Zurück zur Formelsammlung Mathematik
Formeln aus der Trigonometrie der Ebene .
Allgemeingültige Formeln befinden sich in den Abschnitten Winkelfunktionen Arkusfunktionen
Es werden die folgenden Bezeichnungen verwendet: Das Dreieck ABC habe die Seiten
a
=
B
C
¯
,
b
=
C
A
¯
,
c
=
A
B
¯
{\displaystyle a={\overline {BC}}\,,\,b={\overline {CA}}\,,\,c={\overline {AB}}}
Winkel
α
,
β
,
γ
{\displaystyle \alpha ,\beta ,\gamma \,}
Ecken A , B und C . Seien
s
a
,
s
b
,
s
c
{\displaystyle s_{a},s_{b},s_{c}}
w
a
,
w
b
,
w
c
{\displaystyle w_{a},w_{b},w_{c}}
h
a
,
h
b
,
h
c
{\displaystyle h_{a},h_{b},h_{c}}
R der Umkreisradius ,
ρ
{\displaystyle \rho }
Inkreisradius und
ρ
a
,
ρ
b
,
ρ
c
{\displaystyle \rho _{a},\rho _{b},\rho _{c}}
Ankreisradien (und zwar die Radien der Ankreise, die den Ecken A , B bzw. C gegenüberliegen) des Dreiecks ABC . Die Variable s steht für den halben Umfang des Dreiecks:
s
=
(
a
+
b
+
c
)
/
2
{\displaystyle s=(a+b+c)/2}
ABC mit A bezeichnet.
α
+
β
+
γ
=
π
(
=
180
∘
)
{\displaystyle \alpha +\beta +\gamma =\pi \quad (=180^{\circ })}
a
b
=
sin
α
sin
β
,
a
sin
α
=
b
sin
β
=
c
sin
γ
=
2
R
,
a
:
b
:
c
=
sin
α
:
sin
β
:
sin
γ
{\displaystyle {\frac {a}{b}}={\frac {\sin \alpha }{\sin \beta }}\quad ,\quad {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R\quad ,\quad a:b:c=\sin \alpha :\sin \beta :\sin \gamma \,}
Siehe auch: Sinussatz
a
2
=
b
2
+
c
2
−
2
b
c
cos
α
,
cos
α
=
b
2
+
c
2
−
a
2
2
b
c
,
a
2
+
b
c
cos
α
=
a
2
+
b
2
+
c
2
2
{\displaystyle a^{2}=b^{2}+c^{2}-2bc\ \cos \alpha \quad ,\quad \cos \alpha ={\frac {b^{2}+c^{2}-a^{2}}{2bc}}\quad ,\quad a^{2}+bc\,\cos \alpha ={\frac {a^{2}+b^{2}+c^{2}}{2}}}
Siehe auch: Kosinussatz
a
=
b
cos
γ
+
c
cos
β
{\displaystyle a=b\;\cos \gamma +c\;\cos \beta }
b
+
c
a
=
cos
β
−
γ
2
sin
α
2
,
b
−
c
a
=
sin
β
−
γ
2
cos
α
2
{\displaystyle {\frac {b+c}{a}}={\frac {\cos {\frac {\beta -\gamma }{2}}}{\sin {\frac {\alpha }{2}}}}\quad ,\quad {\frac {b-c}{a}}={\frac {\sin {\frac {\beta -\gamma }{2}}}{\cos {\frac {\alpha }{2}}}}}
a
+
b
a
−
b
=
tan
α
+
β
2
tan
α
−
β
2
=
cot
γ
2
tan
α
−
β
2
{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan {\frac {\alpha +\beta }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}={\frac {\cot {\frac {\gamma }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}}
Siehe auch: Tangenssatz
s
−
a
=
b
+
c
−
a
2
,
(
s
−
b
)
+
(
s
−
c
)
=
a
,
(
s
−
a
)
+
(
s
−
b
)
+
(
s
−
c
)
=
s
{\displaystyle s-a={\frac {b+c-a}{2}}\quad ,\quad \left(s-b\right)+\left(s-c\right)=a\quad ,\quad \left(s-a\right)+\left(s-b\right)+\left(s-c\right)=s}
sin
α
2
=
(
s
−
b
)
(
s
−
c
)
b
c
,
cos
α
2
=
s
(
s
−
a
)
b
c
,
tan
α
2
=
(
s
−
b
)
(
s
−
c
)
s
(
s
−
a
)
{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{bc}}}\quad ,\quad \cos {\frac {\alpha }{2}}={\sqrt {\frac {s\left(s-a\right)}{bc}}}\quad ,\quad \tan {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{s\left(s-a\right)}}}}
s
=
4
R
cos
α
2
cos
β
2
cos
γ
2
,
s
−
a
=
4
R
cos
α
2
sin
β
2
sin
γ
2
{\displaystyle s=4R\cos \,{\frac {\alpha }{2}}\,\cos {\frac {\beta }{2}}\,\cos {\frac {\gamma }{2}}\quad ,\quad s-a=4R\,\cos {\frac {\alpha }{2}}\,\sin {\frac {\beta }{2}}\,\sin {\frac {\gamma }{2}}}
Heronsche Formel:
A
=
s
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
=
1
4
(
a
+
b
+
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
(
a
+
b
−
c
)
{\displaystyle A={\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)}}}
A
=
1
4
2
(
b
2
c
2
+
c
2
a
2
+
a
2
b
2
)
−
(
a
4
+
b
4
+
c
4
)
{\displaystyle A={\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)}}}
A
=
1
2
b
c
sin
α
{\displaystyle A={\frac {1}{2}}bc\sin \alpha }
A
=
1
2
a
h
a
{\displaystyle A={\frac {1}{2}}ah_{a}}
ha die Höhe auf der Seite BC ist.
A
=
2
R
2
sin
α
sin
β
sin
γ
{\displaystyle A=2R^{2}\sin \alpha \sin \beta \sin \gamma \,}
A
=
a
b
c
4
R
{\displaystyle A={\frac {abc}{4R}}}
A
=
ρ
s
=
ρ
a
(
s
−
a
)
{\displaystyle A=\rho \,s=\rho _{a}\left(s-a\right)}
A
=
ρ
ρ
a
ρ
b
ρ
c
{\displaystyle A={\sqrt {\rho \rho _{a}\rho _{b}\rho _{c}}}}
a
=
2
R
sin
α
{\displaystyle a=2R\sin \alpha \,}
R
=
a
b
c
4
A
{\displaystyle R={\frac {abc}{4A}}}
ρ
=
(
s
−
a
)
tan
α
2
=
(
s
−
b
)
tan
β
2
=
(
s
−
c
)
tan
γ
2
{\displaystyle \rho =\left(s-a\right)\tan {\frac {\alpha }{2}}=\left(s-b\right)\tan {\frac {\beta }{2}}=\left(s-c\right)\tan {\frac {\gamma }{2}}}
ρ
=
4
R
sin
α
2
sin
β
2
sin
γ
2
=
s
tan
α
2
tan
β
2
tan
γ
2
{\displaystyle \rho =4R\,\sin {\frac {\alpha }{2}}\,\sin {\frac {\beta }{2}}\,\sin {\frac {\gamma }{2}}=s\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}}
ρ
=
R
(
cos
α
+
cos
β
+
cos
γ
−
1
)
{\displaystyle \rho =R\left(\cos \alpha +\cos \beta +\cos \gamma -1\right)}
ρ
=
A
s
=
a
b
c
4
R
s
{\displaystyle \rho ={\frac {A}{s}}={\frac {abc}{4Rs}}}
ρ
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
s
=
1
2
(
b
+
c
−
a
)
(
c
+
a
−
b
)
(
a
+
b
−
c
)
a
+
b
+
c
{\displaystyle \rho ={\sqrt {\frac {\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}}={\frac {1}{2}}{\sqrt {\frac {\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}{a+b+c}}}}
ρ
=
a
cot
β
2
+
cot
γ
2
{\displaystyle \rho ={\frac {a}{\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}}}}
Chapple-Euler-Ungleichung:
2
ρ
≤
R
{\displaystyle 2\rho \leq R}
ABC gleichseitig ist.
ρ
a
=
s
tan
α
2
{\displaystyle \rho _{a}=s\tan {\frac {\alpha }{2}}}
ρ
a
=
4
R
sin
α
2
cos
β
2
cos
γ
2
=
(
s
−
a
)
tan
α
2
cot
β
2
cot
γ
2
{\displaystyle \rho _{a}=4R\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}=\left(s-a\right)\tan {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}}
ρ
a
=
R
(
−
cos
α
+
cos
β
+
cos
γ
+
1
)
{\displaystyle \rho _{a}=R\left(-\cos \alpha +\cos \beta +\cos \gamma +1\right)}
ρ
a
=
A
s
−
a
=
a
b
c
4
R
(
s
−
a
)
{\displaystyle \rho _{a}={\frac {A}{s-a}}={\frac {abc}{4R\left(s-a\right)}}}
ρ
a
=
s
(
s
−
b
)
(
s
−
c
)
s
−
a
=
1
2
(
a
+
b
+
c
)
(
c
+
a
−
b
)
(
a
+
b
−
c
)
b
+
c
−
a
{\displaystyle \rho _{a}={\sqrt {\frac {s\left(s-b\right)\left(s-c\right)}{s-a}}}={\frac {1}{2}}{\sqrt {\frac {\left(a+b+c\right)\left(c+a-b\right)\left(a+b-c\right)}{b+c-a}}}}
1
ρ
=
1
ρ
a
+
1
ρ
b
+
1
ρ
c
{\displaystyle {\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}
h
a
=
b
sin
γ
=
c
sin
β
=
2
A
a
=
2
R
sin
β
sin
γ
{\displaystyle h_{a}=b\sin \gamma =c\sin \beta ={\frac {2A}{a}}=2R\sin \beta \sin \gamma }
h
a
=
a
cot
β
+
cot
γ
{\displaystyle h_{a}={\frac {a}{\cot \beta +\cot \gamma }}}
1
h
a
+
1
h
b
+
1
h
c
=
1
ρ
=
1
ρ
a
+
1
ρ
b
+
1
ρ
c
{\displaystyle {\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}={\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}
Ist
α
=
π
2
(
=
90
∘
)
{\displaystyle \alpha ={\frac {\pi }{2}}\;(=90^{\circ })}
h
a
=
b
c
a
{\displaystyle h_{a}={\frac {b\,c}{a}}}
s
a
=
1
2
2
b
2
+
2
c
2
−
a
2
=
1
2
b
2
+
c
2
+
2
b
c
cos
α
=
a
2
4
+
b
c
cos
α
{\displaystyle s_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\frac {1}{2}}{\sqrt {b^{2}+c^{2}+2bc\cos \alpha }}={\sqrt {{\frac {a^{2}}{4}}+bc\cos \alpha }}}
s
a
2
+
s
b
2
+
s
c
2
=
3
4
(
a
2
+
b
2
+
c
2
)
{\displaystyle s_{a}^{2}+s_{b}^{2}+s_{c}^{2}={\frac {3}{4}}\left(a^{2}+b^{2}+c^{2}\right)}
w
α
=
2
b
c
cos
α
2
b
+
c
=
2
A
a
cos
β
−
γ
2
{\displaystyle w_{\alpha }={\frac {2bc\cos {\frac {\alpha }{2}}}{b+c}}={\frac {2A}{a\cos {\frac {\beta -\gamma }{2}}}}}
