De trigonometriska funktionerna för en vinkel θ kan konstrueras geometriskt med hjälp av en enhetscirkel Lista över trigonometriska identiteter är en lista av ekvationer som involverar trigonometriska funktioner och som är sanna för varje enskilt värde av de förekommande variablerna. De skiljer sig från triangelidentiteter, vilka är identiteter som potentiellt involverar vinklar, men även omfattar sidolängder eller andra längder i en triangel. Endast de förstnämnda behandlas i denna artikel.
Identiteterna är användbara när uttryck som involverar trigonometriska funktioner måste förenklas. En viktig tillämpning är integration av icke-trigonometriska funktioner: en vanlig teknik är att först göra en substitution med en trigonometrisk funktion och sedan förenkla resultatet med hjälp av en trigonometrisk identitet.
cos
-->
(
x
)
=
sin
-->
(
x
+
π π -->
2
)
{\displaystyle \cos(x)=\sin \left(x+{\frac {\pi }{2}}\right)}
tan
-->
(
x
)
=
sin
-->
(
x
)
cos
-->
(
x
)
{\displaystyle \tan(x)={\frac {\sin(x)}{\cos(x)}}}
cot
-->
(
x
)
=
cos
-->
(
x
)
sin
-->
(
x
)
=
tan
-->
(
π π -->
2
− − -->
x
)
{\displaystyle \cot(x)={\frac {\cos(x)}{\sin(x)}}=\tan({\frac {\pi }{2}}-x)}
sec
-->
(
x
)
=
1
cos
-->
(
x
)
{\displaystyle \sec(x)={\frac {1}{\cos(x)}}}
csc
-->
(
x
)
=
1
sin
-->
(
x
)
{\displaystyle \csc(x)={\frac {1}{\sin(x)}}}
Sinus, cosinus, sekant och cosekant har perioden 2π. Tangens och cotangens har perioden π. Om k är ett heltal gäller:
sin
-->
(
x
)
=
sin
-->
(
x
+
2
k
π π -->
)
cos
-->
(
x
)
=
cos
-->
(
x
+
2
k
π π -->
)
tan
-->
(
x
)
=
tan
-->
(
x
+
k
π π -->
)
cot
-->
(
x
)
=
cot
-->
(
x
+
k
π π -->
)
sec
-->
(
x
)
=
sec
-->
(
x
+
2
k
π π -->
)
csc
-->
(
x
)
=
csc
-->
(
x
+
2
k
π π -->
)
{\displaystyle {\begin{aligned}\sin(x)&=\sin(x+2k\pi )\\\cos(x)&=\cos(x+2k\pi )\\\tan(x)&=\tan(x+k\pi )\\\cot(x)&=\cot(x+k\pi )\\\sec(x)&=\sec(x+2k\pi )\\\csc(x)&=\csc(x+2k\pi )\\\end{aligned}}}
sin
-->
(
− − -->
x
)
=
− − -->
sin
-->
(
x
)
sin
-->
(
π π -->
2
− − -->
x
)
=
cos
-->
(
x
)
sin
-->
(
π π -->
− − -->
x
)
=
+
sin
-->
(
x
)
cos
-->
(
− − -->
x
)
=
+
cos
-->
(
x
)
cos
-->
(
π π -->
2
− − -->
x
)
=
sin
-->
(
x
)
cos
-->
(
π π -->
− − -->
x
)
=
− − -->
cos
-->
(
x
)
tan
-->
(
− − -->
x
)
=
− − -->
tan
-->
(
x
)
tan
-->
(
π π -->
2
− − -->
x
)
=
cot
-->
(
x
)
tan
-->
(
π π -->
− − -->
x
)
=
− − -->
tan
-->
(
x
)
cot
-->
(
− − -->
x
)
=
− − -->
cot
-->
(
x
)
cot
-->
(
π π -->
2
− − -->
x
)
=
tan
-->
(
x
)
cot
-->
(
π π -->
− − -->
x
)
=
− − -->
cot
-->
(
x
)
sec
-->
(
− − -->
x
)
=
+
sec
-->
(
x
)
sec
-->
(
π π -->
2
− − -->
x
)
=
csc
-->
(
x
)
sec
-->
(
π π -->
− − -->
x
)
=
− − -->
sec
-->
(
x
)
csc
-->
(
− − -->
x
)
=
− − -->
csc
-->
(
x
)
csc
-->
(
π π -->
2
− − -->
x
)
=
sec
-->
(
x
)
csc
-->
(
π π -->
− − -->
x
)
=
+
csc
-->
(
x
)
{\displaystyle {\begin{aligned}\sin(-x)&=-\sin(x)&\sin \left({\cfrac {\pi }{2}}-x\right)&=\cos(x)&\sin \left(\pi -x\right)&=+\sin(x)\\\cos(-x)&=+\cos(x)&\cos \left({\cfrac {\pi }{2}}-x\right)&=\sin(x)&\cos \left(\pi -x\right)&=-\cos(x)\\\tan(-x)&=-\tan(x)&\tan \left({\cfrac {\pi }{2}}-x\right)&=\cot(x)&\tan \left(\pi -x\right)&=-\tan(x)\\\cot(-x)&=-\cot(x)&\cot \left({\cfrac {\pi }{2}}-x\right)&=\tan(x)&\cot \left(\pi -x\right)&=-\cot(x)\\\sec(-x)&=+\sec(x)&\sec \left({\cfrac {\pi }{2}}-x\right)&=\csc(x)&\sec \left(\pi -x\right)&=-\sec(x)\\\csc(-x)&=-\csc(x)&\csc \left({\cfrac {\pi }{2}}-x\right)&=\sec(x)&\csc \left(\pi -x\right)&=+\csc(x)\\\end{aligned}}}
En funktion f(x) kallas udda om f(-x) = -f(x) och kallas jämn om f(-x) = f(x). Till exempel är cosinusfunktionen jämn och sinus- och tangensfunktionerna är udda.
sin
-->
(
x
+
π π -->
2
)
=
+
cos
-->
(
x
)
sin
-->
(
x
+
π π -->
)
=
− − -->
sin
-->
(
x
)
cos
-->
(
x
+
π π -->
2
)
=
− − -->
sin
-->
(
x
)
cos
-->
(
x
+
π π -->
)
=
− − -->
cos
-->
(
x
)
tan
-->
(
x
+
π π -->
2
)
=
− − -->
cot
-->
(
x
)
tan
-->
(
x
+
π π -->
)
=
+
tan
-->
(
x
)
{\displaystyle {\begin{aligned}\sin \left(x+{\cfrac {\pi }{2}}\right)&=+\cos(x)&\sin \left(x+\pi \right)&=-\sin(x)\\\cos \left(x+{\cfrac {\pi }{2}}\right)&=-\sin(x)&\cos \left(x+\pi \right)&=-\cos(x)\\\tan \left(x+{\cfrac {\pi }{2}}\right)&=-\cot(x)&\tan \left(x+\pi \right)&=+\tan(x)\\\end{aligned}}}
sin
2
-->
(
x
)
+
cos
2
-->
(
x
)
=
1
{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}
sin
-->
(
x
)
=
± ± -->
1
− − -->
cos
2
-->
(
x
)
{\displaystyle \sin(x)=\pm {\sqrt {1-\cos ^{2}(x)}}}
cos
-->
(
x
)
=
± ± -->
1
− − -->
sin
2
-->
(
x
)
{\displaystyle \cos(x)=\pm {\sqrt {1-\sin ^{2}(x)}}}
Relaterade identiteter
1
+
tan
2
-->
θ θ -->
=
sec
2
-->
θ θ -->
{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
1
+
cot
2
-->
θ θ -->
=
csc
2
-->
θ θ -->
{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }
Dubbla vinkeln
sin
-->
(
2
x
)
=
2
sin
-->
(
x
)
cos
-->
(
x
)
cos
-->
(
2
x
)
=
cos
2
-->
(
x
)
− − -->
sin
2
-->
(
x
)
=
=
2
cos
2
-->
(
x
)
− − -->
1
=
=
1
− − -->
2
sin
2
-->
(
x
)
tan
-->
(
2
x
)
=
2
tan
-->
(
x
)
1
− − -->
tan
2
-->
(
x
)
cot
-->
(
2
x
)
=
cot
-->
(
x
)
− − -->
tan
-->
(
x
)
2
{\displaystyle {\begin{aligned}\sin(2x)&=2\sin(x)\cos(x)\\\cos(2x)&=\cos ^{2}(x)-\sin ^{2}(x)=\\&=2\cos ^{2}(x)-1=\\&=1-2\sin ^{2}(x)\\\tan(2x)&={\frac {2\tan(x)}{1-\tan ^{2}(x)}}\\\cot(2x)&={\frac {\cot(x)-\tan(x)}{2}}\\\end{aligned}}}
Tredubbla vinkeln
sin
-->
(
3
x
)
=
3
sin
-->
(
x
)
− − -->
4
sin
3
-->
(
x
)
cos
-->
(
3
x
)
=
4
cos
3
-->
(
x
)
− − -->
3
cos
-->
(
x
)
tan
-->
(
3
x
)
=
3
tan
-->
(
x
)
− − -->
tan
3
-->
(
x
)
1
− − -->
3
tan
2
-->
(
x
)
cot
-->
(
3
x
)
=
cot
3
-->
(
x
)
− − -->
3
cot
-->
(
x
)
3
cot
2
-->
(
x
)
− − -->
1
{\displaystyle {\begin{aligned}\sin(3x)&=3\sin(x)-4\sin ^{3}(x)\\\cos(3x)&=4\cos ^{3}(x)-3\cos(x)\\\tan(3x)&={\frac {3\tan(x)-\tan ^{3}(x)}{1-3\tan ^{2}(x)}}\\\cot(3x)&={\frac {\cot ^{3}(x)-3\cot(x)}{3\cot ^{2}(x)-1}}\\\end{aligned}}}
Halva vinkeln
sin
2
-->
(
x
2
)
=
1
− − -->
cos
-->
(
x
)
2
cos
2
-->
(
x
2
)
=
1
+
cos
-->
(
x
)
2
tan
-->
(
x
2
)
=
sin
-->
(
x
)
1
+
cos
-->
(
x
)
=
=
1
− − -->
cos
-->
(
x
)
sin
-->
(
x
)
tan
2
-->
(
x
2
)
=
1
− − -->
cos
-->
(
x
)
1
+
cos
-->
(
x
)
cot
-->
(
x
2
)
=
sin
-->
(
x
)
1
− − -->
cos
-->
(
x
)
=
=
1
+
cos
-->
(
x
)
sin
-->
(
x
)
cot
2
-->
(
x
2
)
=
1
+
cos
-->
(
x
)
1
− − -->
cos
-->
(
x
)
{\displaystyle {\begin{aligned}\sin ^{2}\left({\frac {x}{2}}\right)&={\frac {1-\cos(x)}{2}}\\\cos ^{2}\left({\frac {x}{2}}\right)&={\frac {1+\cos(x)}{2}}\\\tan \left({\frac {x}{2}}\right)&={\frac {\sin(x)}{1+\cos(x)}}&=\\&={\frac {1-\cos(x)}{\sin(x)}}\\\tan ^{2}\left({\frac {x}{2}}\right)&={\frac {1-\cos(x)}{1+\cos(x)}}\\\cot \left({\frac {x}{2}}\right)&={\frac {\sin(x)}{1-\cos(x)}}&=\\&={\frac {1+\cos(x)}{\sin(x)}}\\\cot ^{2}\left({\frac {x}{2}}\right)&={\frac {1+\cos(x)}{1-\cos(x)}}\\\end{aligned}}}
Potenser
sin
2
-->
θ θ -->
=
1
− − -->
cos
-->
2
θ θ -->
2
{\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}}
cos
2
-->
θ θ -->
=
1
+
cos
-->
2
θ θ -->
2
{\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}}
sin
2
-->
θ θ -->
cos
2
-->
θ θ -->
=
1
− − -->
cos
-->
4
θ θ -->
8
{\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos 4\theta }{8}}}
sin
3
-->
θ θ -->
=
3
sin
-->
θ θ -->
− − -->
sin
-->
3
θ θ -->
4
{\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin 3\theta }{4}}}
cos
3
-->
θ θ -->
=
3
cos
-->
θ θ -->
+
cos
-->
3
θ θ -->
4
{\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos 3\theta }{4}}}
sin
3
-->
θ θ -->
cos
3
-->
θ θ -->
=
3
sin
-->
2
θ θ -->
− − -->
sin
-->
6
θ θ -->
32
{\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin 2\theta -\sin 6\theta }{32}}}
sin
4
-->
θ θ -->
=
3
− − -->
4
cos
-->
2
θ θ -->
+
cos
-->
4
θ θ -->
8
{\displaystyle \sin ^{4}\theta ={\frac {3-4\cos 2\theta +\cos 4\theta }{8}}}
cos
4
-->
θ θ -->
=
3
+
4
cos
-->
2
θ θ -->
+
cos
-->
4
θ θ -->
8
{\displaystyle \cos ^{4}\theta ={\frac {3+4\cos 2\theta +\cos 4\theta }{8}}}
sin
4
-->
θ θ -->
cos
4
-->
θ θ -->
=
3
− − -->
4
cos
-->
4
θ θ -->
+
cos
-->
8
θ θ -->
128
{\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos 4\theta +\cos 8\theta }{128}}}
sin
5
-->
θ θ -->
=
10
sin
-->
θ θ -->
− − -->
5
sin
-->
3
θ θ -->
+
sin
-->
5
θ θ -->
16
{\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin 3\theta +\sin 5\theta }{16}}}
cos
5
-->
θ θ -->
=
10
cos
-->
θ θ -->
+
5
cos
-->
3
θ θ -->
+
cos
-->
5
θ θ -->
16
{\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos 3\theta +\cos 5\theta }{16}}}
sin
5
-->
θ θ -->
cos
5
-->
θ θ -->
=
10
sin
-->
2
θ θ -->
− − -->
5
sin
-->
6
θ θ -->
+
sin
-->
10
θ θ -->
512
{\displaystyle \sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin 2\theta -5\sin 6\theta +\sin 10\theta }{512}}}
sin
-->
(
x
± ± -->
y
)
=
sin
-->
(
x
)
cos
-->
(
y
)
± ± -->
cos
-->
(
x
)
sin
-->
(
y
)
cos
-->
(
x
± ± -->
y
)
=
cos
-->
(
x
)
cos
-->
(
y
)
∓ ∓ -->
sin
-->
(
x
)
sin
-->
(
y
)
tan
-->
(
x
± ± -->
y
)
=
tan
-->
(
x
)
± ± -->
tan
-->
(
y
)
1
∓ ∓ -->
tan
-->
(
x
)
tan
-->
(
y
)
cot
-->
(
x
± ± -->
y
)
=
cot
-->
(
x
)
cot
-->
(
y
)
∓ ∓ -->
1
cot
-->
(
y
)
± ± -->
cot
-->
(
x
)
{\displaystyle {\begin{aligned}\sin(x\pm y)&=\sin(x)\cos(y)\pm \cos(x)\sin(y)\\\cos(x\pm y)&=\cos(x)\cos(y)\mp \sin(x)\sin(y)\\\tan(x\pm y)&={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}\\\cot(x\pm y)&={\frac {\cot(x)\cot(y)\mp 1}{\cot(y)\pm \cot(x)}}\\\end{aligned}}}
Observera att
± ± -->
{\displaystyle \pm }
∓ ∓ -->
{\displaystyle \mp }
x + y ) = cos(x )cos(y ) - sin(x )sin(y ) medan cos(x - y ) = cos(x )cos(y ) + sin(x )sin(y ).
sin
-->
(
x
)
− − -->
sin
-->
(
y
)
sin
-->
(
x
)
+
sin
-->
(
y
)
=
tan
-->
x
− − -->
y
2
tan
-->
x
+
y
2
cos
-->
(
x
)
− − -->
cos
-->
(
y
)
cos
-->
(
x
)
+
cos
-->
(
y
)
=
− − -->
tan
-->
(
x
+
y
2
)
tan
-->
(
x
− − -->
y
2
)
tan
-->
(
x
)
− − -->
tan
-->
(
y
)
tan
-->
(
x
)
+
tan
-->
(
y
)
=
sin
-->
(
x
− − -->
y
)
sin
-->
(
x
+
y
)
cot
-->
(
x
)
− − -->
cot
-->
(
y
)
cot
-->
(
x
)
+
cot
-->
(
y
)
=
− − -->
sin
-->
(
x
− − -->
y
)
sin
-->
(
x
+
y
)
{\displaystyle {\begin{aligned}{\frac {\sin(x)-\sin(y)}{\sin(x)+\sin(y)}}&={\cfrac {\tan {\cfrac {x-y}{2}}}{\tan {\cfrac {x+y}{2}}}}\\{\frac {\cos(x)-\cos(y)}{\cos(x)+\cos(y)}}&=-\tan \left({\cfrac {x+y}{2}}\right)\tan \left({\cfrac {x-y}{2}}\right)\\{\frac {\tan(x)-\tan(y)}{\tan(x)+\tan(y)}}&={\cfrac {\sin(x-y)}{\sin(x+y)}}\\{\frac {\cot(x)-\cot(y)}{\cot(x)+\cot(y)}}&=-{\cfrac {\sin(x-y)}{\sin(x+y)}}\\\end{aligned}}}
Summor
sin
-->
(
x
)
+
sin
-->
(
y
)
=
2
sin
-->
(
x
+
y
2
)
cos
-->
(
x
− − -->
y
2
)
cos
-->
(
x
)
+
cos
-->
(
y
)
=
2
cos
-->
(
x
+
y
2
)
cos
-->
(
x
− − -->
y
2
)
tan
-->
(
x
)
+
tan
-->
(
y
)
=
sin
-->
(
x
+
y
)
cos
-->
(
x
)
cos
-->
(
y
)
cot
-->
(
x
)
+
cot
-->
(
y
)
=
sin
-->
(
x
+
y
)
sin
-->
(
x
)
sin
-->
(
y
)
{\displaystyle {\begin{aligned}\sin(x)+\sin(y)&=2\sin \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)\\\cos(x)+\cos(y)&=2\cos \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)\\\tan(x)+\tan(y)&={\frac {\sin(x+y)}{\cos(x)\cos(y)}}\\\cot(x)+\cot(y)&={\frac {\sin(x+y)}{\sin(x)\sin(y)}}\\\end{aligned}}}
sin
-->
(
x
)
− − -->
sin
-->
(
y
)
=
2
cos
-->
(
x
+
y
2
)
sin
-->
(
x
− − -->
y
2
)
cos
-->
(
x
)
− − -->
cos
-->
(
y
)
=
− − -->
2
sin
-->
(
x
+
y
2
)
sin
-->
(
x
− − -->
y
2
)
tan
-->
(
x
)
− − -->
tan
-->
(
y
)
=
sin
-->
(
x
− − -->
y
)
cos
-->
(
x
)
cos
-->
(
y
)
cot
-->
(
x
)
− − -->
cot
-->
(
y
)
=
− − -->
sin
-->
(
x
− − -->
y
)
sin
-->
(
x
)
sin
-->
(
y
)
{\displaystyle {\begin{aligned}\sin(x)-\sin(y)&=2\cos \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)\\\cos(x)-\cos(y)&=-2\sin \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)\\\tan(x)-\tan(y)&={\frac {\sin(x-y)}{\cos(x)\cos(y)}}\\\cot(x)-\cot(y)&=-{\frac {\sin(x-y)}{\sin(x)\sin(y)}}\\\end{aligned}}}
Produkter
sin
-->
(
x
)
sin
-->
(
y
)
=
cos
-->
(
x
− − -->
y
)
− − -->
cos
-->
(
x
+
y
)
2
sin
-->
(
x
)
cos
-->
(
y
)
=
sin
-->
(
x
− − -->
y
)
+
sin
-->
(
x
+
y
)
2
cos
-->
(
x
)
cos
-->
(
y
)
=
cos
-->
(
x
− − -->
y
)
+
cos
-->
(
x
+
y
)
2
{\displaystyle {\begin{aligned}\sin \left(x\right)\sin \left(y\right)&={\cos \left(x-y\right)-\cos \left(x+y\right) \over 2}\\\sin \left(x\right)\cos \left(y\right)&={\sin \left(x-y\right)+\sin \left(x+y\right) \over 2}\\\cos \left(x\right)\cos \left(y\right)&={\cos \left(x-y\right)+\cos \left(x+y\right) \over 2}\\\end{aligned}}}
Inversa funktioner
sin
-->
(
arcsin
-->
(
x
)
)
=
x
cos
-->
(
arccos
-->
(
x
)
)
=
x
tan
-->
(
arctan
-->
(
x
)
)
=
x
cot
-->
(
arccot
-->
(
x
)
)
=
x
sec
-->
(
arcsec
-->
(
x
)
)
=
x
csc
-->
(
arccsc
-->
(
x
)
)
=
x
{\displaystyle {\begin{aligned}\sin(\arcsin(x))&=x&\quad \cos(\arccos(x))&=x\\\tan(\arctan(x))&=x&\quad \cot(\operatorname {arccot}(x))&=x\\\sec(\operatorname {arcsec}(x))&=x&\quad \csc(\operatorname {arccsc}(x))&=x\\\end{aligned}}}
arcsin
-->
(
sin
-->
(
x
)
)
=
x
,
för
− − -->
π π -->
/
2
≤ ≤ -->
x
≤ ≤ -->
π π -->
/
2
arccos
-->
(
cos
-->
(
x
)
)
=
x
,
för
0
≤ ≤ -->
x
≤ ≤ -->
π π -->
arctan
-->
(
tan
-->
(
x
)
)
=
x
,
för
− − -->
π π -->
/
2
<
x
<
π π -->
/
2
arccot
-->
(
cot
-->
(
x
)
)
=
x
,
för
0
<
x
<
π π -->
arcsec
-->
(
sec
-->
(
x
)
)
=
x
,
för
0
≤ ≤ -->
x
<
π π -->
/
2
eller
π π -->
/
2
<
x
≤ ≤ -->
π π -->
arccsc
-->
(
csc
-->
(
x
)
)
=
x
,
för
− − -->
π π -->
/
2
≤ ≤ -->
x
<
0
eller
0
<
x
≤ ≤ -->
π π -->
/
2
{\displaystyle {\begin{aligned}\arcsin(\sin(x))&=x,{\mbox{ för }}-\pi /2\leq x\leq \pi /2\\\arccos(\cos(x))&=x,{\mbox{ för }}0\leq x\leq \pi \\\arctan(\tan(x))&=x,{\mbox{ för }}-\pi /2<x<\pi /2\\\operatorname {arccot}(\cot(x))&=x,{\mbox{ för }}0<x<\pi \\\operatorname {arcsec}(\sec(x))&=x,{\mbox{ för }}0\leq x<\pi /2{\mbox{ eller }}\pi /2<x\leq \pi \\\operatorname {arccsc}(\csc(x))&=x,{\mbox{ för }}-\pi /2\leq x<0{\mbox{ eller }}0<x\leq \pi /2\\\end{aligned}}}
Kompletterande
arccos
-->
(
x
)
=
π π -->
2
− − -->
arcsin
-->
(
x
)
arccot
-->
(
x
)
=
π π -->
2
− − -->
arctan
-->
(
x
)
arccsc
-->
(
x
)
=
π π -->
2
− − -->
arcsec
-->
(
x
)
{\displaystyle {\begin{aligned}\arccos(x)&={\frac {\pi }{2}}-\arcsin(x)\\\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\\\operatorname {arccsc}(x)&={\frac {\pi }{2}}-\operatorname {arcsec}(x)\\\end{aligned}}}
arcsin
-->
(
− − -->
x
)
=
− − -->
arcsin
-->
(
x
)
arccos
-->
(
− − -->
x
)
=
π π -->
− − -->
arccos
-->
(
x
)
arctan
-->
(
− − -->
x
)
=
− − -->
arctan
-->
(
x
)
arccot
-->
(
− − -->
x
)
=
π π -->
− − -->
arccot
-->
(
x
)
arcsec
-->
(
− − -->
x
)
=
π 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}}}
Reciproka funktioner
arccos
-->
1
x
=
arcsec
-->
(
x
)
arcsin
-->
1
x
=
arccsc
-->
(
x
)
arctan
-->
1
x
=
π π -->
2
− − -->
arctan
-->
(
x
)
=
arccot
-->
(
x
)
,
om
x
>
0
arctan
-->
1
x
=
− − -->
π π -->
2
− − -->
arctan
-->
(
x
)
=
− − -->
π π -->
+
arccot
-->
(
x
)
,
om
x
<
0
arccot
-->
1
x
=
π π -->
2
− − -->
arccot
-->
(
x
)
=
arctan
-->
(
x
)
,
om
x
>
0
arccot
-->
1
x
=
3
π π -->
2
− − -->
arccot
-->
(
x
)
=
π π -->
+
arctan
-->
(
x
)
,
om
x
<
0
arcsec
-->
1
x
=
arccos
-->
(
x
)
arccsc
-->
1
x
=
arcsin
-->
(
x
)
{\displaystyle {\begin{aligned}\arccos {\frac {1}{x}}&=\operatorname {arcsec}(x)\\\arcsin {\frac {1}{x}}&=\operatorname {arccsc}(x)\\\arctan {\frac {1}{x}}&={\frac {\pi }{2}}-\arctan(x)=\operatorname {arccot}(x),{\text{ om }}x>0\\\arctan {\frac {1}{x}}&=-{\frac {\pi }{2}}-\arctan(x)=-\pi +\operatorname {arccot}(x),{\text{ om }}x<0\\\operatorname {arccot} {\frac {1}{x}}&={\frac {\pi }{2}}-\operatorname {arccot}(x)=\arctan(x),{\text{ om }}x>0\\\operatorname {arccot} {\frac {1}{x}}&={\frac {3\pi }{2}}-\operatorname {arccot}(x)=\pi +\arctan(x),{\text{ om }}x<0\\\operatorname {arcsec} {\frac {1}{x}}&=\arccos(x)\\\operatorname {arccsc} {\frac {1}{x}}&=\arcsin(x)\\\end{aligned}}}
arcsin
-->
α α -->
± ± -->
arcsin
-->
β β -->
=
arcsin
-->
(
α α -->
1
− − -->
β β -->
2
± ± -->
β β -->
1
− − -->
α α -->
2
)
arccos
-->
α α -->
± ± -->
arccos
-->
β β -->
=
arccos
-->
(
α α -->
β β -->
∓ ∓ -->
(
1
− − -->
α α -->
2
)
(
1
− − -->
β β -->
2
)
)
arctan
-->
α α -->
± ± -->
arctan
-->
β β -->
=
arctan
-->
(
α α -->
± ± -->
β β -->
1
∓ ∓ -->
α α -->
β β -->
)
{\displaystyle {\begin{aligned}\arcsin \alpha \pm \arcsin \beta &=\arcsin \left(\alpha {\sqrt {1-\beta ^{2}}}\pm \beta {\sqrt {1-\alpha ^{2}}}\right)\\\arccos \alpha \pm \arccos \beta &=\arccos \left(\alpha \beta \mp {\sqrt {(1-\alpha ^{2})(1-\beta ^{2})}}\right)\\\arctan \alpha \pm \arctan \beta &=\arctan \left({\frac {\alpha \pm \beta }{1\mp \alpha \beta }}\right)\end{aligned}}}
