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}}
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}}} 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 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} 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)} 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}}} 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} } 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 }[}
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 }
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]
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}}} 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)} 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}}.}
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]
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