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Arkus sínus a Arkus kosínus Cyklometrická funkcia je matematická funkcia inverzná ku funkciám goniometrickým .
Definícia Medzi cyklometrické funkcie patria:
Aby mohla k ľubovoľnej funkcii existovať inverzná funkcia , daná funkcia musí byť prostá , to znamená: rôznym dvom prvkom musí priraďovať dve rôzne hodnoty. Goniometrické funkcie sú ale periodické, a teda nie sú prosté. Preto ak chceme uvažovať o cyklometrických funkciách musíme najskôr ošetriť ich definičný obor a taktiež aj definičné obory goniometrických funkcií – to znamená, že musíme vybrať len tú podmnožinu definičného oboru danej goniometrickej funkcie, na ktorej je prostá.
Definičné obory cyklometrických a goniometrických funkcií
Goniometrické funkcie
Cyklometrické funkcie
Sínus:
s
i
n
(
x
)
{\displaystyle sin(x)}
x
∈<
−
π
/
2
;
π
/
2
>
{\displaystyle x\in <-\pi /2;\pi /2>}
Arkus sínus:
a
r
c
s
i
n
(
x
)
{\displaystyle arcsin(x)}
x
∈<
−
1
;
1
>
{\displaystyle x\in <-1;1>}
Cosínus:
c
o
s
(
x
)
{\displaystyle cos(x)}
x
∈<
0
,
π
>
{\displaystyle x\in <0,\pi >}
Arkus cosínus:
a
r
c
c
o
s
(
x
)
{\displaystyle arccos(x)}
x
∈<
−
1
;
1
>
{\displaystyle x\in <-1;1>}
Tangens:
t
g
(
x
)
{\displaystyle tg(x)}
x
∈
(
−
π
/
2
;
π
/
2
)
{\displaystyle x\in (-\pi /2;\pi /2)}
Arkus tangens:
a
r
c
t
g
(
x
)
{\displaystyle arctg(x)}
x
∈
R
{\displaystyle x\in R}
Cotangens:
c
t
g
(
x
)
{\displaystyle ctg(x)}
x
∈
(
0
,
π
)
{\displaystyle x\in (0,\pi )}
Arkus cotangens:
a
r
c
c
t
g
(
x
)
{\displaystyle arcctg(x)}
x
∈
R
{\displaystyle x\in R}
Vzťahy medzi cyklometrickými a goniometrickými funkciami sin a arcsin
a
r
c
s
i
n
(
s
i
n
x
)
=
x
{\displaystyle \mathrm {arcsin} (\mathrm {sin} x)=x\!}
|
x
|
≤
π
2
{\displaystyle \ |x|\leq {\frac {\pi }{2}}}
sin
(
a
r
c
s
i
n
x
)
=
x
{\displaystyle \sin(\mathrm {arcsin} x)=x\!}
cos a arccos
a
r
c
c
o
s
(
c
o
s
x
)
=
x
{\displaystyle \mathrm {arccos} (\mathrm {cos} x)=x\!}
0
≤
x
≤
π
{\displaystyle \ 0\leq x\leq \pi }
cos
(
a
r
c
c
o
s
x
)
=
x
{\displaystyle \cos(\mathrm {arccos} x)=x\!}
|
x
|
≤
1
{\displaystyle \ |x|\leq 1}
tg a arctg
a
r
c
t
g
(
t
g
x
)
=
x
{\displaystyle \mathrm {arctg} (\mathrm {tg} x)=x\!}
|
x
|
<
π
2
{\displaystyle \ |x|<{\frac {\pi }{2}}}
t
g
(
a
r
c
t
g
x
)
=
x
{\displaystyle \mathrm {tg} (\mathrm {arctg} x)=x\!}
cotg a arccotg
a
r
c
c
o
t
g
(
c
o
t
g
x
)
=
x
{\displaystyle \mathrm {arccotg} (\mathrm {cotg} x)=x\!}
0
<
x
<
π
{\displaystyle \ 0<x<\pi }
c
o
t
g
(
a
r
c
o
t
g
x
)
=
x
{\displaystyle \mathrm {cotg} (\mathrm {arcotg} x)=x\!}
Vzťahy medzi cyklometrickými funkciami
a
r
c
s
i
n
x
=
π
2
−
a
r
c
c
o
s
x
=
a
r
c
t
g
x
1
−
x
2
=
π
2
−
a
r
c
c
o
t
g
x
1
−
x
2
{\displaystyle \mathrm {arcsin} x={\frac {\pi }{2}}-\mathrm {arccos} x=\mathrm {arctg} {\frac {x}{\sqrt {1-x^{2}}}}={\frac {\pi }{2}}-\mathrm {arccotg} {\frac {x}{\sqrt {1-x^{2}}}}}
a
r
c
c
o
s
x
=
π
2
−
a
r
c
s
i
n
x
=
π
2
−
a
r
c
t
g
x
1
−
x
2
=
a
r
c
c
o
t
g
x
1
−
x
2
{\displaystyle \mathrm {arccos} x={\frac {\pi }{2}}-\mathrm {arcsin} x={\frac {\pi }{2}}-\mathrm {arctg} {\frac {x}{\sqrt {1-x^{2}}}}=\mathrm {arccotg} {\frac {x}{\sqrt {1-x^{2}}}}}
a
r
c
t
g
x
=
a
r
c
s
i
n
x
1
+
x
2
=
π
2
−
a
r
c
c
o
s
x
1
+
x
2
=
π
2
−
a
r
c
c
o
t
g
x
{\displaystyle \mathrm {arctg} x=\mathrm {arcsin} {\frac {x}{\sqrt {1+x^{2}}}}={\frac {\pi }{2}}-\mathrm {arccos} {\frac {x}{\sqrt {1+x^{2}}}}={\frac {\pi }{2}}-\mathrm {arccotg} x}
a
r
c
c
o
t
g
x
=
π
2
−
a
r
c
s
i
n
x
1
+
x
2
=
a
r
c
c
o
s
x
1
+
x
2
=
π
2
−
a
r
c
t
g
x
{\displaystyle \mathrm {arccotg} x={\frac {\pi }{2}}-\mathrm {arcsin} {\frac {x}{\sqrt {1+x^{2}}}}=\mathrm {arccos} {\frac {x}{\sqrt {1+x^{2}}}}={\frac {\pi }{2}}-\mathrm {arctg} x}
Pre
x
>
0
{\displaystyle x>0}
a
r
c
c
o
t
g
x
=
a
r
c
t
g
1
x
{\displaystyle \mathrm {arccotg} x=\mathrm {arctg} {\frac {1}{x}}}
Pre
x
<
0
{\displaystyle x<0}
a
r
c
c
o
t
g
x
=
π
+
a
r
c
t
g
1
x
{\displaystyle \mathrm {arccotg} x=\pi +\mathrm {arctg} {\frac {1}{x}}}
Vzťahy medzi cyklometrickými funkciami so vzájomne opačnými argumentmi
a
r
c
s
i
n
(
−
x
)
=
−
a
r
c
s
i
n
x
{\displaystyle \mathrm {arcsin} (-x)=-\mathrm {arcsin} x\!}
a
r
c
c
o
s
(
−
x
)
=
π
−
a
r
c
c
o
s
x
{\displaystyle \mathrm {arccos} (-x)=\pi -\mathrm {arccos} x\!}
a
r
c
t
g
(
−
x
)
=
−
a
r
c
t
g
x
{\displaystyle \mathrm {arctg} (-x)=-\mathrm {arctg} x\!}
a
r
c
c
o
t
g
(
−
x
)
=
π
−
a
r
c
c
o
t
g
x
{\displaystyle \mathrm {arccotg} (-x)=\pi -\mathrm {arccotg} x\!}
Súčty a rozdiely cyklometrických funkcií arcsin x + arcsin y
arcsin
x
+
arcsin
y
=
arcsin
[
x
1
−
y
2
+
y
1
−
x
2
]
,
{\displaystyle \arcsin x+\arcsin y=\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}
x
y
≤
0
{\displaystyle \ xy\leq 0}
x
2
+
y
2
≤
1
{\displaystyle x^{2}+y^{2}\leq 1}
arcsin
x
+
arcsin
y
=
π
−
arcsin
[
x
1
−
y
2
+
y
1
−
x
2
]
,
{\displaystyle \arcsin x+\arcsin y=\pi -\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}
x
>
0
,
y
>
0
,
x
2
+
y
2
>
1
{\displaystyle \ x>0,y>0,x^{2}+y^{2}>1}
arcsin
x
+
arcsin
y
=
−
π
−
arcsin
[
x
1
−
y
2
+
y
1
−
x
2
]
,
{\displaystyle \arcsin x+\arcsin y=-\pi -\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}
x
<
0
,
y
<
0
,
x
2
+
y
2
>
1
{\displaystyle \ x<0,y<0,x^{2}+y^{2}>1}
arcsin x - arcsin y
arcsin
x
−
arcsin
y
=
arcsin
[
x
1
−
y
2
−
y
1
−
x
2
]
,
{\displaystyle \arcsin x-\arcsin y=\arcsin[x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}],}
x
y
≥
0
{\displaystyle \ xy\geq 0}
x
2
+
y
2
≤
1
{\displaystyle x^{2}+y^{2}\leq 1}
arcsin
x
−
arcsin
y
=
π
−
arcsin
[
x
1
−
y
2
−
y
1
−
x
2
]
,
{\displaystyle \arcsin x-\arcsin y=\pi -\arcsin[x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}],}
x
>
0
,
y
<
0
,
x
2
+
y
2
>
1
{\displaystyle \ x>0,y<0,x^{2}+y^{2}>1}
arcsin
x
−
arcsin
y
=
−
π
−
arcsin
[
x
1
−
y
2
+
y
1
−
x
2
]
,
{\displaystyle \arcsin x-\arcsin y=-\pi -\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}
x
<
0
,
y
>
0
,
x
2
+
y
2
>
1
{\displaystyle \ x<0,y>0,x^{2}+y^{2}>1}
arccos x + arccos y
arccos
x
+
arccos
y
=
arccos
[
x
y
−
1
−
x
2
⋅
1
−
y
2
]
,
{\displaystyle \arccos x+\arccos y=\arccos[xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}
x
+
y
≥
0
{\displaystyle \ x+y\geq 0}
arccos
x
+
arccos
y
=
2
π
−
arccos
[
x
y
−
1
−
x
2
⋅
1
−
y
2
]
,
{\displaystyle \arccos x+\arccos y=2\pi -\arccos[xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}
x
+
y
<
0
{\displaystyle \ x+y<0}
arccos x - arccos y
arccos
x
−
arccos
y
=
−
arccos
[
x
y
+
1
−
x
2
⋅
1
−
y
2
]
,
{\displaystyle \arccos x-\arccos y=-\arccos[xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}
x
≥
y
{\displaystyle \ x\geq y}
arccos
x
−
arccos
y
=
arccos
[
x
y
+
1
−
x
2
⋅
1
−
y
2
]
,
{\displaystyle \arccos x-\arccos y=\arccos[xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}
x
<
y
{\displaystyle \ x<y}
arctg x + arctg y
a
r
c
t
g
x
+
a
r
c
t
g
y
=
a
r
c
t
g
x
+
y
1
−
x
y
,
{\displaystyle \mathrm {arctg} \,x+\mathrm {arctg} \,y=\mathrm {arctg} \,{\frac {x+y}{1-xy}},}
x
y
<
1
{\displaystyle \ xy<1}
a
r
c
t
g
x
+
a
r
c
t
g
y
=
π
+
a
r
c
t
g
x
+
y
1
−
x
y
,
{\displaystyle \mathrm {arctg} \,x+\mathrm {arctg} \,y=\pi +\mathrm {arctg} \,{\frac {x+y}{1-xy}},}
x
>
0
,
x
y
>
1
{\displaystyle \ x>0,xy>1}
a
r
c
t
g
x
+
a
r
c
t
g
y
=
−
π
+
a
r
c
t
g
x
+
y
1
−
x
y
,
{\displaystyle \mathrm {arctg} \,x+\mathrm {arctg} \,y=-\pi +\mathrm {arctg} \,{\frac {x+y}{1-xy}},}
x
<
0
,
x
y
>
1
{\displaystyle \ x<0,xy>1}
arctg x - arctg y
a
r
c
t
g
x
−
a
r
c
t
g
y
=
a
r
c
t
g
x
−
y
1
+
x
y
,
{\displaystyle \mathrm {arctg} x-\mathrm {arctg} y=\mathrm {arctg} {\frac {x-y}{1+xy}},}
x
y
>
−
1
{\displaystyle \ xy>-1}
a
r
c
t
g
x
−
a
r
c
t
g
y
=
π
+
a
r
c
t
g
x
−
y
1
+
x
y
,
{\displaystyle \mathrm {arctg} x-\mathrm {arctg} y=\pi +\mathrm {arctg} {\frac {x-y}{1+xy}},}
x
>
0
,
x
y
<
−
1
{\displaystyle \ x>0,xy<-1}
a
r
c
t
g
x
−
a
r
c
t
g
y
=
−
π
+
a
r
c
t
g
x
−
y
1
+
x
y
,
{\displaystyle \mathrm {arctg} x-\mathrm {arctg} y=-\pi +\mathrm {arctg} {\frac {x-y}{1+xy}},}
x
<
0
,
x
y
<
−
1
{\displaystyle \ x<0,xy<-1}
arccotg x + arccotg y
a
r
c
c
o
t
g
x
+
a
r
c
c
o
t
g
y
=
a
r
c
c
o
t
g
x
y
−
1
x
+
y
,
{\displaystyle \mathrm {arccotg} x+\mathrm {arccotg} y=\mathrm {arccotg} {\frac {xy-1}{x+y}},}
x
>
−
y
{\displaystyle \ x>-y}
a
r
c
c
o
t
g
x
+
a
r
c
c
o
t
g
y
=
a
r
c
c
o
t
g
x
y
−
1
x
+
y
+
π
,
{\displaystyle \mathrm {arccotg} x+\mathrm {arccotg} y=\mathrm {arccotg} {\frac {xy-1}{x+y}}+\pi ,}
x
<
−
y
{\displaystyle \ x<-y}
arcsin x + arccos x
a
r
s
i
n
x
+
a
r
c
o
s
x
=
π
2
,
{\displaystyle \mathrm {arsin} x+\mathrm {arcos} x={\frac {\pi }{2}},}
|
x
|
≤
1
{\displaystyle \ |x|\leq 1}
arctg x + arccotg x
a
r
t
g
x
+
a
r
c
o
t
g
x
=
π
2
{\displaystyle \mathrm {artg} x+\mathrm {arcotg} x={\frac {\pi }{2}}}
Vyjadrenie cyklometrických funkcií v logaritmickom tvare Cyklometrické funkcie sa dajú tiež vyjadriť použitím logaritmov a komplexných čísel:
arcsin
x
=
−
i
log
(
i
x
+
1
−
x
2
)
arccos
x
=
−
i
log
(
x
+
x
2
−
1
)
=
π
2
+
i
log
(
i
x
+
1
−
x
2
)
=
π
2
−
arcsin
x
a
r
c
t
g
x
=
i
2
(
log
(
1
−
i
x
)
−
log
(
1
+
i
x
)
)
=
a
r
c
c
o
t
g
1
x
a
r
c
c
o
t
g
x
=
i
2
(
log
(
1
−
i
x
)
−
log
(
1
+
i
x
)
)
=
a
r
c
t
g
1
x
{\displaystyle {\begin{aligned}\arcsin x&{}=-i\,\log \left(i\,x+{\sqrt {1-x^{2}}}\right)&{}\\\arccos x&{}=-i\,\log \left(x+{\sqrt {x^{2}-1}}\right)={\frac {\pi }{2}}\,+i\log \left(i\,x+{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\arcsin x&{}\\\mathrm {arctg} x&{}={\frac {i}{2}}\left(\log \left(1-i\,x\right)-\log \left(1+i\,x\right)\right)=\mathrm {arccotg} {\frac {1}{x}}\\\mathrm {arccotg} x&{}={\frac {i}{2}}\left(\log \left(1-{\frac {i}{x}}\right)-\log \left(1+{\frac {i}{x}}\right)\right)=\mathrm {arctg} {\frac {1}{x}}\\\end{aligned}}}
Pozri aj Literatúra Rektorys, K. a spol.: Přehled užité matematiky I. , Prometheus, Praha, 2003, 7. vydání. ISBN 80-7196-179-5
Bartch, Hans-Jochen: Matematické vzorce , SNTL, Praha 1987, 2. revidované vydání 
![{\displaystyle \arcsin x+\arcsin y=\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/221eebb74205d19ce08e3d40ceb8bb44f34bbbb3)
![{\displaystyle \arcsin x+\arcsin y=\pi -\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab8b5fc8928ecc527a81b7cd38cf9dc6a781238)
![{\displaystyle \arcsin x+\arcsin y=-\pi -\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8964cbc068ea6fb8c2c53ecbc31c090d2ab4a28c)
![{\displaystyle \arcsin x-\arcsin y=\arcsin[x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/972e7fe37e143e4de42fc2955b9105598a5daf7b)
![{\displaystyle \arcsin x-\arcsin y=\pi -\arcsin[x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d52c9c397e746de2a5e3a84df93dd3ca9f293136)
![{\displaystyle \arcsin x-\arcsin y=-\pi -\arcsin[x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8341e1155b0d78a17125ce972891381b2c6e8e)
![{\displaystyle \arccos x+\arccos y=\arccos[xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/768b4e1e861871f826d37bc5b6ec553d517c6c9a)
![{\displaystyle \arccos x+\arccos y=2\pi -\arccos[xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/493019bfe806dbb5e0e8df8ca962c9bc6e453984)
![{\displaystyle \arccos x-\arccos y=-\arccos[xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b43649132cbc13f23e0a473bc7d7ff8acf6cee2)
![{\displaystyle \arccos x-\arccos y=\arccos[xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b826d85a2a6e719ef00cc608168451190a5fde)
