Sinus versus

Trigonometrija
Oris Zgodovina Uporaba Funkcije inverzne Posplošena trigonometrija
Sklici
Identitete Natančne konstante Tabele Enotski krog
Zakoni in izreki
Sinusni Kosinusni Tangensni Kotangensni Pitagorov izrek
Infinitezimalni račun
Trigonometrične substitucije Integrali inverzne funkcije Odvodi
p p u

Sinus versus (tudi versinus ali obrnjeni sinus) (oznaka ${\displaystyle \operatorname {versine} z\,}$ ali ${\displaystyle \operatorname {vers} z}$) je danes redko uporabljana trigonometrična funkcija, ki je povezana z danes bolj znanimi trigonometričnimi funkcijami na naslednji način:

${\displaystyle \operatorname {versin} (\theta )=1-\cos(\theta )=2\sin ^{2}\left({\frac {\theta }{2}}\right).}$

Grafična oblika versinusa je enaka kosinusoidi, ki je premaknjena za 1 navzgor. Funkcija je povsod definirana. Ničle ima funkcija v točkah ${\displaystyle (2k\pi ,{\text{ }}0)}$. Minimumi so tam kjer so ničle. Maksimumi pa so v točkah ${\displaystyle ((2k+1)\pi ,{\text{ }}2)}$.

V nadaljevanju so naštete vse obrnjene trigonometrične funkcije in njihove povezave z običajnimi (pogosteje uporabljanimi) funkcijami:

${\displaystyle {\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,}$
${\displaystyle {\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,}$
${\displaystyle {\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,}$
${\displaystyle {\textrm {covercosin}}(\theta ):={\textrm {vercosin}}\!\left({\frac {\pi }{2}}-\theta \right)=1+\sin(\theta )\,}$
${\displaystyle {\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}={\frac {1-\cos(\theta )}{2}}\,}$
${\displaystyle {\textrm {havercosin}}(\theta ):={\frac {{\textrm {vercosin}}(\theta )}{2}}={\frac {1+\cos(\theta )}{2}}\,}$
${\displaystyle {\textrm {hacoversin}}(\theta ):={\frac {{\textrm {coversin}}(\theta )}{2}}={\frac {1-\sin(\theta )}{2}}\,}$
${\displaystyle {\textrm {hacovercosin}}(\theta ):={\frac {{\textrm {covercosin}}(\theta )}{2}}={\frac {1+\sin(\theta )}{2}}\,}$

Iz pregleda se vidi, da je znanih več obrnjenih trigonometričnih funkcij

• kosinus versus ali vercosine, ki ga označujemo z ${\displaystyle \operatorname {vercosin} (\theta )}$
• koversinus, kiga označujemo z ${\displaystyle \operatorname {coversin} (\theta )}$ včasih okrajšamo v ${\displaystyle \operatorname {cvs} (\theta )}$
• koverkosinus, ki ga označujemo z ${\displaystyle \operatorname {covercosin} (\theta )}$
• haversinus, ki ga označujemo z ${\displaystyle \operatorname {haversin} (\theta )}$ najbolj znan haversinski obrazec, ki se je nekdaj uporabljal v navigaciji
• haverkosinus ali haverkosinus z oznako ${\displaystyle \operatorname {havercosin} (\theta )}$
• hacoversinus ali hakoversinus ali kohaversinus, ki ga označujemo z ${\displaystyle \operatorname {hacoversin} (\theta )}$
• hakoverzni kosinus ali hakoverkosinus ali kohaverkosinus z oznako ${\displaystyle \operatorname {hacovercosin} (\theta )}$
• ekssekanta, ki jo označujemo z ${\displaystyle \operatorname {exsec} (\theta )}$
• ekskosekanta z oznako ${\displaystyle \operatorname {excosec} (\theta )}$

Trigonometrični funkciji, ki jo označimo z fun, pripadajo sledeče povezave

${\displaystyle {\begin{aligned}\operatorname {verfun} (\theta )&:=&2\left(\operatorname {fun} \!\left({\frac {\theta }{2}}\right)\right)^{2}\\\operatorname {cofun} (\theta )&:=&\operatorname {fun} \!\left({\frac {\pi }{2}}-\theta \right)\\\operatorname {hafun} (\theta )&:=&{\frac {\operatorname {fun} (\theta )}{2}}\\\operatorname {exfun} (\theta )&:=&\operatorname {fun} (\theta )-1\\\end{aligned}}}$.

${\displaystyle {\frac {d}{dx}}\mathrm {versin} (x)=\sin {x}}$	${\displaystyle \int \mathrm {versin} (x)\,dx=x-\sin {x}+C}$
${\displaystyle {\frac {d}{dx}}\mathrm {vercosin} (x)=-\sin {x}}$	${\displaystyle \int \mathrm {vercosin} (x)\,dx=x+\sin {x}+C}$
${\displaystyle {\frac {d}{dx}}\mathrm {coversin} (x)=-\cos {x}}$	${\displaystyle \int \mathrm {coversin} (x)\,dx=x+\cos {x}+C}$
${\displaystyle {\frac {d}{dx}}\mathrm {covercosin} (x)=\cos {x}}$	${\displaystyle \int \mathrm {covercosin} (x)\,dx=x-\cos {x}+C}$
${\displaystyle {\frac {d}{dx}}\mathrm {haversin} (x)={\frac {\sin {x}}{2}}}$	${\displaystyle \int \mathrm {haversin} (x)\,dx={\frac {x-\sin {x}}{2}}+C}$
${\displaystyle {\frac {d}{dx}}\mathrm {havercosin} (x)={\frac {-\sin {x}}{2}}}$	${\displaystyle \int \mathrm {havercosin} (x)\,dx={\frac {x+\sin {x}}{2}}+C}$
${\displaystyle {\frac {d}{dx}}\mathrm {hacoversin} (x)={\frac {-\cos {x}}{2}}}$	${\displaystyle \int \mathrm {hacoversin} (x)\,dx={\frac {x+\cos {x}}{2}}+C}$
${\displaystyle {\frac {d}{dx}}\mathrm {hacovercosin} (x)={\frac {\cos {x}}{2}}}$	${\displaystyle \int \mathrm {hacovercosin} (x)\,dx={\frac {x-\cos {x}}{2}}+C}$

• eksekanta
• kosinus versus

• Versina na MathWorld (angleško)
• Haversina na MathWorld (angleško)