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Trigonometrično funkcijo versin lahko prikažemo geometrijsko z enotskim krogom, ki ima središče v O . Sinus versus (tudi versinus ali obrnjeni sinus) (oznaka
versine
z
{\displaystyle \operatorname {versine} z\,}
vers
z
{\displaystyle \operatorname {vers} z}
trigonometrična funkcija , ki je povezana z danes bolj znanimi trigonometričnimi funkcijami na naslednji način:
versin
(
θ
)
=
1
−
cos
(
θ
)
=
2
sin
2
(
θ
2
)
.
{\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
(
2
k
π
,
0
)
{\displaystyle (2k\pi ,{\text{ }}0)}
(
(
2
k
+
1
)
π
,
2
)
{\displaystyle ((2k+1)\pi ,{\text{ }}2)}
V nadaljevanju so naštete vse obrnjene trigonometrične funkcije in njihove povezave z običajnimi (pogosteje uporabljanimi) funkcijami:
versin
(
θ
)
:=
2
sin
2
(
θ
2
)
=
1
−
cos
(
θ
)
{\displaystyle {\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,}
vercosin
(
θ
)
:=
2
cos
2
(
θ
2
)
=
1
+
cos
(
θ
)
{\displaystyle {\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,}
coversin
(
θ
)
:=
versin
(
π
2
−
θ
)
=
1
−
sin
(
θ
)
{\displaystyle {\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,}
covercosin
(
θ
)
:=
vercosin
(
π
2
−
θ
)
=
1
+
sin
(
θ
)
{\displaystyle {\textrm {covercosin}}(\theta ):={\textrm {vercosin}}\!\left({\frac {\pi }{2}}-\theta \right)=1+\sin(\theta )\,}
haversin
(
θ
)
:=
versin
(
θ
)
2
=
1
−
cos
(
θ
)
2
{\displaystyle {\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}={\frac {1-\cos(\theta )}{2}}\,}
havercosin
(
θ
)
:=
vercosin
(
θ
)
2
=
1
+
cos
(
θ
)
2
{\displaystyle {\textrm {havercosin}}(\theta ):={\frac {{\textrm {vercosin}}(\theta )}{2}}={\frac {1+\cos(\theta )}{2}}\,}
hacoversin
(
θ
)
:=
coversin
(
θ
)
2
=
1
−
sin
(
θ
)
2
{\displaystyle {\textrm {hacoversin}}(\theta ):={\frac {{\textrm {coversin}}(\theta )}{2}}={\frac {1-\sin(\theta )}{2}}\,}
hacovercosin
(
θ
)
:=
covercosin
(
θ
)
2
=
1
+
sin
(
θ
)
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
vercosin
(
θ
)
{\displaystyle \operatorname {vercosin} (\theta )}
koversinus , kiga označujemo z
coversin
(
θ
)
{\displaystyle \operatorname {coversin} (\theta )}
cvs
(
θ
)
{\displaystyle \operatorname {cvs} (\theta )}
koverkosinus , ki ga označujemo z
covercosin
(
θ
)
{\displaystyle \operatorname {covercosin} (\theta )}
haversinus , ki ga označujemo z
haversin
(
θ
)
{\displaystyle \operatorname {haversin} (\theta )}
haversinski obrazec , ki se je nekdaj uporabljal v navigaciji haverkosinus ali haverkosinus z oznako
havercosin
(
θ
)
{\displaystyle \operatorname {havercosin} (\theta )}
hacoversinus ali hakoversinus ali kohaversinus , ki ga označujemo z
hacoversin
(
θ
)
{\displaystyle \operatorname {hacoversin} (\theta )}
hakoverzni kosinus ali hakoverkosinus ali kohaverkosinus z oznako
hacovercosin
(
θ
)
{\displaystyle \operatorname {hacovercosin} (\theta )}
ekssekanta
exsec
(
θ
)
{\displaystyle \operatorname {exsec} (\theta )}
ekskosekanta z oznako
excosec
(
θ
)
{\displaystyle \operatorname {excosec} (\theta )}
Povezave s trigonometričnimi funkcijami [ uredi | uredi kodo ] Trigonometrični funkciji, ki jo označimo z fun , pripadajo sledeče povezave
verfun
(
θ
)
:=
2
(
fun
(
θ
2
)
)
2
cofun
(
θ
)
:=
fun
(
π
2
−
θ
)
hafun
(
θ
)
:=
fun
(
θ
)
2
exfun
(
θ
)
:=
fun
(
θ
)
−
1
{\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}}}
d
d
x
v
e
r
s
i
n
(
x
)
=
sin
x
{\displaystyle {\frac {d}{dx}}\mathrm {versin} (x)=\sin {x}}
∫
v
e
r
s
i
n
(
x
)
d
x
=
x
−
sin
x
+
C
{\displaystyle \int \mathrm {versin} (x)\,dx=x-\sin {x}+C}
d
d
x
v
e
r
c
o
s
i
n
(
x
)
=
−
sin
x
{\displaystyle {\frac {d}{dx}}\mathrm {vercosin} (x)=-\sin {x}}
∫
v
e
r
c
o
s
i
n
(
x
)
d
x
=
x
+
sin
x
+
C
{\displaystyle \int \mathrm {vercosin} (x)\,dx=x+\sin {x}+C}
d
d
x
c
o
v
e
r
s
i
n
(
x
)
=
−
cos
x
{\displaystyle {\frac {d}{dx}}\mathrm {coversin} (x)=-\cos {x}}
∫
c
o
v
e
r
s
i
n
(
x
)
d
x
=
x
+
cos
x
+
C
{\displaystyle \int \mathrm {coversin} (x)\,dx=x+\cos {x}+C}
d
d
x
c
o
v
e
r
c
o
s
i
n
(
x
)
=
cos
x
{\displaystyle {\frac {d}{dx}}\mathrm {covercosin} (x)=\cos {x}}
∫
c
o
v
e
r
c
o
s
i
n
(
x
)
d
x
=
x
−
cos
x
+
C
{\displaystyle \int \mathrm {covercosin} (x)\,dx=x-\cos {x}+C}
d
d
x
h
a
v
e
r
s
i
n
(
x
)
=
sin
x
2
{\displaystyle {\frac {d}{dx}}\mathrm {haversin} (x)={\frac {\sin {x}}{2}}}
∫
h
a
v
e
r
s
i
n
(
x
)
d
x
=
x
−
sin
x
2
+
C
{\displaystyle \int \mathrm {haversin} (x)\,dx={\frac {x-\sin {x}}{2}}+C}
d
d
x
h
a
v
e
r
c
o
s
i
n
(
x
)
=
−
sin
x
2
{\displaystyle {\frac {d}{dx}}\mathrm {havercosin} (x)={\frac {-\sin {x}}{2}}}
∫
h
a
v
e
r
c
o
s
i
n
(
x
)
d
x
=
x
+
sin
x
2
+
C
{\displaystyle \int \mathrm {havercosin} (x)\,dx={\frac {x+\sin {x}}{2}}+C}
d
d
x
h
a
c
o
v
e
r
s
i
n
(
x
)
=
−
cos
x
2
{\displaystyle {\frac {d}{dx}}\mathrm {hacoversin} (x)={\frac {-\cos {x}}{2}}}
∫
h
a
c
o
v
e
r
s
i
n
(
x
)
d
x
=
x
+
cos
x
2
+
C
{\displaystyle \int \mathrm {hacoversin} (x)\,dx={\frac {x+\cos {x}}{2}}+C}
d
d
x
h
a
c
o
v
e
r
c
o
s
i
n
(
x
)
=
cos
x
2
{\displaystyle {\frac {d}{dx}}\mathrm {hacovercosin} (x)={\frac {\cos {x}}{2}}}
∫
h
a
c
o
v
e
r
c
o
s
i
n
(
x
)
d
x
=
x
−
cos
x
2
+
C
{\displaystyle \int \mathrm {hacovercosin} (x)\,dx={\frac {x-\cos {x}}{2}}+C}
