Arcus-funktionerne er omvendte funktioner til de trigonometriske funktioner med restriktioner i deres definitionsmængder for at gøre dem injektive . Der findes arcus-funktioner til sinus , cosinus , tangens , samt deres reciprokke funktioner: cosekans , sekans og cotangens . Arcus-funktioner betegnes også circulære funktioner . De kan anvendes til at beregne en vinkel ud fra kendte forholdstal i en trekant og er hyppigt brugte i ingeniørvidenskab , navigation , astronomi , fysik og geometri .
Nedenstående figurer viser grafer for seks arcus-funktioner.
Grafer for arcsin (rød) og arccos (blå). Grafer for arctan (rød) og arccot (blå).
Grafer for arcsec (rød) og arccsc (blå).
Der er flere notationer i brug for arcus-funktioner.
Det mest almindelige er at anvende "arc-" som et præfiks:
arcsin
(
x
)
{\displaystyle \arcsin(x)}
arccos
(
x
)
{\displaystyle \arccos(x)}
arctan
(
x
)
{\displaystyle \arctan(x)}
ISO 80000-2 [ 1]
Inden for datalogi (programmeringssprog , regneark , og tilsvarende) vil man ofte se forkortede betegnelser:
asin, acos, atan
{\displaystyle {\text{asin, acos, atan}}}
På lommeregnere støder man ofte på skrivemåderne
sin
−
1
(
x
)
{\displaystyle \sin ^{-1}(x)}
cos
−
1
(
x
)
{\displaystyle \cos ^{-1}(x)}
tan
−
1
(
x
)
{\displaystyle \tan ^{-1}(x)}
John Herschel i 1813[ 2] [ 3]
sin
2
(
x
)
{\displaystyle \sin ^{2}(x)}
(
sin
(
x
)
)
2
{\displaystyle {(\sin(x))}^{2}}
sin
(
sin
(
x
)
)
{\displaystyle \sin(\sin(x))}
[ 4]
Da ingen af de seks trigonometriske funktioner er injektive (en-til-en), er det nødvendigt at indføre restriktioner for at kunne danne omvendte funktioner. For eksempel har ligningen
sin
(
x
)
=
0
{\displaystyle \sin(x)=0}
x
=
n
⋅
π
{\displaystyle x=n\cdot \pi }
n
{\displaystyle n}
x
{\displaystyle x}
arcsin
(
0
)
{\displaystyle \arcsin(0)}
For at definere
arcsin
{\displaystyle \arcsin }
R
{\displaystyle \mathbb {R} }
reelle tal ) til det lukkede interval
[
−
π
/
2
;
+
π
/
2
]
{\displaystyle [-\pi /2;+\pi /2]}
arcsin
{\displaystyle \arcsin }
y
=
sin
(
y
)
⇔
x
=
arcsin
(
y
)
{\displaystyle y=\sin(y)\quad \Leftrightarrow \quad x=\arcsin(y)}
Af definitionen fremgår, at hvis
(
x
,
y
)
{\displaystyle (x,y)}
(
y
,
x
)
{\displaystyle (y,x)}
værdimængder lig med de oprindelige funktioners definitionsmængder og omvendt. Definitionsprocessen er illustreret på nedenstående tre figurer gældende for henholdsvis arcussinus, arcuscosinus og arcustangens.
Sinus og arcussinus. Den sorte kurve viser grafen for sinus. Den blå kurve markerer begrænsningen af definitionsmængden til det lukkede interval fra
−
π
/
2
{\displaystyle -\pi /2}
+
π
/
2
{\displaystyle +\pi /2}
y
=
x
{\displaystyle y=x}
Cosinus og arcuscosinus. Den sorte kurve viser grafen for cosinus. Den blå kurve markerer begrænsningen af definitionsmængden til det lukkede interval fra
0
{\displaystyle 0}
π
{\displaystyle \pi }
y
=
x
{\displaystyle y=x}
Tangens og arcustangens. De sorte kurver viser grafen for tangens. Den blå kurve markerer begrænsningen af definitionsmængden til det åbne interval fra
−
π
/
2
{\displaystyle -\pi /2}
+
π
/
2
{\displaystyle +\pi /2}
tan
{\displaystyle \tan }
arctan
{\displaystyle \arctan }
y
=
x
{\displaystyle y=x}
Disse fremgår af nedenstående tabel. Her betegner
R
{\displaystyle \mathbb {R} }
∨
{\displaystyle \lor }
Funktion og definition
Definitionsmængde
Værdimængde (radianer )
Værdimængde (grader )
y
=
arcsin
(
x
)
⇔
x
=
sin
(
y
)
{\displaystyle y=\arcsin(x)\quad \Leftrightarrow \quad x=\sin(y)}
−
1
≦
x
≦
+
1
{\displaystyle -1\leqq x\leqq +1}
−
π
2
≦
y
≦
+
π
2
{\displaystyle -\textstyle {\frac {\pi }{2}}\leqq y\leqq +\textstyle {\frac {\pi }{2}}}
−
90
∘
≦
y
≦
+
90
∘
{\displaystyle -90^{\circ }\leqq y\leqq +90^{\circ }}
y
=
arccos
(
x
)
⇔
x
=
cos
(
y
)
{\displaystyle y=\arccos(x)\quad \Leftrightarrow \quad x=\cos(y)}
−
1
≦
x
≦
+
1
{\displaystyle -1\leqq x\leqq +1}
0
≦
y
≦
π
{\displaystyle 0\leqq y\leqq \pi }
0
∘
≦
y
≦
180
∘
{\displaystyle 0^{\circ }\leqq y\leqq 180^{\circ }}
y
=
arctan
(
x
)
⇔
x
=
tan
(
y
)
{\displaystyle y=\arctan(x)\quad \Leftrightarrow \quad x=\tan(y)}
R
{\displaystyle \mathbb {R} }
−
π
2
<
y
<
+
π
2
{\displaystyle -\textstyle {\frac {\pi }{2}}<y<+\textstyle {\frac {\pi }{2}}}
−
90
∘
<
y
<
+
90
∘
{\displaystyle -90^{\circ }<y<+90^{\circ }}
y
=
arccot
(
x
)
⇔
x
=
cot
(
y
)
{\displaystyle y=\operatorname {arccot}(x)\quad \Leftrightarrow \quad x=\cot(y)}
R
{\displaystyle \mathbb {R} }
0
<
y
<
π
{\displaystyle 0<y<\pi }
0
∘
<
y
<
180
∘
{\displaystyle 0^{\circ }<y<180^{\circ }}
y
=
arcsec
(
x
)
⇔
x
=
sec
(
y
)
{\displaystyle y=\operatorname {arcsec}(x)\quad \Leftrightarrow \quad x=\sec(y)}
x
≦
−
1
∨
x
≧
+
1
{\displaystyle x\leqq -1\lor x\geqq +1}
0
≦
y
<
π
2
∨
π
2
<
y
≦
π
{\displaystyle 0\leqq y<\textstyle {\frac {\pi }{2}}\lor \textstyle {\frac {\pi }{2}}<y\leqq \pi }
0
∘
≦
y
<
90
∘
∨
90
∘
<
y
≦
180
∘
{\displaystyle 0^{\circ }\leqq y<90^{\circ }\lor 90^{\circ }<y\leqq 180^{\circ }}
y
=
arccsc
(
x
)
⇔
x
=
csc
(
y
)
{\displaystyle y=\operatorname {arccsc}(x)\quad \Leftrightarrow \quad x=\csc(y)}
x
≦
−
1
∨
x
≧
+
1
{\displaystyle x\leqq -1\lor x\geqq +1}
−
π
2
≦
y
<
0
∨
0
<
y
≦
π
2
{\displaystyle -\textstyle {\frac {\pi }{2}}\leqq y<0\lor 0<y\leqq \textstyle {\frac {\pi }{2}}}
−
90
∘
≦
y
<
0
∘
∨
0
∘
<
y
≦
90
∘
{\displaystyle -90^{\circ }\leqq y<0^{\circ }\lor 0^{\circ }<y\leqq 90^{\circ }}
Værdier for sinus, cosinus og tangens af arcus-funktioner kan ses i den følgende tabel sammen med diagrammer af retvinklede trekanter , som kan illustrere hvordan man kan udlede disse resultater ved at anvende Pythagoras' sætning og definitionerne af de trigonometriske funktioner.
θ
{\displaystyle \theta }
sin
(
θ
)
{\displaystyle \sin(\theta )}
cos
(
θ
)
{\displaystyle \cos(\theta )}
tan
(
θ
)
{\displaystyle \tan(\theta )}
Diagram
arcsin
(
x
)
{\displaystyle \arcsin(x)}
sin
(
arcsin
(
x
)
)
=
x
{\displaystyle \sin(\arcsin(x))=x}
cos
(
arcsin
(
x
)
)
=
1
−
x
2
{\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}
tan
(
arcsin
(
x
)
)
=
x
1
−
x
2
{\displaystyle \tan(\arcsin(x))={\frac {x}{\sqrt {1-x^{2}}}}}
arccos
(
x
)
{\displaystyle \arccos(x)}
sin
(
arccos
(
x
)
)
=
1
−
x
2
{\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}
cos
(
arccos
(
x
)
)
=
x
{\displaystyle \cos(\arccos(x))=x}
tan
(
arccos
(
x
)
)
=
1
−
x
2
x
{\displaystyle \tan(\arccos(x))={\frac {\sqrt {1-x^{2}}}{x}}}
arctan
(
x
)
{\displaystyle \arctan(x)}
sin
(
arctan
(
x
)
)
=
x
1
+
x
2
{\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}
cos
(
arctan
(
x
)
)
=
1
1
+
x
2
{\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}
tan
(
arctan
(
x
)
)
=
x
{\displaystyle \tan(\arctan(x))=x}
arccsc
(
x
)
{\displaystyle \operatorname {arccsc}(x)}
sin
(
arccsc
(
x
)
)
=
1
x
{\displaystyle \sin(\operatorname {arccsc}(x))={\frac {1}{x}}}
cos
(
arccsc
(
x
)
)
=
x
2
−
1
x
{\displaystyle \cos(\operatorname {arccsc}(x))={\frac {\sqrt {x^{2}-1}}{x}}}
tan
(
arccsc
(
x
)
)
=
1
x
2
−
1
{\displaystyle \tan(\operatorname {arccsc}(x))={\frac {1}{\sqrt {x^{2}-1}}}}
arcsec
(
x
)
{\displaystyle \operatorname {arcsec}(x)}
sin
(
arcsec
(
x
)
)
=
x
2
−
1
x
{\displaystyle \sin(\operatorname {arcsec}(x))={\frac {\sqrt {x^{2}-1}}{x}}}
cos
(
arcsec
(
x
)
)
=
1
x
{\displaystyle \cos(\operatorname {arcsec}(x))={\frac {1}{x}}}
tan
(
arcsec
(
x
)
)
=
x
2
−
1
{\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}}}
arccot
(
x
)
{\displaystyle \operatorname {arccot}(x)}
sin
(
arccot
(
x
)
)
=
1
1
+
x
2
{\displaystyle \sin(\operatorname {arccot}(x))={\frac {1}{\sqrt {1+x^{2}}}}}
cos
(
arccot
(
x
)
)
=
x
1
+
x
2
{\displaystyle \cos(\operatorname {arccot}(x))={\frac {x}{\sqrt {1+x^{2}}}}}
tan
(
arccot
(
x
)
)
=
1
x
{\displaystyle \tan(\operatorname {arccot}(x))={\frac {1}{x}}}
Funktionssammenhænge:
arccos
(
x
)
=
π
2
−
arcsin
(
x
)
arcsin
(
x
)
=
π
2
−
arccos
(
x
)
arccot
(
x
)
=
π
2
−
arctan
(
x
)
arctan
(
x
)
=
π
2
−
arccot
(
x
)
arccsc
(
x
)
=
π
2
−
arcsec
(
x
)
arcsec
(
x
)
=
π
2
−
arccsc
(
x
)
{\displaystyle {\begin{aligned}\arccos(x)&={\frac {\pi }{2}}-\arcsin(x)&\arcsin(x)&={\frac {\pi }{2}}-\arccos(x)\\[0.5em]\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)&\arctan(x)&={\frac {\pi }{2}}-\operatorname {arccot}(x)\\[0.5em]\operatorname {arccsc}(x)&={\frac {\pi }{2}}-\operatorname {arcsec}(x)&\operatorname {arcsec}(x)&={\frac {\pi }{2}}-\operatorname {arccsc}(x)\end{aligned}}}
Fortegnsskift:
arcsin
(
−
x
)
=
−
arcsin
(
x
)
arccos
(
−
x
)
=
π
−
arccos
(
x
)
arctan
(
−
x
)
=
−
arctan
(
x
)
arccot
(
−
x
)
=
π
−
arccot
(
x
)
arcsec
(
−
x
)
=
π
−
arcsec
(
x
)
arccsc
(
−
x
)
=
−
arccsc
(
x
)
{\displaystyle {\begin{aligned}\arcsin(-x)&=-\arcsin(x)\\\arccos(-x)&=\pi -\arccos(x)\\\arctan(-x)&=-\arctan(x)\\\operatorname {arccot}(-x)&=\pi -\operatorname {arccot}(x)\\\operatorname {arcsec}(-x)&=\pi -\operatorname {arcsec}(x)\\\operatorname {arccsc}(-x)&=-\operatorname {arccsc}(x)\end{aligned}}}
Reciprokke argumenter:
arccos
(
1
x
)
=
arcsec
(
x
)
arcsin
(
1
x
)
=
arccsc
(
x
)
{\displaystyle {\begin{aligned}\arccos \left({\frac {1}{x}}\right)&=\operatorname {arcsec}(x)\\[0.3em]\arcsin \left({\frac {1}{x}}\right)&=\operatorname {arccsc}(x)\\[0.3em]\end{aligned}}}
arctan
(
1
x
)
=
{
π
2
−
arctan
(
x
)
=
arccot
(
x
)
hvis
x
>
0
−
π
2
−
arctan
(
x
)
=
arccot
(
x
)
−
π
hvis
x
<
0
{\displaystyle \arctan \left({\frac {1}{x}}\right)={\begin{cases}{\begin{array}{lcl}\ \ \ {\frac {\pi }{2}}-\arctan(x)=\operatorname {arccot}(x)&{\text{ hvis }}x>0&\\-{\frac {\pi }{2}}-\arctan(x)=\operatorname {arccot}(x)-\pi &{\text{ hvis }}x<0\end{array}}\end{cases}}}
arccot
(
1
x
)
=
{
π
2
−
arccot
(
x
)
=
arctan
(
x
)
hvis
x
>
0
3
π
2
−
arccot
(
x
)
=
π
+
arctan
(
x
)
hvis
x
<
0
{\displaystyle \operatorname {arccot} \left({\frac {1}{x}}\right)={\begin{cases}{\begin{array}{lcl}\ {\frac {\pi }{2}}-\operatorname {arccot}(x)=\arctan(x)&{\text{ hvis }}x>0&\\{\frac {3\pi }{2}}-\operatorname {arccot}(x)=\pi +\arctan(x)&{\text{ hvis }}x<0\end{array}}\end{cases}}}
arcsec
(
1
x
)
=
arccos
(
x
)
arccsc
(
1
x
)
=
arcsin
(
x
)
{\displaystyle {\begin{aligned}\operatorname {arcsec} \left({\frac {1}{x}}\right)&=\arccos(x)\\[0.3em]\operatorname {arccsc} \left({\frac {1}{x}}\right)&=\arcsin(x)\end{aligned}}}
I en retvinklet trekan er sidelængderne 5, 12 og 13. Hvor stor er vinklen
v
{\displaystyle v}
sin
(
v
)
=
5
13
⇔
v
=
arcsin
(
5
13
)
=
22.52
∘
{\displaystyle \sin(v)={\tfrac {5}{13}}\quad \Leftrightarrow \quad v=\arcsin({\tfrac {5}{13}})=22.52^{\circ }}
cos
(
v
)
=
12
13
⇔
v
=
arccos
(
12
13
)
=
22.52
∘
{\displaystyle \cos(v)={\tfrac {12}{13}}\quad \Leftrightarrow \quad v=\arccos({\tfrac {12}{13}})=22.52^{\circ }}
tan
(
v
)
=
5
23
⇔
v
=
arcsin
(
5
12
)
=
22.52
∘
{\displaystyle \tan(v)={\tfrac {5}{23}}\quad \Leftrightarrow \quad v=\arcsin({\tfrac {5}{12}})=22.52^{\circ }}
Beregningerne kan udføres på en lommeregner.
Arcus-funktionerne har nedenstående differentialkvotienter , som er gyldige for både reelle og komplekse værdier af
x
{\displaystyle x}
d
d
x
arcsin
(
x
)
=
1
1
−
x
2
;
x
≠
−
1
,
+
1
d
d
x
arccos
(
x
)
=
−
1
1
−
x
2
;
x
≠
−
1
,
+
1
d
d
x
arctan
(
x
)
=
1
1
+
x
2
;
x
≠
−
i
,
+
i
d
d
x
arccot
(
x
)
=
−
1
1
+
x
2
;
x
≠
−
i
,
+
i
d
d
x
arcsec
(
x
)
=
1
x
2
1
−
1
x
2
;
x
≠
−
1
,
0
,
+
1
d
d
x
arccsc
(
x
)
=
−
1
x
2
1
−
1
x
2
;
x
≠
−
1
,
0
,
+
1
{\displaystyle {\begin{aligned}{\frac {\text{d}}{{\text{d}}x}}\arcsin(x)&{}={\frac {1}{\sqrt {1-x^{2}}}}\;;&x&{}\neq -1,+1\\{\frac {\text{d}}{{\text{d}}x}}\arccos(x)&{}=-{\frac {1}{\sqrt {1-x^{2}}}}\;;&x&{}\neq -1,+1\\{\frac {\text{d}}{{\text{d}}x}}\arctan(x)&{}={\frac {1}{1+x^{2}}}\;;&x&{}\neq -i,+i\\{\frac {\text{d}}{{\text{d}}x}}\operatorname {arccot}(x)&{}=-{\frac {1}{1+x^{2}}}\;;&x&{}\neq -i,+i\\{\frac {\text{d}}{{\text{d}}x}}\operatorname {arcsec}(x)&{}={\frac {1}{x^{2}{\sqrt {1-{\frac {1}{x^{2}}}}}}}\;;&x&{}\neq -1,0,+1\\{\frac {\text{d}}{{\text{d}}x}}\operatorname {arccsc}(x)&{}=-{\frac {1}{x^{2}{\sqrt {1-{\frac {1}{x^{2}}}}}}}\;;&x&{}\neq -1,0,+1\end{aligned}}}
For reelle værdier af
x
{\displaystyle x}
d
d
x
arcsec
(
x
)
=
1
|
x
|
x
2
−
1
;
|
x
|
>
1
d
d
x
arccsc
(
x
)
=
−
1
|
x
|
x
2
−
1
;
|
x
|
>
1
{\displaystyle {\begin{aligned}{\frac {\text{d}}{{\text{d}}x}}\operatorname {arcsec}(x)&{}={\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\\{\frac {\text{d}}{{\text{d}}x}}\operatorname {arccsc}(x)&{}=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\end{aligned}}}
Formlerne kan udledes ved hjælp af differentialkvotienterne for de trigonometriske funktioner.
Eksempel for
arcsin
{\displaystyle \arcsin }
Lad
x
=
sin
(
θ
)
{\displaystyle x=\sin(\theta )}
arcsin
(
x
)
=
θ
{\displaystyle \arcsin(x)=\theta }
d
x
d
θ
=
cos
(
θ
)
=
1
−
x
2
{\displaystyle {\frac {{\text{d}}x}{{\text{d}}\theta }}=\cos(\theta )={\sqrt {1-x^{2}}}}
Heraf følger, at
d
d
x
arcsin
(
x
)
=
d
θ
d
x
=
1
d
x
/
d
θ
=
1
1
−
x
2
.
{\displaystyle {\frac {\text{d}}{{\text{d}}x}}\arcsin(x)={\frac {{\text{d}}\theta }{{\text{d}}x}}={\frac {1}{{\text{d}}x/{\text{d}}\theta }}={\frac {1}{\sqrt {1-x^{2}}}}.}
Nedenfor vises stamfunktioner for seks arcus-funktioner. Argumentet
x
{\displaystyle x}
reelt eller komplekst . Størrelsen
C
{\displaystyle C}
∫
arcsin
(
x
)
d
x
=
x
arcsin
(
x
)
+
1
−
x
2
+
C
∫
arccos
(
x
)
d
x
=
x
arccos
(
x
)
−
1
−
x
2
+
C
∫
arctan
(
x
)
d
x
=
x
arctan
(
x
)
−
1
2
ln
(
1
+
x
2
)
+
C
∫
arccot
(
x
)
d
x
=
x
arccot
(
x
)
+
1
2
ln
(
1
+
x
2
)
+
C
∫
arcsec
(
x
)
d
x
=
x
arcsec
(
x
)
−
ln
[
x
(
1
+
x
2
−
1
x
2
)
]
+
C
∫
arccsc
(
x
)
d
x
=
x
arccsc
(
x
)
+
ln
[
x
(
1
+
x
2
−
1
x
2
)
]
+
C
{\displaystyle {\begin{aligned}\int \arcsin(x)\,{\text{d}}x&{}=x\,\arcsin(x)+{\sqrt {1-x^{2}}}+C\\\int \arccos(x)\,{\text{d}}x&{}=x\,\arccos(x)-{\sqrt {1-x^{2}}}+C\\\int \arctan(x)\,{\text{d}}x&{}=x\,\arctan(x)-{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arccot}(x)\,{\text{d}}x&{}=x\,\operatorname {arccot}(x)+{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arcsec}(x)\,{\text{d}}x&{}=x\,\operatorname {arcsec}(x)-\ln \left[x\left(1+{\sqrt {\frac {x^{2}-1}{x^{2}}}}\right)\right]+C\\\int \operatorname {arccsc}(x)\,{\text{d}}x&{}=x\,\operatorname {arccsc}(x)+\ln \left[x\left(1+{\sqrt {\frac {x^{2}-1}{x^{2}}}}\right)\right]+C\end{aligned}}}
For reelle
x
>
1
{\displaystyle x>1}
∫
arcsec
(
x
)
d
x
=
x
arcsec
(
x
)
−
ln
(
x
+
x
2
−
1
)
+
C
∫
arccsc
(
x
)
d
x
=
x
arccsc
(
x
)
+
ln
(
x
+
x
2
−
1
)
+
C
{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,{\text{d}}x&{}=x\,\operatorname {arcsec}(x)-\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\\\int \operatorname {arccsc}(x)\,{\text{d}}x&{}=x\,\operatorname {arccsc}(x)+\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\end{aligned}}}
For reelle
x
>
1
{\displaystyle x>1}
x
<
−
1
{\displaystyle x<-1}
∫
arcsec
(
x
)
d
x
=
x
arcsec
(
x
)
−
sgn
(
x
)
ln
|
x
+
x
2
−
1
|
+
C
∫
arccsc
(
x
)
d
x
=
x
arccsc
(
x
)
+
sgn
(
x
)
ln
|
x
+
x
2
−
1
|
+
C
{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,{\text{d}}x&{}=x\,\operatorname {arcsec}(x)-\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\\\int \operatorname {arccsc}(x)\,{\text{d}}x&{}=x\,\operatorname {arccsc}(x)+\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\end{aligned}}}
Her betegner |•| størrelsens absolutte værdi og
sgn
{\displaystyle \operatorname {sgn} }
signum . Med anvendelse af area-funktioner (inverse hyperbolske funktioner) kan de sidste to formler også skrives således:
∫
arcsec
(
x
)
d
x
=
x
arcsec
(
x
)
−
arcosh
(
|
x
|
)
+
C
∫
arccsc
(
x
)
d
x
=
x
arccsc
(
x
)
+
arcosh
(
|
x
|
)
+
C
{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,{\text{d}}x&{}=x\,\operatorname {arcsec}(x)-\operatorname {arcosh} (|x|)+C\\\int \operatorname {arccsc}(x)\,{\text{d}}x&{}=x\,\operatorname {arccsc}(x)+\operatorname {arcosh} (|x|)+C\\\end{aligned}}}
^ "ISO 80000-2:2019" . International Organization for Standardization . Hentet 2024-09-11 .^ Cajori, Florian (1919). A History of Mathematics The Macmillan Company . s. 272.^ Herschel, John Frederick William (1813). "On a remarkable Application of Cotes's Theorem" . Philosophical Transactions . Royal Society, London. 103 (1): 8.^ Korn, Grandino Arthur; Korn, Theresa M. (2000) [1961]. "21.2.-4. Inverse Trigonometric Functions". Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review Dover Publications, Inc. s. 811 . ISBN 978-0-486-41147-7
![{\displaystyle [-\pi /2;+\pi /2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac1e5fd170d71e04e47150862ecdc2359ff292d)
![{\displaystyle {\begin{aligned}\arccos(x)&={\frac {\pi }{2}}-\arcsin(x)&\arcsin(x)&={\frac {\pi }{2}}-\arccos(x)\\[0.5em]\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)&\arctan(x)&={\frac {\pi }{2}}-\operatorname {arccot}(x)\\[0.5em]\operatorname {arccsc}(x)&={\frac {\pi }{2}}-\operatorname {arcsec}(x)&\operatorname {arcsec}(x)&={\frac {\pi }{2}}-\operatorname {arccsc}(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b53cc76628cfb3eb4dfc20bedc79a295f9699220)
![{\displaystyle {\begin{aligned}\arccos \left({\frac {1}{x}}\right)&=\operatorname {arcsec}(x)\\[0.3em]\arcsin \left({\frac {1}{x}}\right)&=\operatorname {arccsc}(x)\\[0.3em]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/518d5fcdb52e96ae7402dd364d75f501da2f1c5c)
![{\displaystyle {\begin{aligned}\operatorname {arcsec} \left({\frac {1}{x}}\right)&=\arccos(x)\\[0.3em]\operatorname {arccsc} \left({\frac {1}{x}}\right)&=\arcsin(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26e96a18fe7b82fdeb0a8b1593ecf93e9bfd7c1a)
![{\displaystyle {\begin{aligned}\int \arcsin(x)\,{\text{d}}x&{}=x\,\arcsin(x)+{\sqrt {1-x^{2}}}+C\\\int \arccos(x)\,{\text{d}}x&{}=x\,\arccos(x)-{\sqrt {1-x^{2}}}+C\\\int \arctan(x)\,{\text{d}}x&{}=x\,\arctan(x)-{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arccot}(x)\,{\text{d}}x&{}=x\,\operatorname {arccot}(x)+{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arcsec}(x)\,{\text{d}}x&{}=x\,\operatorname {arcsec}(x)-\ln \left[x\left(1+{\sqrt {\frac {x^{2}-1}{x^{2}}}}\right)\right]+C\\\int \operatorname {arccsc}(x)\,{\text{d}}x&{}=x\,\operatorname {arccsc}(x)+\ln \left[x\left(1+{\sqrt {\frac {x^{2}-1}{x^{2}}}}\right)\right]+C\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d46e1dd916be8a230424b52d9494adb2671c47)
