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Beliebt Trigonometrie >

beweisen 4cos^3(a)-3cos(a)=cos(3a)

Lösung

beweisen

4 cos^{3} (a) - 3 cos (a) = cos (3 a)

Lösung

Wahr

Schritte zur Lösung

4 cos^{3} (a) - 3 cos (a) = cos (3 a)

Manipuliere die rechte Seite

cos (3 a)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (3 a)

Schreibe um

= cos (2 a + a)

Benutze die Identität der Winkelsumme:

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

= cos (2 a) cos (a) - sin (2 a) sin (a)

Verwende die Doppelwinkelidentität:

sin (2 a) = 2 sin (a) cos (a)

= cos (2 a) cos (a) - 2 sin (a) cos (a) sin (a)

Vereinfache

cos (2 a) cos (a) - 2 sin (a) cos (a) sin (a) : cos (a) cos (2 a) - 2 sin^{2} (a) cos (a)

cos (2 a) cos (a) - 2 sin (a) cos (a) sin (a)

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

2 sin (a) cos (a) sin (a)

Wende Exponentenregel an:

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

sin (a) sin (a) = sin^{1 + 1} (a)

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

Addiere die Zahlen:

1 + 1 = 2

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

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

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

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

Verwende die Doppelwinkelidentität:

cos (2 a) = 2 cos^{2} (a) - 1

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

Verwende die Pythagoreische Identität:

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

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

= (2 cos^{2} (a) - 1) cos (a) - 2 (1 - cos^{2} (a)) cos (a)

Multipliziere aus

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

(2 cos^{2} (a) - 1) cos (a) - 2 (1 - cos^{2} (a)) cos (a)

= cos (a) (2 cos^{2} (a) - 1) - 2 cos (a) (1 - cos^{2} (a))

Multipliziere aus

cos (a) (2 cos^{2} (a) - 1) : 2 cos^{3} (a) - cos (a)

cos (a) (2 cos^{2} (a) - 1)

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

= cos (a) 2 cos^{2} (a) - cos (a) 1

= 2 cos^{2} (a) cos (a) - 1 cos (a)

Vereinfache

2 cos^{2} (a) cos (a) - 1 \cdot cos (a) : 2 cos^{3} (a) - cos (a)

2 cos^{2} (a) cos (a) - 1 cos (a)

2 cos^{2} (a) cos (a) = 2 cos^{3} (a)

2 cos^{2} (a) cos (a)

Wende Exponentenregel an:

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

cos^{2} (a) cos (a) = cos^{2 + 1} (a)

= 2 cos^{2 + 1} (a)

Addiere die Zahlen:

2 + 1 = 3

= 2 cos^{3} (a)

1 \cdot cos (a) = cos (a)

1 cos (a)

Multipliziere:

1 \cdot cos (a) = cos (a)

= cos (a)

= 2 cos^{3} (a) - cos (a)

= 2 cos^{3} (a) - cos (a)

= 2 cos^{3} (a) - cos (a) - 2 (1 - cos^{2} (a)) cos (a)

Multipliziere aus

- 2 cos (a) (1 - cos^{2} (a)) : - 2 cos (a) + 2 cos^{3} (a)

- 2 cos (a) (1 - cos^{2} (a))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

= - 2 cos (a) 1 - (- 2 cos (a)) cos^{2} (a)

Wende Minus-Plus Regeln an

- (- a) = a

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

Vereinfache

- 2 \cdot 1 \cdot cos (a) + 2 cos^{2} (a) cos (a) : - 2 cos (a) + 2 cos^{3} (a)

- 2 \cdot 1 cos (a) + 2 cos^{2} (a) cos (a)

2 \cdot 1 \cdot cos (a) = 2 cos (a)

2 \cdot 1 cos (a)

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 cos (a)

2 cos^{2} (a) cos (a) = 2 cos^{3} (a)

2 cos^{2} (a) cos (a)

Wende Exponentenregel an:

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

cos^{2} (a) cos (a) = cos^{2 + 1} (a)

= 2 cos^{2 + 1} (a)

Addiere die Zahlen:

2 + 1 = 3

= 2 cos^{3} (a)

= - 2 cos (a) + 2 cos^{3} (a)

= - 2 cos (a) + 2 cos^{3} (a)

= 2 cos^{3} (a) - cos (a) - 2 cos (a) + 2 cos^{3} (a)

Vereinfache

2 cos^{3} (a) - cos (a) - 2 cos (a) + 2 cos^{3} (a) : 4 cos^{3} (a) - 3 cos (a)

2 cos^{3} (a) - cos (a) - 2 cos (a) + 2 cos^{3} (a)

Fasse gleiche Terme zusammen

= 2 cos^{3} (a) + 2 cos^{3} (a) - cos (a) - 2 cos (a)

Addiere gleiche Elemente:

2 cos^{3} (a) + 2 cos^{3} (a) = 4 cos^{3} (a)

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

Addiere gleiche Elemente:

- cos (a) - 2 cos (a) = - 3 cos (a)

= 4 cos^{3} (a) - 3 cos (a)

= 4 cos^{3} (a) - 3 cos (a)

= 4 cos^{3} (a) - 3 cos (a)

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

\Rightarrow Wahr

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