Trigonometria Esempi

cos (8 x)

$\cos (8 x)$

Passaggio 1

Per espandere

cos (8 x)

$\cos (8 x)$ un buon metodo è la formula di de Moivre

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

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

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}

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

Passaggio 2

Espandi il lato destro di

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

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

Espandi:

{(cos (x) + i \cdot sin (x))}^{8}

${(\cos (x) + i \cdot \sin (x))}^{8}$

Passaggio 3

Usa il teorema binomiale.

{cos}^{8} (x) + 8 {cos}^{7} (x) (i sin (x)) + 28 {cos}^{6} (x) {(i sin (x))}^{2} + 56 {cos}^{5} (x) {(i sin (x))}^{3} + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) (i \sin (x)) + 28 \cos^{6} (x) {(i \sin (x))}^{2} + 56 \cos^{5} (x) {(i \sin (x))}^{3} + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4

Semplifica i termini.

Tocca per altri passaggi...

Passaggio 4.1

Semplifica ciascun termine.

Tocca per altri passaggi...

Passaggio 4.1.1

Applica la regola del prodotto a

i sin (x)

$i \sin (x)$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) + 28 {cos}^{6} (x) (i^{2} {sin}^{2} (x)) + 56 {cos}^{5} (x) {(i sin (x))}^{3} + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) + 28 \cos^{6} (x) (i^{2} \sin^{2} (x)) + 56 \cos^{5} (x) {(i \sin (x))}^{3} + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.2

Riscrivi utilizzando la proprietà commutativa della moltiplicazione.

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) + 28 \cdot i^{2} {cos}^{6} (x) {sin}^{2} (x) + 56 {cos}^{5} (x) {(i sin (x))}^{3} + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) + 28 \cdot i^{2} \cos^{6} (x) \sin^{2} (x) + 56 \cos^{5} (x) {(i \sin (x))}^{3} + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.3

Riscrivi

i^{2}

$i^{2}$ come

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) + 28 \cdot - 1 {cos}^{6} (x) {sin}^{2} (x) + 56 {cos}^{5} (x) {(i sin (x))}^{3} + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) + 28 \cdot - 1 \cos^{6} (x) \sin^{2} (x) + 56 \cos^{5} (x) {(i \sin (x))}^{3} + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.4

Moltiplica

28

$28$ per

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) + 56 {cos}^{5} (x) {(i sin (x))}^{3} + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) + 56 \cos^{5} (x) {(i \sin (x))}^{3} + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.5

Applica la regola del prodotto a

i sin (x)

$i \sin (x)$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) + 56 {cos}^{5} (x) (i^{3} {sin}^{3} (x)) + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) + 56 \cos^{5} (x) (i^{3} \sin^{3} (x)) + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.6

Riscrivi utilizzando la proprietà commutativa della moltiplicazione.

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) + 56 \cdot i^{3} {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) + 56 \cdot i^{3} \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.7

Metti in evidenza

i^{2}

$i^{2}$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) + 56 \cdot (i^{2} \cdot i) {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) + 56 \cdot (i^{2} \cdot i) \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.8

Riscrivi

i^{2}

$i^{2}$ come

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) + 56 \cdot (- 1 \cdot i) {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) + 56 \cdot (- 1 \cdot i) \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.9

Riscrivi

- 1 i

$- 1 i$ come

- i

$- i$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) + 56 \cdot (- i) {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) + 56 \cdot (- i) \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.10

Moltiplica

- 1

$- 1$ per

56

$56$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {(i sin (x))}^{4} + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) {(i \sin (x))}^{4} + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.11

Applica la regola del prodotto a

i sin (x)

$i \sin (x)$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) (i^{4} {sin}^{4} (x)) + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) (i^{4} \sin^{4} (x)) + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.12

Riscrivi utilizzando la proprietà commutativa della moltiplicazione.

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 \cdot i^{4} {cos}^{4} (x) {sin}^{4} (x) + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cdot i^{4} \cos^{4} (x) \sin^{4} (x) + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.13

Riscrivi

i^{4}

$i^{4}$ come

1

$1$ .

Tocca per altri passaggi...

Passaggio 4.1.13.1

Riscrivi

i^{4}

$i^{4}$ come

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

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

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 \cdot {(i^{2})}^{2} {cos}^{4} (x) {sin}^{4} (x) + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cdot {(i^{2})}^{2} \cos^{4} (x) \sin^{4} (x) + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.13.2

Riscrivi

i^{2}

$i^{2}$ come

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 \cdot {(- 1)}^{2} {cos}^{4} (x) {sin}^{4} (x) + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cdot {(- 1)}^{2} \cos^{4} (x) \sin^{4} (x) + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.13.3

Eleva

- 1

$- 1$ alla potenza di

2

$2$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 \cdot 1 {cos}^{4} (x) {sin}^{4} (x) + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cdot 1 \cos^{4} (x) \sin^{4} (x) + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 \cdot 1 {cos}^{4} (x) {sin}^{4} (x) + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

Passaggio 4.1.14

Moltiplica

70

$70$ per

1

$1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 {cos}^{3} (x) {(i sin (x))}^{5} + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cos^{3} (x) {(i \sin (x))}^{5} + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.15

Applica la regola del prodotto a

i sin (x)

$i \sin (x)$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 {cos}^{3} (x) (i^{5} {sin}^{5} (x)) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cos^{3} (x) (i^{5} \sin^{5} (x)) + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.16

Riscrivi utilizzando la proprietà commutativa della moltiplicazione.

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 \cdot i^{5} {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cdot i^{5} \cos^{3} (x) \sin^{5} (x) + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.17

Metti in evidenza

i^{4}

$i^{4}$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 \cdot (i^{4} i) {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cdot (i^{4} i) \cos^{3} (x) \sin^{5} (x) + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.18

Riscrivi

i^{4}

$i^{4}$ come

1

$1$ .

Tocca per altri passaggi...

Passaggio 4.1.18.1

Riscrivi

i^{4}

$i^{4}$ come

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

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

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 \cdot ({(i^{2})}^{2} i) {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cdot ({(i^{2})}^{2} i) \cos^{3} (x) \sin^{5} (x) + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.18.2

Riscrivi

i^{2}

$i^{2}$ come

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 \cdot ({(- 1)}^{2} i) {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cdot ({(- 1)}^{2} i) \cos^{3} (x) \sin^{5} (x) + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.18.3

Eleva

- 1

$- 1$ alla potenza di

2

$2$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 \cdot (1 i) {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cdot (1 i) \cos^{3} (x) \sin^{5} (x) + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 \cdot (1 i) {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

Passaggio 4.1.19

Moltiplica

i

$i$ per

1

$1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 \cdot i {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) {(i sin (x))}^{6} + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 \cdot i \cos^{3} (x) \sin^{5} (x) + 28 \cos^{2} (x) {(i \sin (x))}^{6} + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.20

Applica la regola del prodotto a

i sin (x)

$i \sin (x)$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 {cos}^{2} (x) (i^{6} {sin}^{6} (x)) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cos^{2} (x) (i^{6} \sin^{6} (x)) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.21

Riscrivi utilizzando la proprietà commutativa della moltiplicazione.

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot i^{6} {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cdot i^{6} \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.22

Metti in evidenza

i^{4}

$i^{4}$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot (i^{4} i^{2}) {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cdot (i^{4} i^{2}) \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.23

Riscrivi

i^{4}

$i^{4}$ come

1

$1$ .

Tocca per altri passaggi...

Passaggio 4.1.23.1

Riscrivi

i^{4}

$i^{4}$ come

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

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

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot ({(i^{2})}^{2} i^{2}) {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cdot ({(i^{2})}^{2} i^{2}) \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.23.2

Riscrivi

i^{2}

$i^{2}$ come

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot ({(- 1)}^{2} i^{2}) {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cdot ({(- 1)}^{2} i^{2}) \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.23.3

Eleva

- 1

$- 1$ alla potenza di

2

$2$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot (1 i^{2}) {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cdot (1 i^{2}) \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot (1 i^{2}) {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

Passaggio 4.1.24

Moltiplica

i^{2}

$i^{2}$ per

1

$1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot i^{2} {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cdot i^{2} \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.25

Riscrivi

i^{2}

$i^{2}$ come

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) + 28 \cdot - 1 {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) + 28 \cdot - 1 \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.26

Moltiplica

28

$28$ per

- 1

$- 1$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) - 28 {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) {(i sin (x))}^{7} + {(i sin (x))}^{8}

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) - 28 \cos^{2} (x) \sin^{6} (x) + 8 \cos (x) {(i \sin (x))}^{7} + {(i \sin (x))}^{8}$

Passaggio 4.1.27

Applica la regola del prodotto a

i sin (x)

$i \sin (x)$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) - 28 {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) (i^{7} {sin}^{7} (x)) + {(i sin (x))}^{8}

Passaggio 4.1.28

Riscrivi

i^{7}

$i^{7}$ come

i^{4} (i^{2} \cdot i)

$i^{4} (i^{2} \cdot i)$ .

Tocca per altri passaggi...

Passaggio 4.1.28.1

Metti in evidenza

i^{4}

$i^{4}$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) - 28 {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) (i^{4} i^{3} {sin}^{7} (x)) + {(i sin (x))}^{8}

Passaggio 4.1.28.2

Metti in evidenza

i^{2}

$i^{2}$ .

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) - 28 {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) (i^{4} (i^{2} \cdot i) {sin}^{7} (x)) + {(i sin (x))}^{8}

{cos}^{8} (x) + 8 {cos}^{7} (x) i sin (x) - 28 {cos}^{6} (x) {sin}^{2} (x) - 56 i {cos}^{5} (x) {sin}^{3} (x) + 70 {cos}^{4} (x) {sin}^{4} (x) + 56 i {cos}^{3} (x) {sin}^{5} (x) - 28 {cos}^{2} (x) {sin}^{6} (x) + 8 cos (x) (i^{4} (i^{2} \cdot i) {sin}^{7} (x)) + {(i sin (x))}^{8}

Passaggio 4.1.29

Riscrivi

$i^{4}$ come

$1$ .

Tocca per altri passaggi...

Passaggio 4.1.29.1

Riscrivi

$i^{4}$ come

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

Passaggio 4.1.29.2

Riscrivi

$i^{2}$ come

$- 1$ .

Passaggio 4.1.29.3

Eleva

$- 1$ alla potenza di

$2$ .

Passaggio 4.1.30

Moltiplica

$i^{2} \cdot i$ per

$1$ .

Passaggio 4.1.31

Riscrivi

$i^{2}$ come

$- 1$ .

Passaggio 4.1.32

Riscrivi

$- 1 i$ come

$- i$ .

Passaggio 4.1.33

Moltiplica

$- 1$ per

$8$ .

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) - 28 \cos^{2} (x) \sin^{6} (x) - 8 \cos (x) (i \sin^{7} (x)) + {(i \sin (x))}^{8}$

Passaggio 4.1.34

Applica la regola del prodotto a

$i \sin (x)$ .

Passaggio 4.1.35

Riscrivi

$i^{8}$ come

${(i^{4})}^{2}$ .

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) - 28 \cos^{2} (x) \sin^{6} (x) - 8 \cos (x) i \sin^{7} (x) + {(i^{4})}^{2} \sin^{8} (x)$

Passaggio 4.1.36

Riscrivi

$i^{4}$ come

$1$ .

Tocca per altri passaggi...

Passaggio 4.1.36.1

Riscrivi

$i^{4}$ come

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

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) - 28 \cos^{2} (x) \sin^{6} (x) - 8 \cos (x) i \sin^{7} (x) + {({(i^{2})}^{2})}^{2} \sin^{8} (x)$

Passaggio 4.1.36.2

Riscrivi

$i^{2}$ come

$- 1$ .

$\cos^{8} (x) + 8 \cos^{7} (x) i \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) - 28 \cos^{2} (x) \sin^{6} (x) - 8 \cos (x) i \sin^{7} (x) + {({(- 1)}^{2})}^{2} \sin^{8} (x)$

Passaggio 4.1.36.3

Eleva

$- 1$ alla potenza di

$2$ .

Passaggio 4.1.37

Uno elevato a qualsiasi potenza è uno.

Passaggio 4.1.38

Moltiplica

$\sin^{8} (x)$ per

$1$ .

Passaggio 4.2

Riordina i fattori in

$\cos^{8} (x) + 8 i \cos^{7} (x) \sin (x) - 28 \cos^{6} (x) \sin^{2} (x) - 56 i \cos^{5} (x) \sin^{3} (x) + 70 \cos^{4} (x) \sin^{4} (x) + 56 i \cos^{3} (x) \sin^{5} (x) - 28 \cos^{2} (x) \sin^{6} (x) - 8 i \cos (x) \sin^{7} (x) + \sin^{8} (x)$

Passaggio 5

Estrai le espressioni con la parte immaginaria, che sono uguali a

$\cos (8 x)$ . Rimuovi il numero immaginario

$i$ .

$\cos (8 x) = \cos^{8} (x) - 28 \cos^{6} (x) \sin^{2} (x) + 70 \cos^{4} (x) \sin^{4} (x) - 28 \cos^{2} (x) \sin^{6} (x) + \sin^{8} (x)$

Inserisci il TUO problema