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Beliebt Algebra >

vereinfachen (cos(x)+isin(x))^3

Lösung

vereinfachen

(cos (x) + i sin (x))^{3}

Lösung

(4 cos^{3} (x) - 3 cos (x)) + (3 sin (x) - 4 sin^{3} (x)) i

Schritte zur Lösung

(cos (x) + i sin (x))^{3}

Wende Formel für perfekte dritte Potenzen an:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

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

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

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

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

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

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

Rewrite in standard complex form:

(4 cos^{3} (x) - 3 cos (x)) + (3 sin (x) - 4 sin^{3} (x)) i

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

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