Trigonometrie Beispiele

cos (6 x)

$\cos (6 x)$

Schritt 1

Eine gute Methode

cos (6 x)

$\cos (6 x)$ zu entwickeln, ist die Anwendung des Satzes von de Moivre

(r {(cos (x) + i \cdot sin (x))}^{n} = r^{n} (cos (n x) + i \cdot sin (n x)))

$(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . Wenn

r = 1

$r = 1$ ,

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$

Schritt 2

Multipliziere die rechte Seite der Gleichung

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ durch Anwenden des binomischen Lehrsatzes aus.

Multipliziere aus:

{(cos (x) + i \cdot sin (x))}^{6}

${(\cos (x) + i \cdot \sin (x))}^{6}$

Schritt 3

Wende den binomischen Lehrsatz an.

{cos}^{6} (x) + 6 {cos}^{5} (x) (i sin (x)) + 15 {cos}^{4} (x) {(i sin (x))}^{2} + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) (i \sin (x)) + 15 \cos^{4} (x) {(i \sin (x))}^{2} + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4

Vereinfache Terme.

Tippen, um mehr Schritte zu sehen ...

Schritt 4.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 4.1.1

Wende die Produktregel auf

i sin (x)

$i \sin (x)$ an.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) + 15 {cos}^{4} (x) (i^{2} {sin}^{2} (x)) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cos^{4} (x) (i^{2} \sin^{2} (x)) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.2

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) + 15 \cdot i^{2} {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cdot i^{2} \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.3

Schreibe

i^{2}

$i^{2}$ als

- 1

$- 1$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) + 15 \cdot - 1 {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cdot - 1 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.4

Mutltipliziere

15

$15$ mit

- 1

$- 1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.5

Wende die Produktregel auf

i sin (x)

$i \sin (x)$ an.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) (i^{3} {sin}^{3} (x)) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) (i^{3} \sin^{3} (x)) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.6

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot i^{3} {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot i^{3} \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.7

Faktorisiere

i^{2}

$i^{2}$ aus.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot (i^{2} \cdot i) {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (i^{2} \cdot i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.8

Schreibe

i^{2}

$i^{2}$ als

- 1

$- 1$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot (- 1 \cdot i) {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (- 1 \cdot i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.9

Schreibe

- 1 i

$- 1 i$ als

- i

$- i$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot (- i) {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (- i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.10

Mutltipliziere

- 1

$- 1$ mit

20

$20$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.11

Wende die Produktregel auf

i sin (x)

$i \sin (x)$ an.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) (i^{4} {sin}^{4} (x)) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) (i^{4} \sin^{4} (x)) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.12

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot i^{4} {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot i^{4} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.13

Schreibe

i^{4}

$i^{4}$ als

1

$1$ um.

Tippen, um mehr Schritte zu sehen ...

Schritt 4.1.13.1

Schreibe

i^{4}

$i^{4}$ als

{(i^{2})}^{2}

${(i^{2})}^{2}$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot {(i^{2})}^{2} {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot {(i^{2})}^{2} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.13.2

Schreibe

i^{2}

$i^{2}$ als

- 1

$- 1$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot {(- 1)}^{2} {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot {(- 1)}^{2} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.13.3

Potenziere

- 1

$- 1$ mit

2

$2$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot 1 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot 1 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot 1 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot 1 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.14

Mutltipliziere

15

$15$ mit

1

$1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Schritt 4.1.15

Wende die Produktregel auf

i sin (x)

$i \sin (x)$ an.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (i^{5} {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i^{5} \sin^{5} (x)) + {(i \sin (x))}^{6}$

Schritt 4.1.16

Faktorisiere

i^{4}

$i^{4}$ aus.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (i^{4} i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i^{4} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Schritt 4.1.17

Schreibe

i^{4}

$i^{4}$ als

1

$1$ um.

Tippen, um mehr Schritte zu sehen ...

Schritt 4.1.17.1

Schreibe

i^{4}

$i^{4}$ als

{(i^{2})}^{2}

${(i^{2})}^{2}$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) ({(i^{2})}^{2} i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) ({(i^{2})}^{2} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Schritt 4.1.17.2

Schreibe

i^{2}

$i^{2}$ als

- 1

$- 1$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) ({(- 1)}^{2} i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) ({(- 1)}^{2} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Schritt 4.1.17.3

Potenziere

- 1

$- 1$ mit

2

$2$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (1 i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (1 i \sin^{5} (x)) + {(i \sin (x))}^{6}$

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (1 i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (1 i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Schritt 4.1.18

Mutltipliziere

i

$i$ mit

1

$1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Schritt 4.1.19

Wende die Produktregel auf

i sin (x)

$i \sin (x)$ an.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + i^{6} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{6} \sin^{6} (x)$

Schritt 4.1.20

Faktorisiere

i^{4}

$i^{4}$ aus.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + i^{4} i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{4} i^{2} \sin^{6} (x)$

Schritt 4.1.21

Schreibe

i^{4}

$i^{4}$ als

1

$1$ um.

Tippen, um mehr Schritte zu sehen ...

Schritt 4.1.21.1

Schreibe

i^{4}

$i^{4}$ als

{(i^{2})}^{2}

${(i^{2})}^{2}$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + {(i^{2})}^{2} i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + {(i^{2})}^{2} i^{2} \sin^{6} (x)$

Schritt 4.1.21.2

Schreibe

i^{2}

$i^{2}$ als

- 1

$- 1$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + {(- 1)}^{2} i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + {(- 1)}^{2} i^{2} \sin^{6} (x)$

Schritt 4.1.21.3

Potenziere

- 1

$- 1$ mit

2

$2$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + 1 i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + 1 i^{2} \sin^{6} (x)$

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + 1 i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + 1 i^{2} \sin^{6} (x)$

Schritt 4.1.22

Mutltipliziere

i^{2}

$i^{2}$ mit

1

$1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{2} \sin^{6} (x)$

Schritt 4.1.23

Schreibe

i^{2}

$i^{2}$ als

- 1

$- 1$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) - 1 {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - 1 \sin^{6} (x)$

Schritt 4.1.24

Schreibe

- 1 {sin}^{6} (x)

$- 1 \sin^{6} (x)$ als

- {sin}^{6} (x)

$- \sin^{6} (x)$ um.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) - {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - \sin^{6} (x)$

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) - {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - \sin^{6} (x)$

Schritt 4.2

Stelle die Faktoren in

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) - {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - \sin^{6} (x)$ um.

{cos}^{6} (x) + 6 i {cos}^{5} (x) sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 i cos (x) {sin}^{5} (x) - {sin}^{6} (x)

$\cos^{6} (x) + 6 i \cos^{5} (x) \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 i \cos (x) \sin^{5} (x) - \sin^{6} (x)$

{cos}^{6} (x) + 6 i {cos}^{5} (x) sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 i cos (x) {sin}^{5} (x) - {sin}^{6} (x)

$\cos^{6} (x) + 6 i \cos^{5} (x) \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 i \cos (x) \sin^{5} (x) - \sin^{6} (x)$

Schritt 5

Ziehe die Ausdrücke mit dem imaginären Teil heraus, welche gleich

cos (6 x)

$\cos (6 x)$ sind. Entferne die imaginäre Zahl

i

$i$ .

cos (6 x) = {cos}^{6} (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) - {sin}^{6} (x)

$\cos (6 x) = \cos^{6} (x) - 15 \cos^{4} (x) \sin^{2} (x) + 15 \cos^{2} (x) \sin^{4} (x) - \sin^{6} (x)$

Gib DEINE Aufgabe ein