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
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
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
)
cos
(
x
)
=
cos
(
x
+
2
k
π
)
tan
(
x
)
=
tan
(
x
+
k
π
)
cot
(
x
)
=
cot
(
x
+
k
π
)
sec
(
x
)
=
sec
(
x
+
2
k
π
)
csc
(
x
)
=
csc
(
x
+
2
k
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
2
−
x
)
=
cos
(
x
)
sin
(
π
−
x
)
=
+
sin
(
x
)
cos
(
−
x
)
=
+
cos
(
x
)
cos
(
π
2
−
x
)
=
sin
(
x
)
cos
(
π
−
x
)
=
−
cos
(
x
)
tan
(
−
x
)
=
−
tan
(
x
)
tan
(
π
2
−
x
)
=
cot
(
x
)
tan
(
π
−
x
)
=
−
tan
(
x
)
cot
(
−
x
)
=
−
cot
(
x
)
cot
(
π
2
−
x
)
=
tan
(
x
)
cot
(
π
−
x
)
=
−
cot
(
x
)
sec
(
−
x
)
=
+
sec
(
x
)
sec
(
π
2
−
x
)
=
csc
(
x
)
sec
(
π
−
x
)
=
−
sec
(
x
)
csc
(
−
x
)
=
−
csc
(
x
)
csc
(
π
2
−
x
)
=
sec
(
x
)
csc
(
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
2
)
=
+
cos
(
x
)
sin
(
x
+
π
)
=
−
sin
(
x
)
cos
(
x
+
π
2
)
=
−
sin
(
x
)
cos
(
x
+
π
)
=
−
cos
(
x
)
tan
(
x
+
π
2
)
=
−
cot
(
x
)
tan
(
x
+
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)}}}
1
+
tan
2
θ
=
sec
2
θ
{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
1
+
cot
2
θ
=
csc
2
θ
{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }
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}}}
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}}}
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}}}
sin
2
θ
=
1
−
cos
2
θ
2
{\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}}
cos
2
θ
=
1
+
cos
2
θ
2
{\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}}
sin
2
θ
cos
2
θ
=
1
−
cos
4
θ
8
{\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos 4\theta }{8}}}
sin
3
θ
=
3
sin
θ
−
sin
3
θ
4
{\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin 3\theta }{4}}}
cos
3
θ
=
3
cos
θ
+
cos
3
θ
4
{\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos 3\theta }{4}}}
sin
3
θ
cos
3
θ
=
3
sin
2
θ
−
sin
6
θ
32
{\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin 2\theta -\sin 6\theta }{32}}}
sin
4
θ
=
3
−
4
cos
2
θ
+
cos
4
θ
8
{\displaystyle \sin ^{4}\theta ={\frac {3-4\cos 2\theta +\cos 4\theta }{8}}}
cos
4
θ
=
3
+
4
cos
2
θ
+
cos
4
θ
8
{\displaystyle \cos ^{4}\theta ={\frac {3+4\cos 2\theta +\cos 4\theta }{8}}}
sin
4
θ
cos
4
θ
=
3
−
4
cos
4
θ
+
cos
8
θ
128
{\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos 4\theta +\cos 8\theta }{128}}}
sin
5
θ
=
10
sin
θ
−
5
sin
3
θ
+
sin
5
θ
16
{\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin 3\theta +\sin 5\theta }{16}}}
cos
5
θ
=
10
cos
θ
+
5
cos
3
θ
+
cos
5
θ
16
{\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos 3\theta +\cos 5\theta }{16}}}
sin
5
θ
cos
5
θ
=
10
sin
2
θ
−
5
sin
6
θ
+
sin
10
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}}}
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}}}
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}}}
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
/
2
≤
x
≤
π
/
2
arccos
(
cos
(
x
)
)
=
x
,
för
0
≤
x
≤
π
arctan
(
tan
(
x
)
)
=
x
,
för
−
π
/
2
<
x
<
π
/
2
arccot
(
cot
(
x
)
)
=
x
,
för
0
<
x
<
π
arcsec
(
sec
(
x
)
)
=
x
,
för
0
≤
x
<
π
/
2
eller
π
/
2
<
x
≤
π
arccsc
(
csc
(
x
)
)
=
x
,
för
−
π
/
2
≤
x
<
0
eller
0
<
x
≤
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}}}
arccos
(
x
)
=
π
2
−
arcsin
(
x
)
arccot
(
x
)
=
π
2
−
arctan
(
x
)
arccsc
(
x
)
=
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
−
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}}}
arccos
1
x
=
arcsec
(
x
)
arcsin
1
x
=
arccsc
(
x
)
arctan
1
x
=
π
2
−
arctan
(
x
)
=
arccot
(
x
)
,
om
x
>
0
arctan
1
x
=
−
π
2
−
arctan
(
x
)
=
−
π
+
arccot
(
x
)
,
om
x
<
0
arccot
1
x
=
π
2
−
arccot
(
x
)
=
arctan
(
x
)
,
om
x
>
0
arccot
1
x
=
3
π
2
−
arccot
(
x
)
=
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
±
arcsin
β
=
arcsin
(
α
1
−
β
2
±
β
1
−
α
2
)
arccos
α
±
arccos
β
=
arccos
(
α
β
∓
(
1
−
α
2
)
(
1
−
β
2
)
)
arctan
α
±
arctan
β
=
arctan
(
α
±
β
1
∓
α
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}}}
