Sinus versus je trigonometrijska funkcija
y
=
versin
x
=
1
−
cos
x
{\displaystyle y=\operatorname {versin} x=1-\cos x}
Funkcija se naziva i versinus. Ovi nazivi se rijetko upotrebljavaju. Graf versinusa je kosinusoida translirana za jedan gore.
Svugdje je definisana.
Nule su u tackama
(
2
k
π
,
0
)
{\displaystyle \left(2k\pi ,0\right)}
2
π
{\displaystyle 2\pi}
(
(
2
k
+
1
)
π
,
2
)
{\displaystyle ((2k+1)\pi ,2)}
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Sinus versus ugla alfa je odnos
A
B
¯
O
C
¯
{\displaystyle {\frac {\overline {AB}}{\overline {OC}}}}
Zamjenom duzina sa slike dobicemo da je sinus versus ugla alfa
1
−
c
o
s
α
{\displaystyle 1-cos\alpha }
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Pojam sinusa versusa uveden je u XVII vijeku i danas se skoro uopste ne upotrebljava. Ruski matematicar P. L. Cebisev je smatrao da ce sinus versus igrati vaznu ulogu u matematici.
Latinski: sinus - ispupcenost, nadutost; versus - (prije) okrenut; sinvers - (prije) okrenuti sinus.
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}}\,}
d
d
x
v
e
r
s
i
n
(
x
)
=
sin
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {versin} (x)=\sin {x}}
∫
v
e
r
s
i
n
(
x
)
d
x
=
x
−
sin
x
+
C
{\displaystyle \int \mathrm {versin} (x)\,\mathrm {d} x=x-\sin {x}+C}
d
d
x
v
e
r
c
o
s
i
n
(
x
)
=
−
sin
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\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)\,\mathrm {d} x=x+\sin {x}+C}
d
d
x
c
o
v
e
r
s
i
n
(
x
)
=
−
cos
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\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)\,\mathrm {d} x=x+\cos {x}+C}
d
d
x
c
o
v
e
r
c
o
s
i
n
(
x
)
=
cos
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\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)\,\mathrm {d} x=x-\cos {x}+C}
d
d
x
h
a
v
e
r
s
i
n
(
x
)
=
sin
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\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)\,\mathrm {d} x={\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 {\mathrm {d} }{\mathrm {d} x}}\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)\,\mathrm {d} x={\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 {\mathrm {d} }{\mathrm {d} x}}\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)\,\mathrm {d} x={\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 {\mathrm {d} }{\mathrm {d} x}}\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)\,\mathrm {d} x={\frac {x-\cos {x}}{2}}+C}
Kategorija:Trigonometrija
