes analogous to and under vercos something different is understood than under coversin and under covercos something different than under versin. The following table summarizes the functions together with some related trigonometric functions and the graphical function curve: ${\displaystyle \operatorname {versin} \theta =2\sin ^{2}(\theta /2)}$ ${\displaystyle \operatorname {coversin} \theta =\operatorname {versin} (\pi /2-\theta )}$

${\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}}$
${\displaystyle \operatorname {haversin} \theta ={\frac {\operatorname {versin} \theta }{2}}={\frac {1-\cos \theta }{2}}}$
${\displaystyle \operatorname {vercos} \theta =1+\cos \theta =2\cos ^{2}{\frac {\theta }{2}}}$
${\displaystyle \operatorname {havercos} \theta ={\frac {\operatorname {vercos} \theta }{2}}={\frac {1+\cos \theta }{2}}}$
${\displaystyle \operatorname {coversin} \theta =1-\sin \theta =\operatorname {versin} \left({\frac {\pi }{2}}-\theta \right)}$
${\displaystyle \operatorname {hacoversin} \theta ={\frac {\operatorname {coversin} \theta }{2}}={\frac {1-\sin \theta }{2}}}$
${\displaystyle \operatorname {covercos} \theta =1+\sin \theta =\operatorname {vercos} \left({\frac {\pi }{2}}-\theta \right)}$
${\displaystyle \operatorname {hacovercos} \theta ={\frac {\operatorname {covercos} \theta }{2}}={\frac {1+\sin \theta }{2}}}$

The derivatives and the antiderivatives are:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {versin} x=\sin {x}}$	${\displaystyle \int \mathrm {versin} (x)\,\mathrm {d} x=x-\sin {x}+C}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {vercos} x=-\sin {x}}$	${\displaystyle \int \mathrm {vercos} (x)\,\mathrm {d} x=x+\sin {x}+C}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {coversin} x=-\cos {x}}$	${\displaystyle \int \mathrm {coversin} (x)\,\mathrm {d} x=x+\cos {x}+C}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {covercos} x=\cos {x}}$	${\displaystyle \int \mathrm {covercos} (x)\,\mathrm {d} x=x-\cos {x}+C}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {haversin} x={\frac {\sin {x}}{2}}}$	${\displaystyle \int \mathrm {haversin} (x)\,\mathrm {d} x={\frac {x-\sin {x}}{2}}+C}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {havercos} x={\frac {-\sin {x}}{2}}}$	${\displaystyle \int \mathrm {havercos} (x)\,\mathrm {d} x={\frac {x+\sin {x}}{2}}+C}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {hacoversin} x={\frac {-\cos {x}}{2}}}$	${\displaystyle \int \mathrm {hacoversin} (x)\,\mathrm {d} x={\frac {x+\cos {x}}{2}}+C}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {hacovercos} x={\frac {\cos {x}}{2}}}$	${\displaystyle \int \mathrm {hacovercos} (x)\,\mathrm {d} x={\frac {x-\cos {x}}{2}}+C}$

History and use

The side cosine law of spherical trigonometry played an important role in nautical navigation to the stars in earlier times. In order to simplify the necessary multiplications of trigonometric functions by looking up table values, the Semiversus was introduced.

Among other things, this results in the side cosine law:

${\displaystyle {\rm {sem}}(a)={\rm {sem}}(b-c)+\sin(b)\cdot \sin(c)\cdot {\rm {sem}}(\alpha )}$

literature

• M. Abramowitz, IA Stegun: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . New York, Dover 1972, p. 78 ( online ).

Individual evidence

1. ↑ Eric W. Weisstein : Versine . In: MathWorld (English).
2. ↑ Eric W. Weisstein : Haversine . In: MathWorld (English).
3. ↑ Eric W. Weisstein : Coversine . In: MathWorld (English).
4. ↑ Bobby Schenk: Astronavigation: without formulas - practical , 2nd edition, Delius Klasing & Co., Bielefeld 1978.
5. ↑ Otto Fulst: 17-18 . In: Johannes Lütjen, Walter Stein, Gerhard Zwiebler (Hrsg.): Nautische Tafeln , 24th edition, Arthur Geist Verlag, Bremen 1972.