Formelsammlung Mathematik: Trigonometrie

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Formeln aus der Trigonometrie der Ebene.

Allgemeingültige Formeln befinden sich in den Abschnitten Winkelfunktionen und Arkusfunktionen.

Es werden die folgenden Bezeichnungen verwendet: Das Dreieck ABC habe die Seiten ${\displaystyle a={\overline {BC}}\,,\,b={\overline {CA}}\,,\,c={\overline {AB}}}$, die Winkel ${\displaystyle \alpha ,\beta ,\gamma \,}$ bei den Ecken A, B und C. Seien ${\displaystyle s_{a},s_{b},s_{c}}$ die Seitenhalbierenden, ${\displaystyle w_{a},w_{b},w_{c}}$ die Winkelhalbierenden, ${\displaystyle h_{a},h_{b},h_{c}}$ die Höhen, R der Umkreisradius, ${\displaystyle \rho }$ der Inkreisradius und ${\displaystyle \rho _{a},\rho _{b},\rho _{c}}$ die Ankreisradien (und zwar die Radien der Ankreise, die den Ecken A, B bzw. C gegenüberliegen) des Dreiecks ABC. Die Variable s steht für den halben Umfang des Dreiecks: ${\displaystyle s=(a+b+c)/2}$. Schließlich wird die Fläche des Dreiecks ABC mit A bezeichnet.

${\displaystyle \alpha +\beta +\gamma =\pi \quad (=180^{\circ })}$.

${\displaystyle {\frac {a}{b}}={\frac {\sin \alpha }{\sin \beta }}\quad ,\quad {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R\quad ,\quad a:b:c=\sin \alpha :\sin \beta :\sin \gamma \,}$ (Verhältnisgleichung)

Siehe auch: Sinussatz

${\displaystyle a^{2}=b^{2}+c^{2}-2bc\ \cos \alpha \quad ,\quad \cos \alpha ={\frac {b^{2}+c^{2}-a^{2}}{2bc}}\quad ,\quad a^{2}+bc\,\cos \alpha ={\frac {a^{2}+b^{2}+c^{2}}{2}}}$

Siehe auch: Kosinussatz

${\displaystyle a=b\;\cos \gamma +c\;\cos \beta }$

${\displaystyle {\frac {b+c}{a}}={\frac {\cos {\frac {\beta -\gamma }{2}}}{\sin {\frac {\alpha }{2}}}}\quad ,\quad {\frac {b-c}{a}}={\frac {\sin {\frac {\beta -\gamma }{2}}}{\cos {\frac {\alpha }{2}}}}}$

${\displaystyle {\frac {a+b}{a-b}}={\frac {\tan {\frac {\alpha +\beta }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}={\frac {\cot {\frac {\gamma }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}}$

Siehe auch: Tangenssatz

${\displaystyle s-a={\frac {b+c-a}{2}}\quad ,\quad \left(s-b\right)+\left(s-c\right)=a\quad ,\quad \left(s-a\right)+\left(s-b\right)+\left(s-c\right)=s}$ ${\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{bc}}}\quad ,\quad \cos {\frac {\alpha }{2}}={\sqrt {\frac {s\left(s-a\right)}{bc}}}\quad ,\quad \tan {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{s\left(s-a\right)}}}}$ ${\displaystyle s=4R\cos \,{\frac {\alpha }{2}}\,\cos {\frac {\beta }{2}}\,\cos {\frac {\gamma }{2}}\quad ,\quad s-a=4R\,\cos {\frac {\alpha }{2}}\,\sin {\frac {\beta }{2}}\,\sin {\frac {\gamma }{2}}}$

Heronsche Formel: ${\displaystyle A={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}={\frac {1}{4}}{\sqrt {\left(a+b+c\right)\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}}}$ ${\displaystyle A={\frac {1}{4}}{\sqrt {2\left(b^{2}c^{2}+c^{2}a^{2}+a^{2}b^{2}\right)-\left(a^{4}+b^{4}+c^{4}\right)}}}$

${\displaystyle A={\frac {1}{2}}bc\sin \alpha }$

${\displaystyle A={\frac {1}{2}}ah_{a}}$, wobei h_a die Höhe auf der Seite BC ist.

${\displaystyle A=2R^{2}\sin \alpha \sin \beta \sin \gamma \,}$

${\displaystyle A={\frac {abc}{4R}}}$

${\displaystyle A=\rho \,s=\rho _{a}\left(s-a\right)}$

${\displaystyle A={\sqrt {\rho \rho _{a}\rho _{b}\rho _{c}}}}$

${\displaystyle a=2R\sin \alpha \,}$

${\displaystyle R={\frac {abc}{4A}}}$

${\displaystyle \rho =\left(s-a\right)\tan {\frac {\alpha }{2}}=\left(s-b\right)\tan {\frac {\beta }{2}}=\left(s-c\right)\tan {\frac {\gamma }{2}}}$

${\displaystyle \rho =4R\,\sin {\frac {\alpha }{2}}\,\sin {\frac {\beta }{2}}\,\sin {\frac {\gamma }{2}}=s\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}}$

${\displaystyle \rho =R\left(\cos \alpha +\cos \beta +\cos \gamma -1\right)}$

${\displaystyle \rho ={\frac {A}{s}}={\frac {abc}{4Rs}}}$

${\displaystyle \rho ={\sqrt {\frac {\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}}={\frac {1}{2}}{\sqrt {\frac {\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}{a+b+c}}}}$

${\displaystyle \rho ={\frac {a}{\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}}}}$

Chapple-Euler-Ungleichung: ${\displaystyle 2\rho \leq R}$; Gleichheit tritt nur dann ein, wenn das Dreieck ABC gleichseitig ist.

${\displaystyle \rho _{a}=s\tan {\frac {\alpha }{2}}}$

${\displaystyle \rho _{a}=4R\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}=\left(s-a\right)\tan {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}}$

${\displaystyle \rho _{a}=R\left(-\cos \alpha +\cos \beta +\cos \gamma +1\right)}$

${\displaystyle \rho _{a}={\frac {A}{s-a}}={\frac {abc}{4R\left(s-a\right)}}}$

${\displaystyle \rho _{a}={\sqrt {\frac {s\left(s-b\right)\left(s-c\right)}{s-a}}}={\frac {1}{2}}{\sqrt {\frac {\left(a+b+c\right)\left(c+a-b\right)\left(a+b-c\right)}{b+c-a}}}}$

${\displaystyle {\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}$

${\displaystyle h_{a}=b\sin \gamma =c\sin \beta ={\frac {2A}{a}}=2R\sin \beta \sin \gamma }$

${\displaystyle h_{a}={\frac {a}{\cot \beta +\cot \gamma }}}$

${\displaystyle {\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}={\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}$

Ist ${\displaystyle \alpha ={\frac {\pi }{2}}\;(=90^{\circ })}$ dann gilt ${\displaystyle h_{a}={\frac {b\,c}{a}}}$

${\displaystyle s_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\frac {1}{2}}{\sqrt {b^{2}+c^{2}+2bc\cos \alpha }}={\sqrt {{\frac {a^{2}}{4}}+bc\cos \alpha }}}$

${\displaystyle s_{a}^{2}+s_{b}^{2}+s_{c}^{2}={\frac {3}{4}}\left(a^{2}+b^{2}+c^{2}\right)}$

${\displaystyle w_{\alpha }={\frac {2bc\cos {\frac {\alpha }{2}}}{b+c}}={\frac {2A}{a\cos {\frac {\beta -\gamma }{2}}}}}$

Die folgenden Formeln folgen nach längeren Termumformungen aus α + β + γ = 180°, gelten also allgemein für drei beliebige Winkel α, β und γ mit der Eigenschaft α + β + γ = 180°, solange die in den Formeln vorkommenden Funktionen wohldefiniert sind (letzteres betrifft nur die Formeln, in denen Tangens und Kotangens vorkommen).

${\displaystyle \tan \alpha +\tan \beta +\tan \gamma =\tan \alpha \cdot \tan \beta \cdot \tan \gamma \,}$

${\displaystyle \cot \beta \cdot \cot \gamma +\cot \gamma \cdot \cot \alpha +\cot \alpha \cdot \cot \beta =1}$

${\displaystyle \cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}=\cot {\frac {\alpha }{2}}\cdot \cot {\frac {\beta }{2}}\cdot \cot {\frac {\gamma }{2}}}$

${\displaystyle \tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}+\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}=1}$

${\displaystyle \sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}}$

${\displaystyle -\sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}}$

${\displaystyle \cos \alpha +\cos \beta +\cos \gamma =4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1}$

${\displaystyle -\cos \alpha +\cos \beta +\cos \gamma =4\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}-1}$

${\displaystyle \sin \left(2\alpha \right)+\sin \left(2\beta \right)+\sin \left(2\gamma \right)=4\sin \alpha \sin \beta \sin \gamma }$

${\displaystyle -\sin \left(2\alpha \right)+\sin \left(2\beta \right)+\sin \left(2\gamma \right)=4\sin \alpha \cos \beta \cos \gamma }$

${\displaystyle \cos \left(2\alpha \right)+\cos \left(2\beta \right)+\cos \left(2\gamma \right)=-4\cos \alpha \cos \beta \cos \gamma -1}$

${\displaystyle -\cos \left(2\alpha \right)+\cos \left(2\beta \right)+\cos \left(2\gamma \right)=-4\cos \alpha \sin \beta \sin \gamma +1}$

${\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2\cdot \cos \alpha \cdot \cos \beta \cdot \cos \gamma +2}$

${\displaystyle -\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2\cdot \cos \alpha \cdot \sin \beta \cdot \sin \gamma }$

${\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =-2\cdot \cos \alpha \cdot \cos \beta \cdot \cos \gamma +1}$

${\displaystyle -\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =-2\cdot \cos \alpha \cdot \sin \beta \cdot \sin \gamma +1}$

${\displaystyle -\sin ^{2}\left(2\alpha \right)+\sin ^{2}\left(2\beta \right)+\sin ^{2}\left(2\gamma \right)=-2\cos \left(2\alpha \right)\sin \left(2\beta \right)\sin \left(2\gamma \right)}$

${\displaystyle -\cos ^{2}\left(2\alpha \right)+\cos ^{2}\left(2\beta \right)+\cos ^{2}\left(2\gamma \right)=2\cos \left(2\alpha \right)\sin \left(2\beta \right)\sin \left(2\gamma \right)+1}$