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Beliebt Trigonometrie >

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

Lösung

beweisen

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

Lösung

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

cos (3 t)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (3 t)

Schreibe um

= cos (2 t + t)

Benutze die Identität der Winkelsumme:

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

= cos (2 t) cos (t) - sin (2 t) sin (t)

Verwende die Doppelwinkelidentität:

sin (2 t) = 2 sin (t) cos (t)

= cos (2 t) cos (t) - 2 sin (t) cos (t) sin (t)

Vereinfache

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

cos (2 t) cos (t) - 2 sin (t) cos (t) sin (t)

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

2 sin (t) cos (t) sin (t)

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

1 + 1 = 2

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

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

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

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

Verwende die Doppelwinkelidentität:

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

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

Verwende die Pythagoreische Identität:

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

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

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

Multipliziere aus

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

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

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

Vereinfache

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

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

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

2 cos^{2} (t) cos (t)

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

2 + 1 = 3

= 2 cos^{3} (t)

1 \cdot cos (t) = cos (t)

1 cos (t)

Multipliziere:

1 \cdot cos (t) = cos (t)

= cos (t)

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

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

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

Wende Minus-Plus Regeln an

- (- a) = a

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

Vereinfache

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

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

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

2 \cdot 1 cos (t)

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 cos (t)

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

2 cos^{2} (t) cos (t)

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

2 + 1 = 3

= 2 cos^{3} (t)

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

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

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

Vereinfache

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

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

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

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

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

Addiere gleiche Elemente:

- cos (t) - 2 cos (t) = - 3 cos (t)

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

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

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

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

\Rightarrow Wahr

Beliebte Beispiele

beweisen sec(2x)=(sec^2(x)+sec^4(x))/(2+sec^2(x)-sec^4(x))

p ro v e sec (2 x) = \frac{sec ^{2} ( x ) + sec ^{4} ( x )}{2 + sec ^{2} ( x ) - sec ^{4} ( x )}

beweisen 1/(csc(x))= 1/(sin(x))

p ro v e \frac{1}{csc ( x )} = \frac{1}{sin ( x )}

beweisen 1+tan^2(A)=sec^3(A)cos(A)

p ro v e 1 + tan^{2} (A) = sec^{3} (A) cos (A)

beweisen sec(0)-cos(0)=tan(0)sin(0)

p ro v e sec (0) - cos (0) = tan (0) sin (0)

beweisen 1-sin(θ)cos(θ)tan(θ)=cos^2(θ)

p ro v e 1 - sin (θ) cos (θ) tan (θ) = cos^{2} (θ)