erweitern (cos(2pix)+isin(2pix))^3

Lösung

erweitern

(cos (2 π x) + i sin (2 π x))^{3}

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

Schritte zur Lösung

(cos (2 π x) + i sin (2 π x))^{3}

Wende Formel für perfekte dritte Potenzen an:

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

a = cos (2 π x), b = i sin (2 π x)

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

Vereinfache

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

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

Schreibe

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

in der Standard komplexen Form um:

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

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

e x p an d (e + s + k - 2 c) (e + s + k - 2 c)

e x p an d \frac{tan ( x ) ( 3 - tan ^{2} ( x ) )}{1 - 3 tan ^{2} ( x )}

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

e x p an d - \frac{x ^{2} - 6 x + 21}{( x ^{2} - 3 ) ( x + 4 )}

e x p an d \frac{252037 ( x ^{2} - 46 x + 577 )}{365 ( x - 23 ) ^{2}}