vereinfachen (cos(x)+isin(x))^7

Lösung

vereinfachen

(cos (x) + i sin (x))^{7}

(cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + sin (x) (42 cos^{6} (x) + 21 sin^{4} (x) - 22 sin^{6} (x) - 35 cos^{4} (x)) i

Schritte zur Lösung

(cos (x) + i sin (x))^{7}

Schreibe

(cos (x) + i sin (x))^{7}

um:

cos^{7} (x) + 7 i cos^{6} (x) sin (x) - 21 cos^{5} (x) sin^{2} (x) - 35 i cos^{4} (x) sin^{3} (x) + 35 sin^{4} (x) cos^{3} (x) + 21 i sin^{5} (x) cos^{2} (x) - 7 sin^{6} (x) cos (x) - i sin^{7} (x)

= cos^{7} (x) + 7 i cos^{6} (x) sin (x) - 21 cos^{5} (x) sin^{2} (x) - 35 i cos^{4} (x) sin^{3} (x) + 35 sin^{4} (x) cos^{3} (x) + 21 i sin^{5} (x) cos^{2} (x) - 7 sin^{6} (x) cos (x) - i sin^{7} (x)

Rewrite in standard complex form:

(cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + (7 cos^{6} (x) sin (x) - 35 cos^{4} (x) sin^{3} (x) + 21 sin^{5} (x) cos^{2} (x) - sin^{7} (x)) i

= (cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + (7 cos^{6} (x) sin (x) - 35 cos^{4} (x) sin^{3} (x) + 21 sin^{5} (x) cos^{2} (x) - sin^{7} (x)) i

Vereinfache

7 cos^{6} (x) sin (x) - 35 cos^{4} (x) sin^{3} (x) + 21 sin^{5} (x) cos^{2} (x) - sin^{7} (x) : sin (x) (42 cos^{6} (x) + 21 sin^{4} (x) - 22 sin^{6} (x) - 35 cos^{4} (x))

= (cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + sin (x) (42 cos^{6} (x) + 21 sin^{4} (x) - 22 sin^{6} (x) - 35 cos^{4} (x)) i

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

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

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

e x p an d (3 x^{4} - y) d x + (x - 4 x^{2} y^{3}) d y

s im pl i f y (x - \frac{1}{2}) (x + 1)