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beweisen cos(3t)=cos^3(t)-3sin^2(t)cos(t)

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

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

Multipliziere aus

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

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

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

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

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)

= cos^{2 + 1} (t)

Addiere die Zahlen:

2 + 1 = 3

= cos^{3} (t)

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

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

Addiere gleiche Elemente:

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

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

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

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

\Rightarrow Wahr

p ro v e 4 sin (x) cos (x) + 1 = 2 sin (2 x) + 1

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

p ro v e cot^{6} (x) = cot^{4} (x) csc^{2} (x) - cot^{4} (x)

p ro v e 1 + csc (a) = csc (a) (1 + sin (a))

p ro v e \frac{1 + cos ( 2 x )}{sin ^{2} ( x )} = 2 cot^{2} (x)