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Beliebt Trigonometrie >

beweisen sin(3x)=(sin(x))(4cos^2(x)-1)

Lösung

beweisen

sin (3 x) = (sin (x)) (4 cos^{2} (x) - 1)

Lösung

Wahr

Schritte zur Lösung

sin (3 x) = sin (x) (4 cos^{2} (x) - 1)

Manipuliere die linke Seite

sin (3 x)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

sin (3 x)

Verwende die folgenden Identitäten:

sin (3 x) = - sin^{3} (x) + 3 cos^{2} (x) sin (x)

sin (3 x)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

sin (3 x)

Schreibe um

= sin (2 x + x)

Benutze die Identität der Winkelsumme:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (2 x) cos (x) + cos (2 x) sin (x)

Verwende die Doppelwinkelidentität:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) sin (x) + cos (x) 2 sin (x) cos (x)

Verwende die Doppelwinkelidentität:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

= (cos^{2} (x) - sin^{2} (x)) sin (x) + 2 cos (x) cos (x) sin (x)

= (cos^{2} (x) - sin^{2} (x)) sin (x) + 2 cos (x) cos (x) sin (x)

Multipliziere aus

(cos^{2} (x) - sin^{2} (x)) sin (x) + 2 cos (x) cos (x) sin (x) : - sin^{3} (x) + 3 cos^{2} (x) sin (x)

(cos^{2} (x) - sin^{2} (x)) sin (x) + 2 cos (x) cos (x) sin (x)

2 cos (x) cos (x) sin (x) = 2 cos^{2} (x) sin (x)

2 cos (x) cos (x) sin (x)

Wende Exponentenregel an:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= 2 cos^{1 + 1} (x) sin (x)

Addiere die Zahlen:

1 + 1 = 2

= 2 cos^{2} (x) sin (x)

= sin (x) (cos^{2} (x) - sin^{2} (x)) + 2 cos^{2} (x) sin (x)

= sin (x) (cos^{2} (x) - sin^{2} (x)) + 2 cos^{2} (x) sin (x)

Multipliziere aus

sin (x) (cos^{2} (x) - sin^{2} (x)) : cos^{2} (x) sin (x) - sin^{3} (x)

sin (x) (cos^{2} (x) - sin^{2} (x))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = sin (x), b = cos^{2} (x), c = sin^{2} (x)

= sin (x) cos^{2} (x) - sin (x) sin^{2} (x)

= cos^{2} (x) sin (x) - sin^{2} (x) sin (x)

sin^{2} (x) sin (x) = sin^{3} (x)

sin^{2} (x) sin (x)

Wende Exponentenregel an:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= sin^{2 + 1} (x)

Addiere die Zahlen:

2 + 1 = 3

= sin^{3} (x)

= cos^{2} (x) sin (x) - sin^{3} (x)

= cos^{2} (x) sin (x) - sin^{3} (x) + 2 cos^{2} (x) sin (x)

Vereinfache

cos^{2} (x) sin (x) - sin^{3} (x) + 2 cos^{2} (x) sin (x) : - sin^{3} (x) + 3 cos^{2} (x) sin (x)

cos^{2} (x) sin (x) - sin^{3} (x) + 2 cos^{2} (x) sin (x)

Fasse gleiche Terme zusammen

= - sin^{3} (x) + cos^{2} (x) sin (x) + 2 cos^{2} (x) sin (x)

Addiere gleiche Elemente:

cos^{2} (x) sin (x) + 2 cos^{2} (x) sin (x) = 3 cos^{2} (x) sin (x)

= - sin^{3} (x) + 3 cos^{2} (x) sin (x)

= - sin^{3} (x) + 3 cos^{2} (x) sin (x)

= - sin^{3} (x) + 3 cos^{2} (x) sin (x)

= - sin^{3} (x) + 3 cos^{2} (x) sin (x)

= - sin^{3} (x) + 3 cos^{2} (x) sin (x)

Faktorisiere

- sin^{3} (x) + 3 cos^{2} (x) sin (x) : sin (x) (- sin^{2} (x) + 3 cos^{2} (x))

- sin^{3} (x) + 3 cos^{2} (x) sin (x)

Wende Exponentenregel an:

a^{b + c} = a^{b} a^{c}

sin^{3} (x) = sin (x) sin^{2} (x)

= - sin (x) sin^{2} (x) + 3 cos^{2} (x) sin (x)

Klammere gleiche Terme aus

sin (x)

= sin (x) (- sin^{2} (x) + 3 cos^{2} (x))

= (- sin^{2} (x) + 3 cos^{2} (x)) sin (x)

Manipuliere die rechte Seite

sin (x) (4 cos^{2} (x) - 1)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

sin (x) (4 cos^{2} (x) - 1)

Verwende die Pythagoreische Identität:

1 = cos^{2} (x) + sin^{2} (x)

= sin (x) (4 cos^{2} (x) - cos^{2} (x) + sin^{2} (x))

Addiere gleiche Elemente:

4 cos^{2} (x) - cos^{2} (x) = 3 cos^{2} (x)

= sin (x) (3 cos^{2} (x) + sin^{2} (x))

= sin (x) (3 cos^{2} (x) + sin^{2} (x))

Wir haben gezeigt, dass beide Seiten die gleiche Form annehmen können

\Rightarrow Wahr

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