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Benutzer:NickK~dewikibooks/ Seminaraufgaben2

Seminaraufgaben 2

Trigonometrie

siehe PDF-Datei Seminar2

Goniometrische Gleichungen

a) Alle reellen Lösungen!

${\displaystyle \!\cos {3x}={\frac {2}{5}}\,}$

${\displaystyle \!tan2x=cosx\,}$

${\displaystyle \sin \left({\frac {\pi }{6}}\right)\cdot \cos ^{2}\left(x\right)+{\frac {1}{6}}\cdot \sin ^{2}\left(x\right)={\frac {1}{2}}-{\frac {1}{3}}\cdot \sin ^{2}\left(x\right)}$

${\displaystyle \sin(x)\cdot {\sqrt {\frac {1}{2}}}+\cos \left(x\right)\cdot {\sqrt {\frac {1}{2}}}={\sqrt {\frac {3}{4}}}}$

b) Lösungen in ${\displaystyle [{\frac {5\pi }{4}};{\frac {9\pi }{4}}]}$

${\displaystyle cos\left(2x-{\frac {\pi }{2}}\right)=-{\frac {1}{2}}}$

c) Lösungen in ${\displaystyle [0;2\pi [}$

${\displaystyle cos2x={\frac {1}{cosx}}-1}$

Lineare Gleichungssysteme

Bestimmen Sie alle Lösungen!

a)
${\displaystyle \!1x_{1}+3x_{2}+2x_{3}=24}$
${\displaystyle \!2x_{1}+1x_{2}+5x_{3}=31}$
${\displaystyle \!3x_{1}+4x_{2}+2x_{3}=40}$

b)
${\displaystyle \!1x_{1}+2x_{2}+1x_{3}=1}$
${\displaystyle \!2x_{1}+3x_{2}+1x_{3}=1}$
${\displaystyle \!1x_{1}+3x_{2}+2x_{3}=0}$

c)
${\displaystyle \!1x_{1}+2x_{2}+1x_{3}=1}$
${\displaystyle \!2x_{1}+3x_{2}+1x_{3}=1}$
${\displaystyle \!1x_{1}+3x_{2}+2x_{3}=2}$

d)
${\displaystyle \!1x_{1}+2x_{2}+1x_{3}=1}$
${\displaystyle \!2x_{1}+4x_{2}+2x_{3}=2}$
${\displaystyle \!-x_{1}-2x_{2}-1x_{3}=-1}$