vereinfachen (cos(θ)+isin(θ))^3

Lösung

vereinfachen

(cos (θ) + i sin (θ))^{3}

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

Schritte zur Lösung

(cos (θ) + i sin (θ))^{3}

Wende Formel für perfekte dritte Potenzen an:

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

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

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

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

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

(i sin (θ))^{3} = - sin^{3} (θ) i

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

Rewrite in standard complex form:

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

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

s im pl i f y (x - 2 i)^{2}

s im pl i f y (x - 1) (x + 1)

e x p an d (sin (x) + x cos (x))^{2}

e x p an d \frac{d ^{2}}{d x ^{2}} (x^{4}) - 2 x (6 x - x^{2})

e x p an d \frac{6 x ^{2} - 9 x + 1}{( x - 1 ) ^{2} ( x + 1 )}