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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) = 3 sin (x) - 4 sin^{3} (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)

Vereinfache

cos (2 x) sin (x) + cos (x) \cdot 2 sin (x) cos (x) : sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

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

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

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

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

1 + 1 = 2

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

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

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

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

Verwende die Doppelwinkelidentität:

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

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

Verwende die Pythagoreische Identität:

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

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

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

Multipliziere aus

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x) : - 4 sin^{3} (x) + 3 sin (x)

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

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

Multipliziere aus

sin (x) (1 - 2 sin^{2} (x)) : sin (x) - 2 sin^{3} (x)

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = sin (x), b = 1, c = 2 sin^{2} (x)

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

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

Vereinfache

1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : sin (x) - 2 sin^{3} (x)

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

1 \cdot sin (x) = sin (x)

1 sin (x)

Multipliziere:

1 \cdot sin (x) = sin (x)

= sin (x)

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

2 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)

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

Addiere die Zahlen:

2 + 1 = 3

= 2 sin^{3} (x)

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

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

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

Multipliziere aus

2 sin (x) (1 - sin^{2} (x)) : 2 sin (x) - 2 sin^{3} (x)

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = 2 sin (x), b = 1, c = sin^{2} (x)

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

= 2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

Vereinfache

2 \cdot 1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : 2 sin (x) - 2 sin^{3} (x)

2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 sin (x)

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 sin (x)

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

2 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)

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

Addiere die Zahlen:

2 + 1 = 3

= 2 sin^{3} (x)

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

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

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

Vereinfache

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x) : - 4 sin^{3} (x) + 3 sin (x)

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

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

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

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

Addiere gleiche Elemente:

sin (x) + 2 sin (x) = 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= 3 sin (x) - 4 sin^{3} (x)

= 3 sin (x) - 4 sin^{3} (x)

= 3 sin (x) - 4 sin^{3} (x)

Faktorisiere

3 sin (x) - 4 sin^{3} (x) : sin (x) (3 - 4 sin^{2} (x))

3 sin (x) - 4 sin^{3} (x)

Wende Exponentenregel an:

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

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

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

Klammere gleiche Terme aus

sin (x)

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

(3 - 4 sin^{2} (x)) sin (x)

Verwende die Pythagoreische Identität:

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

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

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

Multipliziere aus

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

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

Multipliziere aus

- 4 (1 - cos^{2} (x)) : - 4 + 4 cos^{2} (x)

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = - 4, b = 1, c = cos^{2} (x)

= - 4 \cdot 1 - (- 4) cos^{2} (x)

Wende Minus-Plus Regeln an

- (- a) = a

= - 4 \cdot 1 + 4 cos^{2} (x)

Multipliziere die Zahlen:

4 \cdot 1 = 4

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

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

Subtrahiere die Zahlen:

3 - 4 = - 1

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

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

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

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

\Rightarrow Wahr

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