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Beliebt Algebra >

vereinfachen (cos(x)+isin(x))^5

Lösung

vereinfachen

(cos (x) + i sin (x))^{5}

Lösung

(cos^{5} (x) - 10 cos^{3} (x) sin^{2} (x) + 5 sin^{4} (x) cos (x)) + (5 cos^{4} (x) sin (x) - 10 sin^{3} (x) cos^{2} (x) + sin^{5} (x)) i

Schritte zur Lösung

(cos (x) + i sin (x))^{5}

Wende Binomialsetz an:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = cos (x), b = i sin (x)

= i = 0 \sum 5 (i 5) cos^{(5 - i)} (x) (i sin (x))^{i}

Erweitere Summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} cos^{5} (x) (i sin (x))^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} cos^{4} (x) (i sin (x))^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} cos^{3} (x) (i sin (x))^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} cos^{2} (x) (i sin (x))^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} cos^{1} (x) (i sin (x))^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} cos^{0} (x) (i sin (x))^{5}

Vereinfache

\frac{5 !}{0 ! ( 5 - 0 ) !} cos^{5} (x) (i sin (x))^{0} : cos^{5} (x)

Vereinfache

\frac{5 !}{1 ! ( 5 - 1 ) !} cos^{4} (x) (i sin (x))^{1} : 5 i cos^{4} (x) sin (x)

Vereinfache

\frac{5 !}{2 ! ( 5 - 2 ) !} cos^{3} (x) (i sin (x))^{2} : 10 i^{2} cos^{3} (x) sin^{2} (x)

Vereinfache

\frac{5 !}{3 ! ( 5 - 3 ) !} cos^{2} (x) (i sin (x))^{3} : 10 i^{3} sin^{3} (x) cos^{2} (x)

Vereinfache

\frac{5 !}{4 ! ( 5 - 4 ) !} cos^{1} (x) (i sin (x))^{4} : 5 i^{4} sin^{4} (x) cos (x)

Vereinfache

\frac{5 !}{5 ! ( 5 - 5 ) !} cos^{0} (x) (i sin (x))^{5} : i^{5} sin^{5} (x)

= cos^{5} (x) + 5 i cos^{4} (x) sin (x) + 10 i^{2} cos^{3} (x) sin^{2} (x) + 10 i^{3} sin^{3} (x) cos^{2} (x) + 5 i^{4} sin^{4} (x) cos (x) + i^{5} sin^{5} (x)

10 i^{2} cos^{3} (x) sin^{2} (x) = - 10 cos^{3} (x) sin^{2} (x)

10 i^{3} sin^{3} (x) cos^{2} (x) = - 10 i sin^{3} (x) cos^{2} (x)

5 i^{4} sin^{4} (x) cos (x) = 5 sin^{4} (x) cos (x)

i^{5} = i

= cos^{5} (x) + 5 i cos^{4} (x) sin (x) - 10 cos^{3} (x) sin^{2} (x) - 10 i sin^{3} (x) cos^{2} (x) + 5 sin^{4} (x) cos (x) + i sin^{5} (x)

Rewrite in standard complex form:

(cos^{5} (x) - 10 cos^{3} (x) sin^{2} (x) + 5 sin^{4} (x) cos (x)) + (5 cos^{4} (x) sin (x) - 10 sin^{3} (x) cos^{2} (x) + sin^{5} (x)) i

= (cos^{5} (x) - 10 cos^{3} (x) sin^{2} (x) + 5 sin^{4} (x) cos (x)) + (5 cos^{4} (x) sin (x) - 10 sin^{3} (x) cos^{2} (x) + sin^{5} (x)) i

Beliebte Beispiele

vereinfachen (x+yi)^4

s im pl i f y (x + y i)^{4}

vereinfachen (sqrt(3)+i)^7

s im pl i f y (3 + i)^{7}

vereinfachen (x+sqrt(2))^2

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

vereinfachen (3+3i)^5

s im pl i f y (3 + 3 i)^{5}

erweitern (1-sin(θ))^2

e x p an d (1 - sin (θ))^{2}