Dreieckberechnung 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 }} Formel 1:
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}}} Formel 2:
wenn α = 90 ∘ {\displaystyle \alpha =90^{\circ }}
sin β = b a {\displaystyle \sin \beta ={\frac {b}{a}}} sin γ = c a {\displaystyle \sin \gamma ={\frac {c}{a}}} wenn β = 90 ∘ {\displaystyle \beta =90^{\circ }}
sin α = a b {\displaystyle \sin \alpha ={\frac {a}{b}}} sin γ = c b {\displaystyle \sin \gamma ={\frac {c}{b}}} wenn γ = 90 ∘ {\displaystyle \gamma =90^{\circ }}
sin α = a c {\displaystyle \sin \alpha ={\frac {a}{c}}} sin β = b c {\displaystyle \sin \beta ={\frac {b}{c}}} Formel 1:
a 2 = b 2 + c 2 − 2 b c cos α {\displaystyle a^{2}=b^{2}+c^{2}-2bc\ \cos \alpha } b 2 = c 2 + a 2 − 2 c a cos β {\displaystyle b^{2}=c^{2}+a^{2}-2ca\ \cos \beta } c 2 = a 2 + b 2 − 2 a b cos γ {\displaystyle c^{2}=a^{2}+b^{2}-2ab\ \cos \gamma } Formel 2:
wenn α = 90 ∘ {\displaystyle \alpha =90^{\circ }}
cos β = c a {\displaystyle \cos \beta ={\frac {c}{a}}} cos γ = b a {\displaystyle \cos \gamma ={\frac {b}{a}}} wenn β = 90 ∘ {\displaystyle \beta =90^{\circ }}
cos α = c b {\displaystyle \cos \alpha ={\frac {c}{b}}} cos γ = a b {\displaystyle \cos \gamma ={\frac {a}{b}}} wenn γ = 90 ∘ {\displaystyle \gamma =90^{\circ }}
a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} Satz des Pythagoras )cos α = b c {\displaystyle \cos \alpha ={\frac {b}{c}}} cos β = a c {\displaystyle \cos \beta ={\frac {a}{c}}}
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}}}}} Formel 1:
b + c b − c = tan β + γ 2 tan β − γ 2 = cot α 2 tan β − γ 2 {\displaystyle {\frac {b+c}{b-c}}={\frac {\tan {\frac {\beta +\gamma }{2}}}{\tan {\frac {\beta -\gamma }{2}}}}={\frac {\cot {\frac {\alpha }{2}}}{\tan {\frac {\beta -\gamma }{2}}}}} Analoge Formeln gelten für a + b a − b {\displaystyle {\frac {a+b}{a-b}}} c + a c − a {\displaystyle {\frac {c+a}{c-a}}}
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}}}}} c + a c − a = tan γ + α 2 tan γ − α 2 = cot β 2 tan γ − α 2 {\displaystyle {\frac {c+a}{c-a}}={\frac {\tan {\frac {\gamma +\alpha }{2}}}{\tan {\frac {\gamma -\alpha }{2}}}}={\frac {\cot {\frac {\beta }{2}}}{\tan {\frac {\gamma -\alpha }{2}}}}} Wegen tan ( − x ) = − tan ( x ) {\displaystyle \tan(-x)=-\tan(x)}
a + c a − c = tan α + γ 2 tan α − γ 2 = cot β 2 tan α − γ 2 {\displaystyle {\frac {a+c}{a-c}}={\frac {\tan {\frac {\alpha +\gamma }{2}}}{\tan {\frac {\alpha -\gamma }{2}}}}={\frac {\cot {\frac {\beta }{2}}}{\tan {\frac {\alpha -\gamma }{2}}}}} Formel 2:
wenn α = 90 ∘ {\displaystyle \alpha =90^{\circ }}
tan β = b c {\displaystyle \tan \beta ={\frac {b}{c}}} tan γ = c b {\displaystyle \tan \gamma ={\frac {c}{b}}} wenn β = 90 ∘ {\displaystyle \beta =90^{\circ }}
tan α = a c {\displaystyle \tan \alpha ={\frac {a}{c}}} tan γ = c a {\displaystyle \tan \gamma ={\frac {c}{a}}} wenn γ = 90 ∘ {\displaystyle \gamma =90^{\circ }}
tan α = a b {\displaystyle \tan \alpha ={\frac {a}{b}}} tan β = b a {\displaystyle \tan \beta ={\frac {b}{a}}}
Im Folgenden bedeutet s {\displaystyle s} A B C {\displaystyle ABC} s = a + b + c 2 {\displaystyle s={\frac {a+b+c}{2}}}
s − a = b + c − a 2 {\displaystyle s-a={\frac {b+c-a}{2}}} s − b = c + a − b 2 {\displaystyle s-b={\frac {c+a-b}{2}}} s − c = a + b − c 2 {\displaystyle s-c={\frac {a+b-c}{2}}} ( s − b ) + ( s − c ) = a {\displaystyle \left(s-b\right)+\left(s-c\right)=a} ( s − c ) + ( s − a ) = b {\displaystyle \left(s-c\right)+\left(s-a\right)=b} ( s − a ) + ( s − b ) = c {\displaystyle \left(s-a\right)+\left(s-b\right)=c} ( s − a ) + ( s − b ) + ( s − c ) = s {\displaystyle \left(s-a\right)+\left(s-b\right)+\left(s-c\right)=s} sin α 2 = ( s − b ) ( s − c ) b c {\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{bc}}}} sin β 2 = ( s − c ) ( s − a ) c a {\displaystyle \sin {\frac {\beta }{2}}={\sqrt {\frac {\left(s-c\right)\left(s-a\right)}{ca}}}} sin γ 2 = ( s − a ) ( s − b ) a b {\displaystyle \sin {\frac {\gamma }{2}}={\sqrt {\frac {\left(s-a\right)\left(s-b\right)}{ab}}}} cos α 2 = s ( s − a ) b c {\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s\left(s-a\right)}{bc}}}} cos β 2 = s ( s − b ) c a {\displaystyle \cos {\frac {\beta }{2}}={\sqrt {\frac {s\left(s-b\right)}{ca}}}} cos γ 2 = s ( s − c ) a b {\displaystyle \cos {\frac {\gamma }{2}}={\sqrt {\frac {s\left(s-c\right)}{ab}}}} tan α 2 = ( s − b ) ( s − c ) s ( s − a ) {\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{s\left(s-a\right)}}}} tan β 2 = ( s − c ) ( s − a ) s ( s − b ) {\displaystyle \tan {\frac {\beta }{2}}={\sqrt {\frac {\left(s-c\right)\left(s-a\right)}{s\left(s-b\right)}}}} tan γ 2 = ( s − a ) ( s − b ) s ( s − c ) {\displaystyle \tan {\frac {\gamma }{2}}={\sqrt {\frac {\left(s-a\right)\left(s-b\right)}{s\left(s-c\right)}}}} s = 4 r cos α 2 cos β 2 cos γ 2 {\displaystyle s=4r\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}} s − a = 4 r cos α 2 sin β 2 sin γ 2 {\displaystyle s-a=4r\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\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}}}
In- und Ankreisradien In diesem Abschnitt werden Formeln aufgelistet, in denen der Inkreisradius ρ {\displaystyle \rho } Ankreisradien ρ a {\displaystyle \rho _{a}} ρ b {\displaystyle \rho _{b}} ρ c {\displaystyle \rho _{c}} A B C {\displaystyle ABC}
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 = F s = a b c 4 r s {\displaystyle \rho ={\frac {F}{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 = b cot γ 2 + cot α 2 = c cot α 2 + cot β 2 {\displaystyle \rho ={\frac {a}{\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}}}={\frac {b}{\cot {\frac {\gamma }{2}}+\cot {\frac {\alpha }{2}}}}={\frac {c}{\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}}}} a ⋅ b + b ⋅ c + c ⋅ a = s 2 + ρ 2 + 4 ⋅ ρ ⋅ r {\displaystyle a\cdot b+b\cdot c+c\cdot a=s^{2}+\rho ^{2}+4\cdot \rho \cdot r} [ 1] Wichtige Ungleichung: 2 ρ ≤ r {\displaystyle 2\rho \leq r} A B C {\displaystyle ABC}
ρ a = s tan α 2 = ( s − b ) cot γ 2 = ( s − c ) cot β 2 {\displaystyle \rho _{a}=s\tan {\frac {\alpha }{2}}=\left(s-b\right)\cot {\frac {\gamma }{2}}=\left(s-c\right)\cot {\frac {\beta }{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 = F s − a = a b c 4 r ( s − a ) {\displaystyle \rho _{a}={\frac {F}{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}}}} Die Ankreise sind gleichberechtigt: Jede Formel für ρ a {\displaystyle \rho _{a}} ρ b {\displaystyle \rho _{b}} ρ c {\displaystyle \rho _{c}}
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}}}}
Die Längen der von A {\displaystyle A} B {\displaystyle B} C {\displaystyle C} A B C {\displaystyle ABC} h a {\displaystyle h_{a}} h b {\displaystyle h_{b}} h c {\displaystyle h_{c}}
h a = b sin γ = c sin β = 2 F a = 2 r sin β sin γ = 2 r ( cos α + cos β cos γ ) {\displaystyle h_{a}=b\sin \gamma =c\sin \beta ={\frac {2F}{a}}=2r\sin \beta \sin \gamma =2r\left(\cos \alpha +\cos \beta \cos \gamma \right)} h b = c sin α = a sin γ = 2 F b = 2 r sin γ sin α = 2 r ( cos β + cos α cos γ ) {\displaystyle h_{b}=c\sin \alpha =a\sin \gamma ={\frac {2F}{b}}=2r\sin \gamma \sin \alpha =2r\left(\cos \beta +\cos \alpha \cos \gamma \right)} h c = a sin β = b sin α = 2 F c = 2 r sin α sin β = 2 r ( cos γ + cos α cos β ) {\displaystyle h_{c}=a\sin \beta =b\sin \alpha ={\frac {2F}{c}}=2r\sin \alpha \sin \beta =2r\left(\cos \gamma +\cos \alpha \cos \beta \right)} h a = a cot β + cot γ ; h b = b cot γ + cot α ; h c = c cot α + cot β {\displaystyle h_{a}={\frac {a}{\cot \beta +\cot \gamma }};\;\;\;\;\;h_{b}={\frac {b}{\cot \gamma +\cot \alpha }};\;\;\;\;\;h_{c}={\frac {c}{\cot \alpha +\cot \beta }}} 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}} 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}}}} Hat das Dreieck A B C {\displaystyle ABC} C {\displaystyle C} γ = 90 ∘ {\displaystyle \gamma =90^{\circ }}
h c = a b c {\displaystyle h_{c}={\frac {ab}{c}}} h a = b {\displaystyle h_{a}=b} h b = a {\displaystyle h_{b}=a}
Seitenhalbierende Die Längen der von A {\displaystyle A} B {\displaystyle B} C {\displaystyle C} A B C {\displaystyle ABC} s a {\displaystyle s_{a}} s b {\displaystyle s_{b}} s c {\displaystyle s_{c}}
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 b = 1 2 2 c 2 + 2 a 2 − b 2 = 1 2 c 2 + a 2 + 2 c a cos β = b 2 4 + c a cos β {\displaystyle s_{b}={\frac {1}{2}}{\sqrt {2c^{2}+2a^{2}-b^{2}}}={\frac {1}{2}}{\sqrt {c^{2}+a^{2}+2ca\cos \beta }}={\sqrt {{\frac {b^{2}}{4}}+ca\cos \beta }}} s c = 1 2 2 a 2 + 2 b 2 − c 2 = 1 2 a 2 + b 2 + 2 a b cos γ = c 2 4 + a b cos γ {\displaystyle s_{c}={\frac {1}{2}}{\sqrt {2a^{2}+2b^{2}-c^{2}}}={\frac {1}{2}}{\sqrt {a^{2}+b^{2}+2ab\cos \gamma }}={\sqrt {{\frac {c^{2}}{4}}+ab\cos \gamma }}} 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)}
Winkelhalbierende Wir bezeichnen mit w α {\displaystyle w_{\alpha }} w β {\displaystyle w_{\beta }} w γ {\displaystyle w_{\gamma }} A {\displaystyle A} B {\displaystyle B} C {\displaystyle C} A B C {\displaystyle ABC}
w α = 2 b c cos α 2 b + c = 2 F a cos β − γ 2 = b c ( b + c − a ) ( a + b + c ) b + c {\displaystyle w_{\alpha }={\frac {2bc\cos {\frac {\alpha }{2}}}{b+c}}={\frac {2F}{a\cos {\frac {\beta -\gamma }{2}}}}={\frac {\sqrt {bc(b+c-a)(a+b+c)}}{b+c}}} w β = 2 c a cos β 2 c + a = 2 F b cos γ − α 2 = c a ( c + a − b ) ( a + b + c ) c + a {\displaystyle w_{\beta }={\frac {2ca\cos {\frac {\beta }{2}}}{c+a}}={\frac {2F}{b\cos {\frac {\gamma -\alpha }{2}}}}={\frac {\sqrt {ca(c+a-b)(a+b+c)}}{c+a}}} w γ = 2 a b cos γ 2 a + b = 2 F c cos α − β 2 = a b ( a + b − c ) ( a + b + c ) a + b {\displaystyle w_{\gamma }={\frac {2ab\cos {\frac {\gamma }{2}}}{a+b}}={\frac {2F}{c\cos {\frac {\alpha -\beta }{2}}}}={\frac {\sqrt {ab(a+b-c)(a+b+c)}}{a+b}}}
Allgemeine Trigonometrie in der Ebene 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 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 }
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}}.}
Additionstheoreme Figur 1 Figur 2 Für Sinus und Kosinus lassen sich die Additionstheoreme aus der Verkettung zweier Drehungen um den Winkel x {\displaystyle x} y {\displaystyle y} Produkt zweier Drehmatrizen der Ebene R 2 {\displaystyle \mathbb {R} ^{2}} . Alternativ folgen die Additionstheoreme aus der Anwendung der Eulerschen Formel auf die Beziehung e i ( x + y ) = e i x ⋅ e i y {\displaystyle \textstyle e^{i(x+y)}=e^{ix}\cdot e^{iy}} Symmetrien .[ 3]
sin ( x ± y ) = sin x ⋅ cos y ± cos x ⋅ sin y {\displaystyle \sin(x\pm y)=\sin x\cdot \cos y\pm \cos x\cdot \sin y} [ 4] cos ( x ± y ) = cos x ⋅ cos y ∓ sin x ⋅ sin y {\displaystyle \cos(x\pm y)=\cos x\cdot \cos y\mp \sin x\cdot \sin y} [ 4] Geometrische Herleitungen sind in Figur 1 und Figur 2 für Winkel α {\displaystyle \alpha } β {\displaystyle \beta } [ 5]
Zu Figur 1:
sin ( α + β ) = sin α ⋅ cos β + cos α ⋅ sin β {\displaystyle \sin(\alpha +\beta )=\sin \alpha \cdot \cos \beta +\cos \alpha \cdot \sin \beta } cos ( α + β ) = cos α ⋅ cos β − sin α ⋅ sin β {\displaystyle \cos(\alpha +\beta )=\cos \alpha \cdot \cos \beta -\sin \alpha \cdot \sin \beta } Zu Figur 2:
sin ( α − β ) = sin α ⋅ cos β − cos α ⋅ sin β {\displaystyle \sin(\alpha -\beta )=\sin \alpha \cdot \cos \beta -\cos \alpha \cdot \sin \beta } cos ( α − β ) = cos α ⋅ cos β + sin α ⋅ sin β {\displaystyle \cos(\alpha -\beta )=\cos \alpha \cdot \cos \beta +\sin \alpha \cdot \sin \beta } Durch Erweiterung mit 1 cos x cos y {\displaystyle \textstyle {1 \over \cos x\cos y}} 1 sin x sin y {\displaystyle \textstyle {1 \over \sin x\sin y}}
tan ( x ± y ) = sin ( x ± y ) cos ( x ± y ) = tan x ± tan y 1 ∓ tan x tan y {\displaystyle \tan(x\pm y)={\frac {\sin(x\pm y)}{\cos(x\pm y)}}={\frac {\tan x\pm \tan y}{1\mp \tan x\;\tan y}}} cot ( x ± y ) = cos ( x ± y ) sin ( x ± y ) = cot x cot y ∓ 1 cot y ± cot x {\displaystyle \cot(x\pm y)={\frac {\cos(x\pm y)}{\sin(x\pm y)}}={\frac {\cot x\cot y\mp 1}{\cot y\pm \cot x}}} Für x = y {\displaystyle x=y} Doppelwinkelfunktionen , für y = π / 2 {\displaystyle y=\pi /2} Phasenverschiebungen .
sin ( x + y ) ⋅ sin ( x − y ) = cos 2 y − cos 2 x = sin 2 x − sin 2 y {\displaystyle \sin(x+y)\cdot \sin(x-y)=\cos ^{2}y-\cos ^{2}x=\sin ^{2}x-\sin ^{2}y} cos ( x + y ) ⋅ cos ( x − y ) = cos 2 y − sin 2 x = cos 2 x − sin 2 y {\displaystyle \cos(x+y)\cdot \cos(x-y)=\cos ^{2}y-\sin ^{2}x=\cos ^{2}x-\sin ^{2}y}
Für die Arkusfunktionen gelten folgende Additionstheoreme[ 6]
Summanden Summenformel Gültigkeitsbereich arcsin x + arcsin y = {\displaystyle \arcsin x+\arcsin y=} arcsin ( x 1 − y 2 + y 1 − x 2 ) {\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)} x y ≤ 0 {\displaystyle xy\leq 0} x 2 + y 2 ≤ 1 {\displaystyle x^{2}+y^{2}\leq 1} π − arcsin ( x 1 − y 2 + y 1 − x 2 ) {\displaystyle \pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)} x > 0 {\displaystyle x>0} y > 0 {\displaystyle y>0} x 2 + y 2 > 1 {\displaystyle x^{2}+y^{2}>1} − π − arcsin ( x 1 − y 2 + y 1 − x 2 ) {\displaystyle -\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)} x < 0 {\displaystyle x<0} y < 0 {\displaystyle y<0} x 2 + y 2 > 1 {\displaystyle x^{2}+y^{2}>1} arcsin x − arcsin y = {\displaystyle \arcsin x-\arcsin y=} arcsin ( x 1 − y 2 − y 1 − x 2 ) {\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)} x y ≥ 0 {\displaystyle xy\geq 0} x 2 + y 2 ≤ 1 {\displaystyle x^{2}+y^{2}\leq 1} π − arcsin ( x 1 − y 2 − y 1 − x 2 ) {\displaystyle \pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)} x > 0 {\displaystyle x>0} y < 0 {\displaystyle y<0} x 2 + y 2 > 1 {\displaystyle x^{2}+y^{2}>1} − π − arcsin ( x 1 − y 2 − y 1 − x 2 ) {\displaystyle -\pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)} x < 0 {\displaystyle x<0} y > 0 {\displaystyle y>0} x 2 + y 2 > 1 {\displaystyle x^{2}+y^{2}>1} arccos x + arccos y = {\displaystyle \arccos x+\arccos y=} arccos ( x y − 1 − x 2 1 − y 2 ) {\displaystyle \arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)} x + y ≥ 0 {\displaystyle x+y\geq 0} 2 π − arccos ( x y − 1 − x 2 1 − y 2 ) {\displaystyle 2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)} x + y < 0 {\displaystyle x+y<0} arccos x − arccos y = {\displaystyle \arccos x-\arccos y=} − arccos ( x y + 1 − x 2 1 − y 2 ) {\displaystyle -\arccos \left(xy+{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)} x ≥ y {\displaystyle x\geq y} arccos ( x y + 1 − x 2 1 − y 2 ) {\displaystyle \arccos \left(xy+{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)} x < y {\displaystyle x<y} arctan x + arctan y = {\displaystyle \arctan x+\arctan y=} arctan ( x + y 1 − x y ) {\displaystyle \arctan \left({\frac {x+y}{1-xy}}\right)} x y < 1 {\displaystyle xy<1} π + arctan ( x + y 1 − x y ) {\displaystyle \pi +\arctan \left({\frac {x+y}{1-xy}}\right)} x > 0 {\displaystyle x>0} x y > 1 {\displaystyle xy>1} − π + arctan ( x + y 1 − x y ) {\displaystyle -\pi +\arctan \left({\frac {x+y}{1-xy}}\right)} x < 0 {\displaystyle x<0} x y > 1 {\displaystyle xy>1} arctan x − arctan y = {\displaystyle \arctan x-\arctan y=} arctan ( x − y 1 + x y ) {\displaystyle \arctan \left({\frac {x-y}{1+xy}}\right)} x y > − 1 {\displaystyle xy>-1} π + arctan ( x − y 1 + x y ) {\displaystyle \pi +\arctan \left({\frac {x-y}{1+xy}}\right)} x > 0 {\displaystyle x>0} x y < − 1 {\displaystyle xy<-1} − π + arctan ( x − y 1 + x y ) {\displaystyle -\pi +\arctan \left({\frac {x-y}{1+xy}}\right)} x < 0 {\displaystyle x<0} x y < − 1 {\displaystyle xy<-1}
Doppelwinkelfunktionen Figur 3 sin ( 2 x ) = 2 sin x ⋅ cos x = 2 tan x 1 + tan 2 x {\displaystyle \sin(2x)=2\sin x\cdot \;\cos x={\frac {2\tan x}{1+\tan ^{2}x}}} Eine geometrische Herleitung ist in Figur 3 für Winkel α {\displaystyle \alpha } β {\displaystyle \beta } [ 7]
Zu Figur 3:
Aus der Berechnung der Flächeninhalte der beiden grauen Dreiecke ergibt sich 1 2 ⋅ 2 sin α ⋅ 2 cos α = 1 2 ⋅ 2 ⋅ sin ( 2 α ) {\displaystyle {\frac {1}{2}}\cdot 2\sin \alpha \cdot 2\cos \alpha ={\frac {1}{2}}\cdot 2\cdot \sin(2\alpha )} 2 sin α ⋅ cos α = sin ( 2 α ) {\displaystyle 2\sin \alpha \cdot \cos \alpha =\sin(2\alpha )} Weitere Beziehungen:
cos ( 2 x ) = cos 2 x − sin 2 x = 1 − 2 sin 2 x = 2 cos 2 x − 1 = 1 − tan 2 x 1 + tan 2 x {\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x=1-2\sin ^{2}x=2\cos ^{2}x-1={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}}} tan ( 2 x ) = 2 tan x 1 − tan 2 x = 2 cot x − tan x {\displaystyle \tan(2x)={\frac {2\tan x}{1-\tan ^{2}x}}={\frac {2}{\cot x-\tan x}}} cot ( 2 x ) = cot 2 x − 1 2 cot x = cot x − tan x 2 {\displaystyle \cot(2x)={\frac {\cot ^{2}x-1}{2\cot x}}={\frac {\cot x-\tan x}{2}}}
Die Formeln für Vielfache berechnen sich normalerweise über die komplexen Zahlen aus der Euler-Formel z = r ( cos ϕ + i sin ϕ ) ⟺ z n = r n ( cos ϕ + i sin ϕ ) n {\displaystyle z=r\left(\cos \phi +i\sin \phi \right)\iff z^{n}=r^{n}\left(\cos \phi +i\sin \phi \right)^{n}} z n = r n ( cos ( n ϕ ) + i sin ( n ϕ ) ) {\displaystyle z^{n}=r^{n}\left(\cos \left(n\phi \right)+i\sin \left(n\phi \right)\right)} cos ( n ϕ ) + i sin ( n ϕ ) = ( cos ϕ + i sin ϕ ) n {\displaystyle \cos \left(n\phi \right)+i\sin \left(n\phi \right)=\left(\cos \phi +i\sin \phi \right)^{n}} cos {\displaystyle \cos } sin {\displaystyle \sin }
Die Formel für cos ( n x ) {\displaystyle \cos(nx)} T n ( cos x ) = cos ( n x ) {\displaystyle T_{n}(\cos x)=\cos(nx)} [ 8] Tschebyschow-Polynomen in Beziehung.
sin ( 3 x ) = 3 sin x − 4 sin 3 x {\displaystyle \sin(3x)=3\sin x-4\sin ^{3}x\,} [ 9] = sin x ( 4 cos 2 x − 1 ) {\displaystyle =\;\sin x\left(4\cos ^{2}x-1\right)} sin ( 4 x ) = 8 sin x cos 3 x − 4 sin x cos x {\displaystyle \sin(4x)=8\sin x\;\cos ^{3}x-4\sin x\;\cos x} [ 10] = sin x ( 8 cos 3 x − 4 cos x ) {\displaystyle =\;\sin x\left(8\cos ^{3}x-4\cos x\right)} sin ( 5 x ) = 5 sin x − 20 sin 3 x + 16 sin 5 x {\displaystyle \sin(5x)=5\sin x-20\sin ^{3}x+16\sin ^{5}x\;} [ 11] = sin x ( 16 cos 4 x − 12 cos 2 x + 1 ) {\displaystyle =\;\sin x\left(16\cos ^{4}x-12\cos ^{2}x+1\right)} sin ( n x ) = n sin x cos n − 1 x − ( n 3 ) sin 3 x cos n − 3 x + ( n 5 ) sin 5 x cos n − 5 x − … + … {\displaystyle \sin(nx)=n\;\sin x\;\cos ^{n-1}x-{n \choose 3}\sin ^{3}x\;\cos ^{n-3}x+{n \choose 5}\sin ^{5}x\;\cos ^{n-5}x\;-\ldots +\ldots } [ 12] [ 13] = ∑ j = 0 ⌊ n − 1 2 ⌋ ( − 1 ) j ( n 2 j + 1 ) sin 2 j + 1 x cos n − 2 j − 1 x {\displaystyle =\;\sum _{j=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }(-1)^{j}{n \choose 2j+1}\sin ^{2j+1}x\;\cos ^{n-2j-1}x} = sin x ∑ k = 0 ⌊ n − 1 2 ⌋ ( − 1 ) k ( n − k − 1 k ) 2 n − 2 k − 1 cos n − 2 k − 1 x {\displaystyle =\;\sin x\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }(-1)^{k}{n-k-1 \choose k}2^{n-2k-1}\cos ^{n-2k-1}x} sin ( n x ) sin ( x ) = cos ( ( n − 1 ) x ) − cos ( ( n + 1 ) x ) 2 {\displaystyle \sin(nx)\;\sin(x)={\frac {\cos((n-1)\,x)-\cos((n+1)\,x)}{2}}} cos ( 3 x ) = 4 cos 3 x − 3 cos x {\displaystyle \cos(3x)=4\cos ^{3}x-3\cos x\,} [ 14] cos ( 4 x ) = 8 cos 4 x − 8 cos 2 x + 1 {\displaystyle \cos(4x)=8\cos ^{4}x-8\cos ^{2}x+1\,} [ 15] cos ( 5 x ) = 16 cos 5 x − 20 cos 3 x + 5 cos x {\displaystyle \cos(5x)=16\cos ^{5}x-20\cos ^{3}x+5\cos x\,} [ 16] cos ( 6 x ) = 32 cos 6 x − 48 cos 4 x + 18 cos 2 x − 1 {\displaystyle \cos(6x)=32\cos ^{6}x-48\cos ^{4}x+18\cos ^{2}x-1\,} [ 17] cos ( n x ) = cos n x − ( n 2 ) sin 2 x cos n − 2 x + ( n 4 ) sin 4 x cos n − 4 x − … + … {\displaystyle \cos(nx)=\cos ^{n}x-{n \choose 2}\sin ^{2}x\;\cos ^{n-2}x+{n \choose 4}\sin ^{4}x\;\cos ^{n-4}x\;-\ldots +\ldots } [ 13] [ 18] = ∑ j = 0 ⌊ n 2 ⌋ ( − 1 ) j ( n 2 j ) sin 2 j x cos n − 2 j x {\displaystyle =\;\sum _{j=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{j}{n \choose 2j}\sin ^{2j}x\;\cos ^{n-2j}x} cos ( n x ) cos ( x ) = cos ( ( n − 1 ) x ) + cos ( ( n + 1 ) x ) 2 {\displaystyle \cos(nx)\;\cos(x)={\frac {\cos((n-1)\,x)+\cos((n+1)\,x)}{2}}} tan ( 3 x ) = 3 tan x − tan 3 x 1 − 3 tan 2 x {\displaystyle \tan(3x)={\frac {3\tan x-\tan ^{3}x}{1-3\tan ^{2}x}}} [ 13] tan ( 4 x ) = 4 tan x − 4 tan 3 x 1 − 6 tan 2 x + tan 4 x {\displaystyle \tan(4x)={\frac {4\tan x-4\tan ^{3}x}{1-6\tan ^{2}x+\tan ^{4}x}}} [ 13] cot ( 3 x ) = cot 3 x − 3 cot x 3 cot 2 x − 1 {\displaystyle \cot(3x)={\frac {\cot ^{3}x-3\cot x}{3\cot ^{2}x-1}}} [ 13] cot ( 4 x ) = cot 4 x − 6 cot 2 x + 1 4 cot 3 x − 4 cot x {\displaystyle \cot(4x)={\frac {\cot ^{4}x-6\cot ^{2}x+1}{4\cot ^{3}x-4\cot x}}} [ 13]
Figur 4 Zur Berechnung des Funktionswertes des halben Arguments dienen die Halbwinkelformeln [ 13]
sin x 2 = 1 − cos x 2 für x ∈ [ 0 , 2 π ] {\displaystyle \sin {\frac {x}{2}}={\sqrt {\frac {1-\cos x}{2}}}\quad {\text{für}}\quad x\in \left[0,2\pi \right]} cos x 2 = 1 + cos x 2 für x ∈ [ − π , π ] {\displaystyle \cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}\quad {\text{für}}\quad x\in \left[-\pi ,\pi \right]} tan x 2 = 1 − cos x sin x = sin x 1 + cos x für x ∈ R ∖ π ( 2 Z + 1 ) {\displaystyle \tan {\frac {x}{2}}={\frac {1-\cos x}{\sin x}}={\frac {\sin x}{1+\cos x}}\quad {\text{für}}\quad x\in \mathbb {R} \setminus \pi (2\mathbb {Z} +1)} cot x 2 = 1 + cos x sin x = sin x 1 − cos x für x ∈ R ∖ 2 π Z {\displaystyle \cot {\frac {x}{2}}={\frac {1+\cos x}{\sin x}}={\frac {\sin x}{1-\cos x}}\quad {\text{für}}\quad x\in \mathbb {R} \setminus 2\pi \mathbb {Z} } Eine geometrische Herleitung der dritten Formel ist in Figur 4 für Winkel α {\displaystyle \alpha } β {\displaystyle \beta } [ 19] tan ( α 2 ) = sin α 1 + cos α = 1 − cos α sin α {\displaystyle \tan \left({\frac {\alpha }{2}}\right)={\frac {\sin \alpha }{1+\cos \alpha }}={\frac {1-\cos \alpha }{\sin \alpha }}}
Außerdem gilt:
tan x 2 = tan x 1 + 1 + tan 2 x für x ∈ ] − π 2 , π 2 [ {\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[} cot x 2 = cot x + 1 + cot 2 x für x ∈ ] 0 , π [ {\displaystyle \cot {\frac {x}{2}}=\cot x+{\sqrt {1+\cot ^{2}x}}\quad {\text{für}}\quad x\in \left]0,\pi \right[} Siehe auch: Halbwinkelsatz
Aus den Additionstheoremen lassen sich Identitäten ableiten, mit deren Hilfe die Summe zweier trigonometrischer Funktionen als Produkt dargestellt werden kann:[ 13]
sin x + sin y = 2 sin x + y 2 cos x − y 2 sin x − sin y = 2 cos x + y 2 sin x − y 2 cos x + cos y = 2 cos x + y 2 cos x − y 2 cos x − cos y = − 2 sin x + y 2 sin x − y 2 {\displaystyle {\begin{aligned}\sin x+\sin y&=2\sin {\frac {x+y}{2}}\cos {\frac {x-y}{2}}\\\sin x-\sin y&=2\cos {\frac {x+y}{2}}\sin {\frac {x-y}{2}}\\\cos x+\cos y&=2\cos {\frac {x+y}{2}}\cos {\frac {x-y}{2}}\\\cos x-\cos y&=-2\sin {\frac {x+y}{2}}\sin {\frac {x-y}{2}}\end{aligned}}} tan x + tan y = sin ( x + y ) cos x cos y tan x − tan y = sin ( x − y ) cos x cos y } ⇒ tan x ± tan y = sin ( x ± y ) cos x cos y {\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}}} cot x + cot y = sin ( y + x ) sin x sin y cot x − cot y = sin ( y − x ) sin x sin y } ⇒ cot x ± cot y = sin ( y ± x ) sin x sin y {\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}}} Daraus ergeben sich noch Spezialfälle:
cos x + sin x = 2 ⋅ sin ( x + π 4 ) = 2 ⋅ cos ( x − π 4 ) cos x − sin x = 2 ⋅ cos ( x + π 4 ) = − 2 ⋅ sin ( x − π 4 ) {\displaystyle {\begin{aligned}\cos x+\sin x&={\sqrt {2}}\cdot \sin \left(x+{\frac {\pi }{4}}\right)={\sqrt {2}}\cdot \cos \left(x-{\frac {\pi }{4}}\right)\\\cos x-\sin x&={\sqrt {2}}\cdot \cos \left(x+{\frac {\pi }{4}}\right)=-{\sqrt {2}}\cdot \sin \left(x-{\frac {\pi }{4}}\right)\end{aligned}}}
Produkte der Winkelfunktionen Produkte der trigonometrischen Funktionen lassen sich mit folgenden Formeln berechnen:[ 13]
sin x sin y = 1 2 ( cos ( x − y ) − cos ( x + y ) ) {\displaystyle \sin x\;\sin y={\frac {1}{2}}{\Big (}\cos(x-y)-\cos(x+y){\Big )}} cos x cos y = 1 2 ( cos ( x − y ) + cos ( x + y ) ) {\displaystyle \cos x\;\cos y={\frac {1}{2}}{\Big (}\cos(x-y)+\cos(x+y){\Big )}} sin x cos y = 1 2 ( sin ( x − y ) + sin ( x + y ) ) {\displaystyle \sin x\;\cos y={\frac {1}{2}}{\Big (}\sin(x-y)+\sin(x+y){\Big )}} tan x tan y = tan x + tan y cot x + cot y = − tan x − tan y cot x − cot y {\displaystyle \tan x\;\tan y={\frac {\tan x+\tan y}{\cot x+\cot y}}=-{\frac {\tan x-\tan y}{\cot x-\cot y}}} cot x cot y = cot x + cot y tan x + tan y = − cot x − cot y tan x − tan y {\displaystyle \cot x\;\cot y={\frac {\cot x+\cot y}{\tan x+\tan y}}=-{\frac {\cot x-\cot y}{\tan x-\tan y}}} tan x cot y = tan x + cot y cot x + tan y = − tan x − cot y cot x − tan y {\displaystyle \tan x\;\cot y={\frac {\tan x+\cot y}{\cot x+\tan y}}=-{\frac {\tan x-\cot y}{\cot x-\tan y}}} sin x sin y sin z = 1 4 ( sin ( x + y − z ) + sin ( y + z − x ) + sin ( z + x − y ) − sin ( x + y + z ) ) {\displaystyle \sin x\;\sin y\;\sin z={\frac {1}{4}}{\Big (}\sin(x+y-z)+\sin(y+z-x)+\sin(z+x-y)-\sin(x+y+z){\Big )}} cos x cos y cos z = 1 4 ( cos ( x + y − z ) + cos ( y + z − x ) + cos ( z + x − y ) + cos ( x + y + z ) ) {\displaystyle \cos x\;\cos y\;\cos z={\frac {1}{4}}{\Big (}\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)+\cos(x+y+z){\Big )}} sin x sin y cos z = 1 4 ( − cos ( x + y − z ) + cos ( y + z − x ) + cos ( z + x − y ) − cos ( x + y + z ) ) {\displaystyle \sin x\;\sin y\;\cos z={\frac {1}{4}}{\Big (}-\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)-\cos(x+y+z){\Big )}} sin x cos y cos z = 1 4 ( sin ( x + y − z ) − sin ( y + z − x ) + sin ( z + x − y ) + sin ( x + y + z ) ) {\displaystyle \sin x\;\cos y\;\cos z={\frac {1}{4}}{\Big (}\sin(x+y-z)-\sin(y+z-x)+\sin(z+x-y)+\sin(x+y+z){\Big )}} ∏ m = 1 n cos ( x m ) = 1 2 n ∑ k 1 = 1 2 ⋯ ∑ k n = 1 2 [ exp ( i ∑ ν = 1 n ( − 1 ) k ν x ν ) ] = 1 2 n − 1 ∑ k 2 = 1 2 ⋯ ∑ k n = 1 2 [ cos ( x 1 + ∑ ν = 2 n ( − 1 ) k ν x ν ) ] {\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]} ∏ m = 1 n sin ( x m ) = 1 ( 2 i ) n ∑ k 1 = 1 2 ⋯ ∑ k n = 1 2 [ ∏ μ = 1 n ( − 1 ) k μ ⋅ exp ( i ∑ ν = 1 n ( − 1 ) k ν x ν ) ] = 1 2 n − 1 ∑ k 2 = 1 2 ⋯ ∑ k n = 1 2 [ ∏ μ = 2 n ( − 1 ) k μ ⋅ { ( − 1 ) n / 2 ⋅ cos ( x 1 + ∑ ν = 2 n ( − 1 ) k ν x ν ) gerade n ( − 1 ) ( n − 1 ) / 2 ⋅ sin ( x 1 + ∑ ν = 2 n ( − 1 ) k ν x ν ) ungerade n ] {\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]} Aus der Doppelwinkelfunktion für sin ( 2 x ) {\displaystyle \sin(2x)}
sin x cos x = 1 2 sin ( 2 x ) {\displaystyle \sin x\;\cos x={\frac {1}{2}}\sin(2x)}
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}}}
Die folgenden Formeln gelten für beliebige ebene Dreiecke und folgen nach längeren Termumformungen aus α + β + γ = 180 ∘ {\displaystyle \alpha +\beta +\gamma =180^{\circ }}
tan α + tan β + tan γ = tan α tan β tan γ cot α 2 + cot β 2 + cot γ 2 = cot α 2 cot β 2 cot γ 2 {\displaystyle {\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \\\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}&=\cot {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}\end{aligned}}} cot β cot γ + cot γ cot α + cot α cot β = 1 tan β 2 tan γ 2 + tan γ 2 tan α 2 + tan α 2 tan β 2 = 1 {\displaystyle {\begin{aligned}\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta &=1\\\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}+\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}&=1\end{aligned}}} sin α + sin β + sin γ = 4 cos α 2 cos β 2 cos γ 2 − sin α + sin β + sin γ = 4 cos α 2 sin β 2 sin γ 2 {\displaystyle {\begin{aligned}\sin \alpha +\sin \beta +\sin \gamma &=4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}\\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}\end{aligned}}} cos α + cos β + cos γ = 4 sin α 2 sin β 2 sin γ 2 + 1 − cos α + cos β + cos γ = 4 sin α 2 cos β 2 cos γ 2 − 1 {\displaystyle {\begin{aligned}\cos \alpha +\cos \beta +\cos \gamma &=4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1\\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}-1\end{aligned}}} sin 2 α + sin 2 β + sin 2 γ = 4 sin α sin β sin γ − sin 2 α + sin 2 β + sin 2 γ = 4 sin α cos β cos γ {\displaystyle {\begin{aligned}\sin 2\alpha +\sin 2\beta +\sin 2\gamma &=4\sin \alpha \sin \beta \sin \gamma \\-\sin 2\alpha +\sin 2\beta +\sin 2\gamma &=4\sin \alpha \cos \beta \cos \gamma \end{aligned}}} cos 2 α + cos 2 β + cos 2 γ = − 4 cos α cos β cos γ − 1 − cos 2 α + cos 2 β + cos 2 γ = − 4 cos α sin β sin γ + 1 {\displaystyle {\begin{aligned}\cos 2\alpha +\cos 2\beta +\cos 2\gamma &=-4\cos \alpha \cos \beta \cos \gamma -1\\-\cos 2\alpha +\cos 2\beta +\cos 2\gamma &=-4\cos \alpha \sin \beta \sin \gamma +1\end{aligned}}} sin 2 α + sin 2 β + sin 2 γ = 2 cos α cos β cos γ + 2 − sin 2 α + sin 2 β + sin 2 γ = 2 cos α sin β sin γ {\displaystyle {\begin{aligned}\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \end{aligned}}} cos 2 α + cos 2 β + cos 2 γ = − 2 cos α cos β cos γ + 1 − cos 2 α + cos 2 β + cos 2 γ = − 2 cos α sin β sin γ + 1 {\displaystyle {\begin{aligned}\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\end{aligned}}} sin 2 2 α + sin 2 2 β + sin 2 2 γ = − 2 cos 2 α cos 2 β cos 2 γ + 2 − sin 2 2 α + sin 2 2 β + sin 2 2 γ = − 2 cos 2 α sin 2 β sin 2 γ {\displaystyle {\begin{aligned}\sin ^{2}2\alpha +\sin ^{2}2\beta +\sin ^{2}2\gamma &=-2\cos 2\alpha \cos 2\beta \cos 2\gamma +2\\-\sin ^{2}2\alpha +\sin ^{2}2\beta +\sin ^{2}2\gamma &=-2\cos 2\alpha \sin 2\beta \sin 2\gamma \end{aligned}}} cos 2 2 α + cos 2 2 β + cos 2 2 γ = 2 cos 2 α cos 2 β cos 2 γ + 1 − cos 2 2 α + cos 2 2 β + cos 2 2 γ = 2 cos 2 α sin 2 β sin 2 γ + 1 {\displaystyle {\begin{aligned}\cos ^{2}2\alpha +\cos ^{2}2\beta +\cos ^{2}2\gamma &=2\cos 2\alpha \cos 2\beta \cos 2\gamma +1\\-\cos ^{2}2\alpha +\cos ^{2}2\beta +\cos ^{2}2\gamma &=2\cos 2\alpha \sin 2\beta \sin 2\gamma +1\end{aligned}}} sin 2 α 2 + sin 2 β 2 + sin 2 γ 2 = − 2 sin α 2 sin β 2 sin γ 2 + 1 − sin 2 α 2 + sin 2 β 2 + sin 2 γ 2 = − 2 sin α 2 cos β 2 cos γ 2 + 1 {\displaystyle {\begin{aligned}\sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}&=-2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1\\-\sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}&=-2\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}+1\end{aligned}}} cos 2 α 2 + cos 2 β 2 + cos 2 γ 2 = 2 sin α 2 sin β 2 sin γ 2 + 2 − cos 2 α 2 + cos 2 β 2 + cos 2 γ 2 = 2 sin α 2 cos β 2 cos γ 2 {\displaystyle {\begin{aligned}\cos ^{2}{\frac {\alpha }{2}}+\cos ^{2}{\frac {\beta }{2}}+\cos ^{2}{\frac {\gamma }{2}}&=2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+2\\-\cos ^{2}{\frac {\alpha }{2}}+\cos ^{2}{\frac {\beta }{2}}+\cos ^{2}{\frac {\gamma }{2}}&=2\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}\end{aligned}}}
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).}
Ableitungen und Stammfunktionen Siehe Formelsammlung Ableitungen und Stammfunktionen
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}}} tan x = ∑ n = 1 ∞ ( − 1 ) n 2 2 n ( 1 − 2 2 n ) β 2 n ( 2 n ) ! x 2 n − 1 = ∑ n = 1 ∞ ( − 1 ) n − 1 2 2 n ( 2 2 n − 1 ) B 2 n ( 2 n ) ! x 2 n − 1 = x + 1 3 x 3 + 2 15 x 5 + 17 315 x 7 + 62 2835 x 9 + ⋯ | x | < π 2 {\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] cot x = 1 x − ∑ n = 1 ∞ ( − 1 ) n − 1 2 2 n β 2 n ( 2 n ) ! x 2 n − 1 = 1 x − ∑ n = 1 ∞ ( − 1 ) n − 1 2 2 n B 2 n ( 2 n ) ! x 2 n − 1 = 1 x − 1 3 x − 1 45 x 3 − 2 945 x 5 − 1 4725 x 7 − ⋯ , 0 < | x | < π {\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 sin ( x ) = x ∏ k = 1 ∞ ( 1 − x 2 k 2 π 2 ) {\displaystyle \sin(x)=x\prod _{k=1}^{\infty }\left(1-{\frac {x^{2}}{k^{2}\pi ^{2}}}\right)} cos ( x ) = ∏ k = 1 ∞ ( 1 − 4 x 2 ( 2 k − 1 ) 2 π 2 ) {\displaystyle \cos(x)=\prod _{k=1}^{\infty }\left(1-{\frac {4x^{2}}{(2k-1)^{2}\pi ^{2}}}\right)} sin ( x ) = ∏ n = − ∞ ∞ ( x + n π π 2 + n π ) {\displaystyle \sin(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{{\frac {\pi }{2}}+n\pi }}\right)} cos ( x ) = ∏ n = − ∞ ∞ ( x + n π + π 2 π 2 + n π ) {\displaystyle \cos(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{{\frac {\pi }{2}}+n\pi }}\right)} tan ( x ) = ∏ n = − ∞ ∞ ( x + n π x + n π + π 2 ) {\displaystyle \tan(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)} csc ( x ) = ∏ n = − ∞ ∞ ( π 2 + n π x + n π ) {\displaystyle \csc(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi }}\right)} sec ( x ) = ∏ n = − ∞ ∞ ( π 2 + n π x + n π + π 2 ) {\displaystyle \sec(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)} cot ( x ) = ∏ n = − ∞ ∞ ( x + n π + π 2 x + n π ) {\displaystyle \cot(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{x+n\pi }}\right)}
Zusammenhang mit der komplexen Exponentialfunktion Ferner besteht zwischen den Funktionen sin x {\displaystyle \sin x} cos x {\displaystyle \cos x} komplexen Exponentialfunktion exp ( i x ) {\displaystyle \exp(\mathrm {i} x)}
exp ( ± i x ) = cos x ± i sin x = e ± i x {\displaystyle \exp(\pm \mathrm {i} x)=\cos x\pm \mathrm {i} \sin x=e^{\pm \mathrm {i} x}} Eulersche Formel )Weiterhin wird cos x + i sin x =: cis ( x ) {\displaystyle \cos {x}+\mathrm {i} \sin {x}=:\operatorname {cis} (x)} [ 33]
Auf Grund der oben genannten Symmetrien gilt weiter:
cos x = exp ( i x ) + exp ( − i x ) 2 {\displaystyle \cos x={\frac {\exp(\mathrm {i} x)+\exp(-\mathrm {i} x)}{2}}} sin x = exp ( i x ) − exp ( − i x ) 2 i {\displaystyle \sin x={\frac {\exp(\mathrm {i} x)-\exp(-\mathrm {i} x)}{2\mathrm {i} }}} Mit diesen Beziehungen können einige Additionstheoreme besonders einfach und elegant hergeleitet werden.
Eine Formelsammlung für das rechtwinklige und das allgemeine Dreieck auf der Kugeloberfläche findet sich in einem eigenen Kapitel.
Einzelnachweise ↑ Die Wurzel 2006/04+05, 104ff., ohne Beweis↑ Joachim Mohr: Kosinus-, Sinus und Tangenswerte ↑ Ausführliche Beweise in Wikibooks Beweisarchiv. ↑ a b Otto Forster: Analysis 1. Differential- und Integralrechnung einer Veränderlichen. vieweg 1983, Seite 87. ↑ Roger B. Nelsen: Beweise ohne Worte , Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, Springer Spektrum , Springer-Verlag Berlin Heidelberg 2016, ISBN 978-3-662-50330-0 , Seite 44 ↑ I. N. Bronstein, K. A. Semendjajew: Taschenbuch der Mathematik . 19. Auflage, 1979. B.G. Teubner Verlagsgesellschaft, Leipzig. S. 237. ↑ Roger B. Nelsen: Beweise ohne Worte , Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, Springer Spektrum , Springer-Verlag Berlin Heidelberg 2016, ISBN 978-3-662-50330-0 , Seite 46 ↑ Milton Abramowitz and Irene A. Stegun, 22.3.15 ↑ Milton Abramowitz and Irene A. Stegun, 4.3.27 ↑ Milton Abramowitz and Irene A. Stegun, 4.3.29 ↑ I. S. Gradshteyn and I. M. Ryzhik , Table of Integrals, Series, and Products , Academic Press, 5th edition (1994), ISBN 0-12-294755-X 1.333.4 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.331.3 (Bei dieser Formel enthält Gradshteyn/Ryzhik allerdings einen Vorzeichenfehler) ↑ a b c d e f g h i j k l m n o I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik , B. G. Teubner Verlagsgesellschaft Leipzig. 19. Auflage 1979. 2.5.2.1.3 ↑ Milton Abramowitz and Irene A. Stegun, 4.3.28 ↑ Milton Abramowitz and Irene A. Stegun, 4.3.30 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.335.4 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.335.5 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.331.3 ↑ Roger B. Nelsen: Beweise ohne Worte , Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, Springer Spektrum , Springer-Verlag Berlin Heidelberg 2016, ISBN 978-3-662-50330-0 , Seite 49 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.1 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.2 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.3 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.4 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.5 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.1 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.2 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.3 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.4 ↑ I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.5 ↑ Weisstein, Eric W.: Harmonic Addition Theorem. Abgerufen am 20. Januar 2018 (englisch). ↑ Milton Abramowitz and Irene A. Stegun, 4.3.67 ↑ Milton Abramowitz and Irene A. Stegun, 4.3.70 ↑ Herbert Amann, Joachim Escher: Analysis I, Birkhäuser Verlag, Basel 2006, 3. Auflage, S. 292 und 298 
