Ein Dreieck mit den üblichen Bezeichnungen Die folgende Liste enthält die meisten bekannten Formeln aus der Trigonometrie in der Ebene . Die meisten dieser Beziehungen verwenden trigonometrische Funktionen .
Dabei werden die folgenden Bezeichnungen verwendet: Das Dreieck
A
B
C
{\displaystyle ABC}
a
=
B
C
{\displaystyle a=BC}
b
=
C
A
{\displaystyle b=CA}
c
=
A
B
{\displaystyle c=AB}
Winkel
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
γ
{\displaystyle \gamma }
Ecken
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
r
{\displaystyle r}
Umkreisradius ,
ρ
{\displaystyle \rho }
Inkreisradius und
ρ
a
{\displaystyle \rho _{a}}
ρ
b
{\displaystyle \rho _{b}}
ρ
c
{\displaystyle \rho _{c}}
Ankreisradien (und zwar die Radien der Ankreise, die den Ecken
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
A
B
C
{\displaystyle ABC}
s
{\displaystyle s}
A
B
C
{\displaystyle ABC}
s
=
a
+
b
+
c
2
{\displaystyle s={\frac {a+b+c}{2}}}
Schließlich wird die Fläche des Dreiecks
A
B
C
{\displaystyle ABC}
F
{\displaystyle F}
Es ist zu beachten, dass hier die Bezeichnungen für den Umkreisradius
r
{\displaystyle r}
ρ
{\displaystyle \rho }
ρ
a
{\displaystyle \rho _{a}}
ρ
b
{\displaystyle \rho _{b}}
ρ
c
{\displaystyle \rho _{c}}
R
{\displaystyle R}
r
{\displaystyle r}
r
a
{\displaystyle r_{a}}
r
b
{\displaystyle r_{b}}
r
c
{\displaystyle r_{c}}
Winkelsumme
α
+
β
+
γ
=
180
∘
{\displaystyle \alpha +\beta +\gamma =180^{\circ }}
Projektionssatz
a
=
b
cos
γ
+
c
cos
β
{\displaystyle a=b\,\cos \gamma +c\,\cos \beta }
b
=
c
cos
α
+
a
cos
γ
{\displaystyle b=c\,\cos \alpha +a\,\cos \gamma }
c
=
a
cos
β
+
b
cos
α
{\displaystyle c=a\,\cos \beta +b\,\cos \alpha }
b
+
c
a
=
cos
β
−
γ
2
sin
α
2
,
c
+
a
b
=
cos
γ
−
α
2
sin
β
2
,
a
+
b
c
=
cos
α
−
β
2
sin
γ
2
{\displaystyle {\frac {b+c}{a}}={\frac {\cos {\frac {\beta -\gamma }{2}}}{\sin {\frac {\alpha }{2}}}},\quad {\frac {c+a}{b}}={\frac {\cos {\frac {\gamma -\alpha }{2}}}{\sin {\frac {\beta }{2}}}},\quad {\frac {a+b}{c}}={\frac {\cos {\frac {\alpha -\beta }{2}}}{\sin {\frac {\gamma }{2}}}}}
b
−
c
a
=
sin
β
−
γ
2
cos
α
2
,
c
−
a
b
=
sin
γ
−
α
2
cos
β
2
,
a
−
b
c
=
sin
α
−
β
2
cos
γ
2
{\displaystyle {\frac {b-c}{a}}={\frac {\sin {\frac {\beta -\gamma }{2}}}{\cos {\frac {\alpha }{2}}}},\quad {\frac {c-a}{b}}={\frac {\sin {\frac {\gamma -\alpha }{2}}}{\cos {\frac {\beta }{2}}}},\quad {\frac {a-b}{c}}={\frac {\sin {\frac {\alpha -\beta }{2}}}{\cos {\frac {\gamma }{2}}}}}
Der Flächeninhalt des Dreiecks wird hier mit
F
{\displaystyle F}
A
{\displaystyle A}
A
{\displaystyle A}
Heronsche Formel:
F
=
s
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
=
1
4
(
a
+
b
+
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
(
a
+
b
−
c
)
{\displaystyle F={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}={\frac {1}{4}}{\sqrt {\left(a+b+c\right)\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}}}
F
=
1
4
2
(
b
2
c
2
+
c
2
a
2
+
a
2
b
2
)
−
(
a
4
+
b
4
+
c
4
)
{\displaystyle F={\frac {1}{4}}{\sqrt {2\left(b^{2}c^{2}+c^{2}a^{2}+a^{2}b^{2}\right)-\left(a^{4}+b^{4}+c^{4}\right)}}}
Weitere Flächenformeln:
F
=
1
2
b
c
sin
α
=
1
2
c
a
sin
β
=
1
2
a
b
sin
γ
{\displaystyle F={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta ={\frac {1}{2}}ab\sin \gamma }
F
=
1
2
a
h
a
=
1
2
b
h
b
=
1
2
c
h
c
{\displaystyle F={\frac {1}{2}}ah_{a}={\frac {1}{2}}bh_{b}={\frac {1}{2}}ch_{c}}
h
a
{\displaystyle h_{a}}
h
b
{\displaystyle h_{b}}
h
c
{\displaystyle h_{c}}
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
A
B
C
{\displaystyle ABC}
F
=
2
r
2
sin
α
sin
β
sin
γ
{\displaystyle F=2r^{2}\sin \,\alpha \,\sin \,\beta \,\sin \,\gamma }
F
=
a
b
c
4
r
{\displaystyle F={\frac {abc}{4r}}}
F
=
ρ
s
=
ρ
a
(
s
−
a
)
=
ρ
b
(
s
−
b
)
=
ρ
c
(
s
−
c
)
{\displaystyle F=\rho s=\rho _{a}\left(s-a\right)=\rho _{b}\left(s-b\right)=\rho _{c}\left(s-c\right)}
F
=
ρ
ρ
a
ρ
b
ρ
c
{\displaystyle F={\sqrt {\rho \rho _{a}\rho _{b}\rho _{c}}}}
F
=
4
ρ
r
cos
α
2
cos
β
2
cos
γ
2
{\displaystyle F=4\rho r\cos \,{\frac {\alpha }{2}}\,\cos \,{\frac {\beta }{2}}\,\cos \,{\frac {\gamma }{2}}}
F
=
s
2
tan
α
2
tan
β
2
tan
γ
2
{\displaystyle F=s^{2}\tan \,{\frac {\alpha }{2}}\,\tan \,{\frac {\beta }{2}}\,\tan \,{\frac {\gamma }{2}}}
F
=
ρ
2
h
a
h
b
h
c
(
h
a
−
2
ρ
)
(
h
b
−
2
ρ
)
(
h
c
−
2
ρ
)
{\displaystyle F=\rho ^{2}{\sqrt {\dfrac {h_{a}\,h_{b}\,h_{c}}{(h_{a}-2\rho )(h_{b}-2\rho )(h_{c}-2\rho )}}}}
1
ρ
=
1
h
a
+
1
h
b
+
1
h
c
{\displaystyle {\dfrac {1}{\rho }}={\dfrac {1}{h_{a}}}+{\dfrac {1}{h_{b}}}+{\dfrac {1}{h_{c}}}}
F
=
r
h
a
h
b
h
c
2
{\displaystyle F={\sqrt {\dfrac {r\,h_{a}\,h_{b}\,h_{c}}{2}}}}
F
=
h
a
h
b
h
c
2
ρ
(
sin
α
+
sin
β
+
sin
γ
)
{\displaystyle F={\dfrac {\,h_{a}\,h_{b}\,h_{c}}{2\rho \,{(\sin \alpha +\sin \beta +\sin \gamma )}}}}
Erweiterter Sinussatz:
a
sin
α
=
b
sin
β
=
c
sin
γ
=
2
r
=
a
b
c
2
F
{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2r={\frac {abc}{2F}}}
a
=
2
r
sin
α
{\displaystyle a=2r\,\sin \alpha }
b
=
2
r
sin
β
{\displaystyle b=2r\,\sin \beta }
c
=
2
r
sin
γ
{\displaystyle c=2r\,\sin \gamma }
r
=
a
b
c
4
F
{\displaystyle r={\frac {abc}{4F}}}
Die trigonometrischen Funktionen am Einheitskreis :
C
P
¯
=
sin
b
{\displaystyle {\overline {CP}}=\sin b}
S
P
¯
=
cos
b
{\displaystyle {\overline {SP}}=\cos b}
D
T
¯
=
tan
b
{\displaystyle {\overline {DT}}=\tan b}
E
K
¯
=
cot
b
{\displaystyle {\overline {EK}}=\cot b}
O
T
¯
=
sec
b
{\displaystyle {\overline {OT}}=\operatorname {sec} \,b}
O
K
¯
=
csc
b
{\displaystyle {\overline {OK}}=\operatorname {csc} \,b}
sin
x
=
sin
(
x
+
2
n
π
)
;
n
∈
Z
{\displaystyle \sin x\quad =\quad \sin(x+2n\pi );\quad n\in \mathbb {Z} }
cos
x
=
cos
(
x
+
2
n
π
)
;
n
∈
Z
{\displaystyle \cos x\quad =\quad \cos(x+2n\pi );\quad n\in \mathbb {Z} }
tan
x
=
tan
(
x
+
n
π
)
;
n
∈
Z
{\displaystyle \tan x\quad =\quad \tan(x+n\pi );\quad n\in \mathbb {Z} }
cot
x
=
cot
(
x
+
n
π
)
;
n
∈
Z
{\displaystyle \cot x\quad =\quad \cot(x+n\pi );\quad n\in \mathbb {Z} }
Gegenseitige Darstellung
Die trigonometrischen Funktionen lassen sich ineinander umwandeln oder gegenseitig darstellen. Es gelten folgende Zusammenhänge:
tan
x
=
sin
x
cos
x
{\displaystyle \tan x={\frac {\sin x}{\cos x}}}
sin
2
x
+
cos
2
x
=
1
{\displaystyle \sin ^{2}x+\cos ^{2}x=1}
(„Trigonometrischer Pythagoras “)
1
+
tan
2
x
=
1
cos
2
x
=
sec
2
x
{\displaystyle 1+\tan ^{2}x={\frac {1}{\cos ^{2}x}}=\sec ^{2}x}
1
+
cot
2
x
=
1
sin
2
x
=
csc
2
x
{\displaystyle 1+\cot ^{2}x={\frac {1}{\sin ^{2}x}}=\csc ^{2}x}
(Siehe auch den Abschnitt Phasenverschiebungen .)
Mittels dieser Gleichungen lassen sich die drei vorkommenden Funktionen durch eine der beiden anderen darstellen:
sin
x
=
1
−
cos
2
x
{\displaystyle \sin x\;=\;{\sqrt {1-\cos ^{2}x}}}
für
x
∈
[
0
,
π
[
=
[
0
∘
,
180
∘
[
{\displaystyle x\in \left[0,\pi \right[\quad =\quad [0^{\circ },180^{\circ }[}
sin
x
=
−
1
−
cos
2
x
{\displaystyle \sin x\;=\;-{\sqrt {1-\cos ^{2}x}}}
für
x
∈
[
π
,
2
π
[
=
[
180
∘
,
360
∘
[
{\displaystyle x\in \left[\pi ,2\pi \right[\quad =\quad [180^{\circ },360^{\circ }[}
sin
x
=
tan
x
1
+
tan
2
x
{\displaystyle \sin x\;=\;{\frac {\tan x}{\sqrt {1+\tan ^{2}x}}}}
für
x
∈
[
0
,
π
2
[
∪
]
3
π
2
,
2
π
[
=
[
0
∘
,
90
∘
[
∪
]
270
∘
,
360
∘
[
{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}
sin
x
=
−
tan
x
1
+
tan
2
x
{\displaystyle \sin x\;=\;-{\frac {\tan x}{\sqrt {1+\tan ^{2}x}}}}
für
x
∈
]
π
2
,
3
π
2
[
=
]
90
∘
,
270
∘
[
{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}
cos
x
=
1
−
sin
2
x
{\displaystyle \cos x\;=\;{\sqrt {1-\sin ^{2}x}}}
für
x
∈
[
0
,
π
2
[
∪
[
3
π
2
,
2
π
[
=
[
0
∘
,
90
∘
[
∪
[
270
∘
,
360
∘
[
{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left[{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;[270^{\circ },360^{\circ }[}
cos
x
=
−
1
−
sin
2
x
{\displaystyle \cos x\;=\;-{\sqrt {1-\sin ^{2}x}}}
für
x
∈
[
π
2
,
3
π
2
[
=
[
90
∘
,
270
∘
[
{\displaystyle x\in \left[{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad [90^{\circ },270^{\circ }[}
cos
x
=
1
1
+
tan
2
x
{\displaystyle \cos x={\frac {1}{\sqrt {1+\tan ^{2}x}}}}
für
x
∈
[
0
,
π
2
[
∪
]
3
π
2
,
2
π
[
=
[
0
∘
,
90
∘
[
∪
]
270
∘
,
360
∘
[
{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}
cos
x
=
−
1
1
+
tan
2
x
{\displaystyle \cos x=-{\frac {1}{\sqrt {1+\tan ^{2}x}}}}
für
x
∈
]
π
2
,
3
π
2
[
=
]
90
∘
,
270
∘
[
{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}
tan
x
=
1
−
cos
2
x
cos
x
{\displaystyle \tan x={\frac {\sqrt {1-\cos ^{2}x}}{\cos x}}}
für
x
∈
[
0
,
π
2
[
∪
]
π
2
,
π
[
=
[
0
∘
,
90
∘
[
∪
]
90
∘
,
180
∘
[
{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {\pi }{2}},\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]90^{\circ },180^{\circ }[}
tan
x
=
−
1
−
cos
2
x
cos
x
{\displaystyle \tan x=-{\frac {\sqrt {1-\cos ^{2}x}}{\cos x}}}
für
x
∈
[
π
,
3
π
2
[
∪
]
3
π
2
,
2
π
[
=
[
180
∘
,
270
∘
[
∪
]
270
∘
,
360
∘
[
{\displaystyle x\in \left[\pi ,{\frac {3\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [180^{\circ },270^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}
tan
x
=
sin
x
1
−
sin
2
x
{\displaystyle \tan x={\frac {\sin x}{\sqrt {1-\sin ^{2}x}}}}
für
x
∈
[
0
,
π
2
[
∪
]
3
π
2
,
2
π
[
=
[
0
∘
,
90
∘
[
∪
]
270
∘
,
360
∘
[
{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}
tan
x
=
−
sin
x
1
−
sin
2
x
{\displaystyle \tan x=-{\frac {\sin x}{\sqrt {1-\sin ^{2}x}}}}
für
x
∈
]
π
2
,
3
π
2
[
=
]
90
∘
,
270
∘
[
{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}
Vorzeichen der Winkelfunktionen
sin
x
>
0
für
x
∈
]
0
∘
,
180
∘
[
{\displaystyle \sin x>0\quad {\text{für}}\quad x\in \left]0^{\circ },180^{\circ }\right[}
sin
x
<
0
für
x
∈
]
180
∘
,
360
∘
[
{\displaystyle \sin x<0\quad {\text{für}}\quad x\in \left]180^{\circ },360^{\circ }\right[}
cos
x
>
0
für
x
∈
[
0
∘
,
90
∘
[
∪
]
270
∘
,
360
∘
]
{\displaystyle \cos x>0\quad {\text{für}}\quad x\in \left[0^{\circ },90^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right]}
cos
x
<
0
für
x
∈
]
90
∘
,
270
∘
[
{\displaystyle \cos x<0\quad {\text{für}}\quad x\in \left]90^{\circ },270^{\circ }\right[}
tan
x
>
0
für
x
∈
]
0
∘
,
90
∘
[
∪
]
180
∘
,
270
∘
[
{\displaystyle \tan x>0\quad {\text{für}}\quad x\in \left]0^{\circ },90^{\circ }\right[\cup \left]180^{\circ },270^{\circ }\right[}
tan
x
<
0
für
x
∈
]
90
∘
,
180
∘
[
∪
]
270
∘
,
360
∘
[
{\displaystyle \tan x<0\quad {\text{für}}\quad x\in \left]90^{\circ },180^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right[}
Die Vorzeichen von
cot
{\displaystyle \cot }
sec
{\displaystyle \sec }
csc
{\displaystyle \csc }
tan
{\displaystyle \tan }
cos
{\displaystyle \cos }
sin
{\displaystyle \sin }
Wichtige Funktionswerte
Darstellung wichtiger Funktionswerte von Kosinus (1. Klammerwert) und Sinus (2. Klammerwert) auf dem Einheitskreis Weitere Informationen
α
{\displaystyle \alpha }
α
{\displaystyle \alpha }
sin
α
{\displaystyle \sin \alpha }
cos
α
{\displaystyle \cos \alpha }
tan
α
{\displaystyle \tan \alpha }
cot
α
{\displaystyle \cot \alpha }
0
∘
{\displaystyle 0^{\circ }}
0
{\displaystyle \,0}
0
{\displaystyle \,0}
1
{\displaystyle \,1}
0
{\displaystyle \,0}
±
∞
{\displaystyle \pm \infty }
15
∘
{\displaystyle 15^{\circ }}
π
12
{\displaystyle {\tfrac {\pi }{12}}}
1
4
(
6
−
2
)
{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}
1
4
(
6
+
2
)
{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
18
∘
{\displaystyle 18^{\circ }}
π
10
{\displaystyle {\tfrac {\pi }{10}}}
1
4
(
5
−
1
)
{\displaystyle {\tfrac {1}{4}}\left({\sqrt {5}}-1\right)}
1
4
10
+
2
5
{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}
1
5
25
−
10
5
{\displaystyle {\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}
5
+
2
5
{\displaystyle {\sqrt {5+2{\sqrt {5}}}}}
30
∘
{\displaystyle 30^{\circ }}
π
6
{\displaystyle {\tfrac {\pi }{6}}}
1
2
{\displaystyle {\tfrac {1}{2}}}
1
2
3
{\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}
1
3
3
{\displaystyle {\tfrac {1}{3}}{\sqrt {3}}}
3
{\displaystyle {\sqrt {3}}}
36
∘
{\displaystyle 36^{\circ }}
π
5
{\displaystyle {\tfrac {\pi }{5}}}
1
4
10
−
2
5
{\displaystyle {\tfrac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}}
1
4
(
1
+
5
)
{\displaystyle {\tfrac {1}{4}}\left(1+{\sqrt {5}}\right)}
5
−
2
5
{\displaystyle {\sqrt {5-2{\sqrt {5}}}}}
1
5
25
+
10
5
{\displaystyle {\tfrac {1}{5}}{\sqrt {25+10{\sqrt {5}}}}}
45
∘
{\displaystyle 45^{\circ }}
π
4
{\displaystyle {\tfrac {\pi }{4}}}
1
2
2
{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}
1
2
2
{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
54
∘
{\displaystyle 54^{\circ }}
3
π
10
{\displaystyle {\tfrac {3\pi }{10}}}
1
4
(
1
+
5
)
{\displaystyle {\tfrac {1}{4}}\left(1+{\sqrt {5}}\right)}
1
4
10
−
2
5
{\displaystyle {\tfrac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}}
1
5
25
+
10
5
{\displaystyle {\tfrac {1}{5}}{\sqrt {25+10{\sqrt {5}}}}}
5
−
2
5
{\displaystyle {\sqrt {5-2{\sqrt {5}}}}}
60
∘
{\displaystyle 60^{\circ }}
π
3
{\displaystyle {\tfrac {\pi }{3}}}
1
2
3
{\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}
1
2
{\displaystyle {\tfrac {1}{2}}}
3
{\displaystyle {\sqrt {3}}}
1
3
3
{\displaystyle {\tfrac {1}{3}}{\sqrt {3}}}
72
∘
{\displaystyle 72^{\circ }}
2
π
5
{\displaystyle {\tfrac {2\pi }{5}}}
1
4
10
+
2
5
{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}
1
4
(
5
−
1
)
{\displaystyle {\tfrac {1}{4}}\left({\sqrt {5}}-1\right)}
5
+
2
5
{\displaystyle {\sqrt {5+2{\sqrt {5}}}}}
1
5
25
−
10
5
{\displaystyle {\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}
75
∘
{\displaystyle 75^{\circ }}
5
π
12
{\displaystyle {\tfrac {5\pi }{12}}}
1
4
(
6
+
2
)
{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}
1
4
(
6
−
2
)
{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
90
∘
{\displaystyle 90^{\circ }}
π
2
{\displaystyle {\tfrac {\pi }{2}}}
1
{\displaystyle \,1}
0
{\displaystyle \,0}
±
∞
{\displaystyle \pm \infty }
0
{\displaystyle \,0}
108
∘
{\displaystyle 108^{\circ }}
3
π
5
{\displaystyle {\tfrac {3\pi }{5}}}
1
4
10
+
2
5
{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}
1
4
(
1
−
5
)
{\displaystyle {\tfrac {1}{4}}\left(1-{\sqrt {5}}\right)}
−
5
+
2
5
{\displaystyle -{\sqrt {5+2{\sqrt {5}}}}}
−
1
5
25
−
10
5
{\displaystyle -{\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}
120
∘
{\displaystyle 120^{\circ }}
2
π
3
{\displaystyle {\tfrac {2\pi }{3}}}
1
2
3
{\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}
−
1
2
{\displaystyle -{\tfrac {1}{2}}}
−
3
{\displaystyle -{\sqrt {3}}}
−
1
3
3
{\displaystyle -{\tfrac {1}{3}}{\sqrt {3}}}
135
∘
{\displaystyle 135^{\circ }}
3
π
4
{\displaystyle {\tfrac {3\pi }{4}}}
1
2
2
{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}
−
1
2
2
{\displaystyle -{\tfrac {1}{2}}{\sqrt {2}}}
−
1
{\displaystyle -1\,}
−
1
{\displaystyle -1\,}
180
∘
{\displaystyle 180^{\circ }}
π
{\displaystyle \pi \,}
0
{\displaystyle \,0}
−
1
{\displaystyle \,-1}
0
{\displaystyle \,0}
±
∞
{\displaystyle \pm \infty }
270
∘
{\displaystyle 270^{\circ }}
3
π
2
{\displaystyle {\tfrac {3\pi }{2}}}
−
1
{\displaystyle \,-1}
0
{\displaystyle \,0}
±
∞
{\displaystyle \pm \infty }
0
{\displaystyle \,0}
360
∘
{\displaystyle 360^{\circ }}
2
π
{\displaystyle 2\pi }
0
{\displaystyle \,0}
1
{\displaystyle \,1}
0
{\displaystyle \,0}
±
∞
{\displaystyle \pm \infty }
Schließen
Mit Hilfe der Additionstheoreme sind noch viele weitere Werte durch algebraische Ausdrücke (ggfs. mit verschachtelten Quadratwurzeln) darstellbar, insbesondere alle ganzzahligen Vielfachen von
3
∘
{\displaystyle 3^{\circ }}
[2]
Symmetrien
Die trigonometrischen Funktionen haben einfache Symmetrien:
sin
(
−
x
)
=
−
sin
(
x
)
cos
(
−
x
)
=
+
cos
(
x
)
tan
(
−
x
)
=
−
tan
(
x
)
cot
(
−
x
)
=
−
cot
(
x
)
sec
(
−
x
)
=
+
sec
(
x
)
csc
(
−
x
)
=
−
csc
(
x
)
{\displaystyle {\begin{aligned}\sin(-x)&=-\sin(x)\\\cos(-x)&=+\cos(x)\\\tan(-x)&=-\tan(x)\\\cot(-x)&=-\cot(x)\\\sec(-x)&=+\sec(x)\\\csc(-x)&=-\csc(x)\\\end{aligned}}}
Phasenverschiebungen
sin
(
x
+
π
2
)
=
cos
x
bzw.
sin
(
x
+
90
∘
)
=
cos
x
{\displaystyle \sin \left(x+{\frac {\pi }{2}}\right)=\cos x\;\quad {\text{bzw.}}\quad \sin \left(x+90^{\circ }\right)=\cos x\;}
cos
(
x
+
π
2
)
=
−
sin
x
bzw.
cos
(
x
+
90
∘
)
=
−
sin
x
{\displaystyle \cos \left(x+{\frac {\pi }{2}}\right)=-\sin x\;\quad {\text{bzw.}}\quad \cos \left(x+90^{\circ }\right)=-\sin x\;}
tan
(
x
+
π
2
)
=
−
cot
x
bzw.
tan
(
x
+
90
∘
)
=
−
cot
x
{\displaystyle \tan \left(x+{\frac {\pi }{2}}\right)=-\cot x\;\quad {\text{bzw.}}\quad \tan \left(x+90^{\circ }\right)=-\cot x\;}
cot
(
x
+
π
2
)
=
−
tan
x
bzw.
cot
(
x
+
90
∘
)
=
−
tan
x
{\displaystyle \cot \left(x+{\frac {\pi }{2}}\right)=-\tan x\;\quad {\text{bzw.}}\quad \cot \left(x+90^{\circ }\right)=-\tan x\;}
sin
x
=
sin
(
π
−
x
)
bzw.
sin
x
=
sin
(
180
∘
−
x
)
{\displaystyle \sin x\ \;=\;\;\;\sin \left(\pi -x\right)\,\quad {\text{bzw.}}\quad \sin x\ =\;\;\;\sin \left(180^{\circ }-x\right)}
cos
x
=
−
cos
(
π
−
x
)
bzw.
cos
x
=
−
cos
(
180
∘
−
x
)
{\displaystyle \cos x\ \,=-\cos \left(\pi -x\right)\quad {\text{bzw.}}\quad \cos x\ =-\cos \left(180^{\circ }-x\right)}
tan
x
=
−
tan
(
π
−
x
)
bzw.
tan
x
=
−
tan
(
180
∘
−
x
)
{\displaystyle \tan x\ =-\tan \left(\pi -x\right)\quad {\text{bzw.}}\quad \tan x\ =-\tan \left(180^{\circ }-x\right)}
Darstellung durch den Tangens des halben Winkels
Mit der Bezeichnung
t
=
tan
x
2
{\displaystyle t=\tan {\tfrac {x}{2}}}
x
{\displaystyle x}
sin
x
=
2
t
1
+
t
2
,
{\displaystyle \sin x={\frac {2t}{1+t^{2}}},}
cos
x
=
1
−
t
2
1
+
t
2
,
{\displaystyle \cos x={\frac {1-t^{2}}{1+t^{2}}},}
tan
x
=
2
t
1
−
t
2
,
{\displaystyle \tan x={\frac {2t}{1-t^{2}}},}
cot
x
=
1
−
t
2
2
t
,
{\displaystyle \cot x={\frac {1-t^{2}}{2t}},}
sec
x
=
1
+
t
2
1
−
t
2
,
{\displaystyle \sec x={\frac {1+t^{2}}{1-t^{2}}},}
csc
x
=
1
+
t
2
2
t
.
{\displaystyle \csc x={\frac {1+t^{2}}{2t}}.}
Für die Arkusfunktionen gelten folgende Additionstheoreme[6]
Potenzen der Winkelfunktionen
Sinus
sin
2
x
=
1
2
(
1
−
cos
(
2
x
)
)
{\displaystyle \sin ^{2}x={\frac {1}{2}}\ {\Big (}1-\cos(2x){\Big )}}
[13] [20]
sin
3
x
=
1
4
(
3
sin
x
−
sin
(
3
x
)
)
{\displaystyle \sin ^{3}x={\frac {1}{4}}\ {\Big (}3\,\sin x-\sin(3x){\Big )}}
[13] [21]
sin
4
x
=
1
8
(
3
−
4
cos
(
2
x
)
+
cos
(
4
x
)
)
{\displaystyle \sin ^{4}x={\frac {1}{8}}\ {\Big (}3-4\,\cos(2x)+\cos(4x){\Big )}}
[13] [22]
sin
5
x
=
1
16
(
10
sin
x
−
5
sin
(
3
x
)
+
sin
(
5
x
)
)
{\displaystyle \sin ^{5}x={\frac {1}{16}}\ {\Big (}10\,\sin x-5\,\sin(3x)+\sin(5x){\Big )}}
[23]
sin
6
x
=
1
32
(
10
−
15
cos
(
2
x
)
+
6
cos
(
4
x
)
−
cos
(
6
x
)
)
{\displaystyle \sin ^{6}x={\frac {1}{32}}\ {\Big (}10-15\,\cos(2x)+6\,\cos(4x)-\cos(6x){\Big )}}
[24]
sin
n
x
=
1
2
n
∑
k
=
0
n
(
n
k
)
cos
(
(
n
−
2
k
)
(
x
−
π
2
)
)
;
n
∈
N
{\displaystyle \sin ^{n}x={\frac {1}{2^{n}}}\,\sum _{k=0}^{n}{n \choose k}\,\cos \left((n-2k)\left(x-{\frac {\pi }{2}}\right)\right)\ ;\quad n\in \mathbb {N} }
sin
n
x
=
1
2
n
(
n
n
2
)
+
1
2
n
−
1
∑
k
=
0
n
2
−
1
(
−
1
)
n
2
−
k
(
n
k
)
cos
(
(
n
−
2
k
)
x
)
;
n
∈
N
und
n
gerade
{\displaystyle \sin ^{n}x={\frac {1}{2^{n}}}\,{n \choose {\frac {n}{2}}}+{\frac {1}{2^{n-1}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{{\frac {n}{2}}-k}\,{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ gerade }}}
sin
n
x
=
1
2
n
−
1
∑
k
=
0
n
−
1
2
(
−
1
)
n
−
1
2
−
k
(
n
k
)
sin
(
(
n
−
2
k
)
x
)
;
n
∈
N
und
n
ungerade
{\displaystyle \sin ^{n}x={\frac {1}{2^{n-1}}}\,\sum _{k=0}^{\frac {n-1}{2}}(-1)^{{\frac {n-1}{2}}-k}\,{n \choose k}\,\sin {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ ungerade}}}
Kosinus
cos
2
x
=
1
2
(
1
+
cos
(
2
x
)
)
{\displaystyle \cos ^{2}x={\frac {1}{2}}\ {\Big (}1+\cos(2x){\Big )}}
[13] [25]
cos
3
x
=
1
4
(
3
cos
x
+
cos
(
3
x
)
)
{\displaystyle \cos ^{3}x={\frac {1}{4}}\ {\Big (}3\,\cos x+\cos(3x){\Big )}}
[13] [26]
cos
4
x
=
1
8
(
3
+
4
cos
(
2
x
)
+
cos
(
4
x
)
)
{\displaystyle \cos ^{4}x={\frac {1}{8}}\ {\Big (}3+4\,\cos(2x)+\cos(4x){\Big )}}
[13] [27]
cos
5
x
=
1
16
(
10
cos
x
+
5
cos
(
3
x
)
+
cos
(
5
x
)
)
{\displaystyle \cos ^{5}x={\frac {1}{16}}\ {\Big (}10\,\cos x+5\,\cos(3x)+\cos(5x){\Big )}}
[28]
cos
6
x
=
1
32
(
10
+
15
cos
(
2
x
)
+
6
cos
(
4
x
)
+
cos
(
6
x
)
)
{\displaystyle \cos ^{6}x={\frac {1}{32}}\ {\Big (}10+15\,\cos(2x)+6\,\cos(4x)+\cos(6x){\Big )}}
[29]
cos
n
x
=
1
2
n
∑
k
=
0
n
(
n
k
)
cos
(
(
n
−
2
k
)
x
)
;
n
∈
N
{\displaystyle \cos ^{n}x={\frac {1}{2^{n}}}\,\sum _{k=0}^{n}{n \choose k}\,\cos((n-2k)x);\quad n\in \mathbb {N} }
cos
n
x
=
1
2
n
(
n
n
2
)
+
1
2
n
−
1
∑
k
=
0
n
2
−
1
(
n
k
)
cos
(
(
n
−
2
k
)
x
)
;
n
∈
N
und
n
gerade
{\displaystyle \cos ^{n}x={\frac {1}{2^{n}}}\,{n \choose {\frac {n}{2}}}+{\frac {1}{2^{n-1}}}\sum _{k=0}^{{\frac {n}{2}}-1}{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ gerade }}}
cos
n
x
=
1
2
n
−
1
∑
k
=
0
n
−
1
2
(
n
k
)
cos
(
(
n
−
2
k
)
x
)
;
n
∈
N
und
n
ungerade
{\displaystyle \cos ^{n}x={\frac {1}{2^{n-1}}}\,\sum _{k=0}^{\frac {n-1}{2}}{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ ungerade}}}
Tangens
tan
2
x
=
1
−
cos
(
2
x
)
1
+
cos
(
2
x
)
=
sec
2
(
x
)
−
1
{\displaystyle \tan ^{2}x={\frac {1-\cos(2x)}{1+\cos(2x)}}=\sec ^{2}(x)-1}
Umrechnung in andere trigonometrische Funktionen
sin
(
arccos
x
)
=
cos
(
arcsin
x
)
=
1
−
x
2
{\displaystyle \sin(\arccos x)=\cos(\arcsin x)={\sqrt {1-x^{2}}}}
sin
(
arctan
x
)
=
cos
(
arccot
x
)
=
x
1
+
x
2
{\displaystyle \sin(\arctan x)=\cos(\operatorname {arccot} x)={\frac {x}{\sqrt {1+x^{2}}}}}
sin
(
arccot
x
)
=
cos
(
arctan
x
)
=
1
1
+
x
2
{\displaystyle \sin(\operatorname {arccot} x)=\cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}
tan
(
arcsin
x
)
=
cot
(
arccos
x
)
=
x
1
−
x
2
{\displaystyle \tan(\arcsin x)=\cot(\arccos x)={\frac {x}{\sqrt {1-x^{2}}}}}
tan
(
arccos
x
)
=
cot
(
arcsin
x
)
=
1
−
x
2
x
{\displaystyle \tan(\arccos x)=\cot(\arcsin x)={\frac {\sqrt {1-x^{2}}}{x}}}
tan
(
arccot
x
)
=
cot
(
arctan
x
)
=
1
x
{\displaystyle \tan(\operatorname {arccot} x)=\cot(\arctan x)={\frac {1}{x}}}
Sinusoid und Linearkombination mit gleicher Phase
a
sin
α
+
b
cos
α
=
{
a
2
+
b
2
sin
(
α
+
arctan
(
b
a
)
)
, für alle
a
>
0
a
2
+
b
2
cos
(
α
−
arctan
(
a
b
)
)
, für alle
b
>
0
{\displaystyle {\begin{aligned}a\sin \alpha +b\cos \alpha =&{\begin{cases}{\sqrt {a^{2}+b^{2}}}\sin \left(\alpha +\arctan \left({\tfrac {b}{a}}\right)\right)&{\text{, für alle }}a>0\\{\sqrt {a^{2}+b^{2}}}\cos \left(\alpha -\arctan \left({\tfrac {a}{b}}\right)\right)&{\text{, für alle }}b>0\end{cases}}\end{aligned}}}
a
cos
α
+
b
sin
α
=
sgn
(
a
)
a
2
+
b
2
cos
(
α
+
arctan
(
−
b
a
)
)
{\displaystyle {\begin{aligned}a\cos \alpha +b\sin \alpha =\operatorname {sgn}(a){\sqrt {a^{2}+b^{2}}}\cos \left(\alpha +\arctan \left(-{\tfrac {b}{a}}\right)\right)\end{aligned}}}
[30]
a
sin
(
x
+
α
)
+
b
sin
(
x
+
β
)
=
a
2
+
b
2
+
2
a
b
cos
(
α
−
β
)
⋅
sin
(
x
+
δ
)
,
{\displaystyle a\sin(x+\alpha )+b\sin(x+\beta )={\sqrt {a^{2}+b^{2}+2ab\cos(\alpha -\beta )}}\cdot \sin(x+\delta ),}
wobei
δ
=
atan2
(
a
sin
α
+
b
sin
β
,
a
cos
α
+
b
cos
β
)
.
{\displaystyle \delta =\operatorname {atan2} (a\sin \alpha +b\sin \beta ,a\cos \alpha +b\cos \beta ).}
Allgemeiner ist
∑
i
a
i
sin
(
x
+
δ
i
)
=
a
sin
(
x
+
δ
)
,
{\displaystyle \sum _{i}a_{i}\sin(x+\delta _{i})=a\sin(x+\delta ),}
wobei
a
2
=
∑
i
,
j
a
i
a
j
cos
(
δ
i
−
δ
j
)
{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\delta _{i}-\delta _{j})}
und
δ
=
atan2
(
∑
i
a
i
sin
δ
i
,
∑
i
a
i
cos
δ
i
)
.
{\displaystyle \delta =\operatorname {atan2} \left(\sum _{i}a_{i}\sin \delta _{i},\sum _{i}a_{i}\cos \delta _{i}\right).}
Bestimmte Integrale
Die Lösungen der nachfolgenden bestimmten Integrale stehen im Zusammenhang mit der Euler’schen Betafunktion , welche weiterhin mit der Gammafunktion verknüpft ist. Das zweite Integral ist z. B. in der Physik bei der Berechnung von Kräften zwischen zylinderförmigen Dauermagneten unter Verwendung der sogenannten Multipol-Entwicklung hilfreich.
∫
0
π
/
2
cos
ν
1
φ
sin
ν
2
φ
d
φ
=
1
2
⋅
B
(
ν
1
+
1
2
,
ν
2
+
1
2
)
,
Re
(
ν
j
+
1
2
)
>
0
{\displaystyle \int _{0}^{\pi /2}\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi ={\frac {1}{2}}\cdot \operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\;,\quad {\text{Re}}\left({\frac {\nu _{j}+1}{2}}\right)>0}
∫
0
π
cos
ν
1
φ
sin
ν
2
φ
d
φ
=
B
(
ν
1
+
1
2
,
ν
2
+
1
2
)
⋅
1
+
(
−
1
)
ν
1
2
,
ν
j
=
0
,
1
,
2
,
3
,
…
{\displaystyle \int _{0}^{\pi }\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi =\operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\cdot {\frac {1+(-1)^{\nu _{1}}}{2}},\quad \nu _{j}=0,1,2,3,\dots }
∫
0
2
π
cos
ν
1
φ
sin
ν
2
φ
d
φ
=
2
⋅
B
(
ν
1
+
1
2
,
ν
2
+
1
2
)
⋅
1
+
(
−
1
)
ν
1
2
⋅
1
+
(
−
1
)
ν
2
2
,
ν
j
=
0
,
1
,
2
,
3
,
…
{\displaystyle \int _{0}^{2\pi }\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi =2\cdot \operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\cdot {\frac {1+(-1)^{\nu _{1}}}{2}}\cdot {\frac {1+(-1)^{\nu _{2}}}{2}},\quad \nu _{j}=0,1,2,3,\dots }
Reihenentwicklung
Der Sinus (rot) verglichen mit seinem 7. Taylorpolynom (grün) Wie auch sonst in der Analysis werden alle Winkel im Bogenmaß angegeben.
Man kann zeigen, dass der Kosinus die Ableitung des Sinus darstellt und die Ableitung des Kosinus der negative Sinus ist. Hat man diese Ableitungen, kann man die Taylorreihe entwickeln (am einfachsten mit dem Entwicklungspunkt
x
=
0
{\displaystyle x=0}
x
{\displaystyle x}
reellen Zahlen gelten. Mit diesen Reihen werden die trigonometrischen Funktionen für komplexe Argumente definiert (
B
n
{\displaystyle B_{n}}
β
n
{\displaystyle \beta _{n}}
Bernoulli-Zahlen ):
sin
x
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
!
x
2
n
+
1
=
x
−
x
3
3
!
+
x
5
5
!
−
x
7
7
!
±
⋯
,
|
x
|
<
∞
{\displaystyle {\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}\pm \cdots \;,\qquad |x|<\infty \end{aligned}}}
cos
x
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
)
!
x
2
n
=
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
±
⋯
,
|
x
|
<
∞
{\displaystyle {\begin{aligned}\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}\pm \cdots \;,\qquad |x|<\infty \end{aligned}}}
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{\displaystyle {\begin{aligned}\tan x&=\sum _{n=1}^{\infty }(-1)^{n}{\frac {2^{2n}(1-2^{2n})\beta _{2n}}{(2n)!}}x^{2n-1}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}x^{2n-1}\\&=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+{\frac {62}{2835}}x^{9}+\,\cdots \qquad |x|<{\tfrac {\pi }{2}}\end{aligned}}}
[31]
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{\displaystyle {\begin{aligned}\cot x&={\frac {1}{x}}-\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n-1}2^{2n}\beta _{2n}}{(2n)!}}x^{2n-1}={\frac {1}{x}}-\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\&={\frac {1}{x}}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-{\frac {1}{4725}}x^{7}-\,\cdots ,\qquad 0<|x|<\pi \end{aligned}}}
[32] Produktentwicklung
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{\displaystyle \sin(x)=x\prod _{k=1}^{\infty }\left(1-{\frac {x^{2}}{k^{2}\pi ^{2}}}\right)}
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{\displaystyle \cos(x)=\prod _{k=1}^{\infty }\left(1-{\frac {4x^{2}}{(2k-1)^{2}\pi ^{2}}}\right)}
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{\displaystyle \sin(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{{\frac {\pi }{2}}+n\pi }}\right)}
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{\displaystyle \cos(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{{\frac {\pi }{2}}+n\pi }}\right)}
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{\displaystyle \tan(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)}
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{\displaystyle \csc(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi }}\right)}
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{\displaystyle \sec(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)}
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{\displaystyle \cot(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{x+n\pi }}\right)}
Eine Formelsammlung für das rechtwinklige und das allgemeine Dreieck auf der Kugeloberfläche findet sich in einem eigenen Kapitel.
![{\displaystyle {\sqrt[{n}]{x}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7b3ba2638d05cd9ed8dafae7e34986399e48ea99)
![{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ac3f92c0f2af860ff485f97d43f23613e86774da)
![{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cc2b5907a1bc2124087735dbfce5637d5a74dcd4)
![{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {\pi }{2}},\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]90^{\circ },180^{\circ }[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2c17abe7fc5ec6ba7cc8e641f49b22ba3e063b63)
![{\displaystyle x\in \left[\pi ,{\frac {3\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [180^{\circ },270^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1ef7ace2e13e827d94d551099dc2d14a8e786fb6)
![{\displaystyle \sin x>0\quad {\text{für}}\quad x\in \left]0^{\circ },180^{\circ }\right[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/fdac0f290fd61f2d4f5b9a9696bf3169b7cd0918)
![{\displaystyle \sin x<0\quad {\text{für}}\quad x\in \left]180^{\circ },360^{\circ }\right[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/170b6c1d4a0f5062cb5c97c5c1157b4e2535e625)
![{\displaystyle \cos x>0\quad {\text{für}}\quad x\in \left[0^{\circ },90^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/af94605331a13a2721b46d99396163d49faf800d)
![{\displaystyle \cos x<0\quad {\text{für}}\quad x\in \left]90^{\circ },270^{\circ }\right[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/518e3b67bc5d734254ae99debc89eb0aee804f3e)
![{\displaystyle \tan x>0\quad {\text{für}}\quad x\in \left]0^{\circ },90^{\circ }\right[\cup \left]180^{\circ },270^{\circ }\right[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/28fa7ed5c5b9ee66db9fa769dd73c51984ab1557)
![{\displaystyle \tan x<0\quad {\text{für}}\quad x\in \left]90^{\circ },180^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/330c10e3c7a41f96ff875db1ce209871c195f4b8)
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![{\displaystyle \sin {\frac {x}{2}}={\sqrt {\frac {1-\cos x}{2}}}\quad {\text{für}}\quad x\in \left[0,2\pi \right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c05b1bef0e3ec010ffd9aeb20f6b5fcd5facc1c9)
![{\displaystyle \cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}\quad {\text{für}}\quad x\in \left[-\pi ,\pi \right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6f10157e35e43a63caa173267e34c4729ab69247)
![{\displaystyle \tan {\frac {x}{2}}={\frac {\tan x}{1+{\sqrt {1+\tan ^{2}x}}}}\quad {\text{für}}\quad x\in \left]-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6b5e598d507ce768e3efd5bfb8bf974e60b004bb)
![{\displaystyle \cot {\frac {x}{2}}=\cot x+{\sqrt {1+\cot ^{2}x}}\quad {\text{für}}\quad x\in \left]0,\pi \right[}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ab72611c37657b6aedab9984b9976053a5213dab)
![{\displaystyle \left.{\begin{matrix}\tan x+\tan y={\dfrac {\sin(x+y)}{\cos x\cos y}}\\[1em]\tan x-\tan y={\dfrac {\sin(x-y)}{\cos x\cos y}}\end{matrix}}\right\}\Rightarrow \tan x\pm \tan y={\frac {\sin(x\pm y)}{\cos x\cos y}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c82f5271400544871570c9e8e755362c0c27390c)
![{\displaystyle \left.{\begin{matrix}\cot x+\cot y={\dfrac {\sin(y+x)}{\sin x\sin y}}\\[1em]\cot x-\cot y={\dfrac {\sin(y-x)}{\sin x\sin y}}\end{matrix}}\right\}\Rightarrow \cot x\pm \cot y={\frac {\sin(y\pm x)}{\sin x\sin y}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/324f139ea94aaeceb07ee500aa35eaa16d0755c8)
![{\displaystyle \prod _{m=1}^{n}\cos(x_{m})={\frac {1}{2^{n}}}\sum _{k_{1}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\exp \left({\text{i}}\sum _{\nu =1}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]={\frac {1}{2^{n-1}}}\sum _{k_{2}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\cos \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/857634cd97a722c95294a0a74547c8f081596355)
![{\displaystyle \prod _{m=1}^{n}\sin(x_{m})={\frac {1}{(2{\text{i}})^{n}}}\sum _{k_{1}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\prod _{\mu =1}^{n}(-1)^{k_{\mu }}\cdot \exp \left({\text{i}}\sum _{\nu =1}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]={\frac {1}{2^{n-1}}}\sum _{k_{2}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\prod _{\mu =2}^{n}(-1)^{k_{\mu }}\cdot {\begin{cases}\displaystyle (-1)^{n/2}\cdot \cos \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)&{\text{gerade}}\;n\\\displaystyle (-1)^{(n-1)/2}\cdot \sin \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)&{\text{ungerade}}\;n\end{cases}}\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/20ec1339958b378bdfab93d89ae5a470552a1c12)
