Trigonometria Exemplos

sin (7 x)

$\sin (7 x)$

Etapa 1

Um bom método para expandir

sin (7 x)

$\sin (7 x)$ é usando o teorema de 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}

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

Etapa 2

Expanda o lado direito de

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 o teorema binomial.

Expandir:

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

${(\cos (x) + i \cdot \sin (x))}^{7}$

Etapa 3

Use o teorema binomial.

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

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

Etapa 4

Simplifique os termos.

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Etapa 4.1

Simplifique cada termo.

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Etapa 4.1.1

Aplique a regra do produto a

i sin (x)

$i \sin (x)$ .

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

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

Etapa 4.1.2

Reescreva usando a propriedade comutativa da multiplicação.

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

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

Etapa 4.1.3

Reescreva

i^{2}

$i^{2}$ como

- 1

$- 1$ .

{cos}^{7} (x) + 7 {cos}^{6} (x) i sin (x) + 21 \cdot - 1 {cos}^{5} (x) {sin}^{2} (x) + 35 {cos}^{4} (x) {(i sin (x))}^{3} + 35 {cos}^{3} (x) {(i sin (x))}^{4} + 21 {cos}^{2} (x) {(i sin (x))}^{5} + 7 cos (x) {(i sin (x))}^{6} + {(i sin (x))}^{7}

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) + 21 \cdot - 1 \cos^{5} (x) \sin^{2} (x) + 35 \cos^{4} (x) {(i \sin (x))}^{3} + 35 \cos^{3} (x) {(i \sin (x))}^{4} + 21 \cos^{2} (x) {(i \sin (x))}^{5} + 7 \cos (x) {(i \sin (x))}^{6} + {(i \sin (x))}^{7}$

Etapa 4.1.4

Multiplique

21

$21$ por

- 1

$- 1$ .

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

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

Etapa 4.1.5

Aplique a regra do produto a

i sin (x)

$i \sin (x)$ .

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

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

Etapa 4.1.6

Reescreva usando a propriedade comutativa da multiplicação.

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

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

Etapa 4.1.7

Fatore

i^{2}

$i^{2}$ .

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

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

Etapa 4.1.8

Reescreva

$i^{2}$ como

$- 1$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) + 35 \cdot (- 1 \cdot i) \cos^{4} (x) \sin^{3} (x) + 35 \cos^{3} (x) {(i \sin (x))}^{4} + 21 \cos^{2} (x) {(i \sin (x))}^{5} + 7 \cos (x) {(i \sin (x))}^{6} + {(i \sin (x))}^{7}$

Etapa 4.1.9

Reescreva

$- 1 i$ como

$- i$ .

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

Etapa 4.1.10

Multiplique

$- 1$ por

$35$ .

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

Etapa 4.1.11

Aplique a regra do produto a

$i \sin (x)$ .

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

Etapa 4.1.12

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 4.1.13

Reescreva

$i^{4}$ como

$1$ .

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Etapa 4.1.13.1

Reescreva

$i^{4}$ como

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

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

Etapa 4.1.13.2

Reescreva

$i^{2}$ como

$- 1$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) - 35 i \cos^{4} (x) \sin^{3} (x) + 35 \cdot {(- 1)}^{2} \cos^{3} (x) \sin^{4} (x) + 21 \cos^{2} (x) {(i \sin (x))}^{5} + 7 \cos (x) {(i \sin (x))}^{6} + {(i \sin (x))}^{7}$

Etapa 4.1.13.3

Eleve

$- 1$ à potência de

$2$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) - 35 i \cos^{4} (x) \sin^{3} (x) + 35 \cdot 1 \cos^{3} (x) \sin^{4} (x) + 21 \cos^{2} (x) {(i \sin (x))}^{5} + 7 \cos (x) {(i \sin (x))}^{6} + {(i \sin (x))}^{7}$

Etapa 4.1.14

Multiplique

$35$ por

$1$ .

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

Etapa 4.1.15

Aplique a regra do produto a

$i \sin (x)$ .

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

Etapa 4.1.16

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 4.1.17

Fatore

$i^{4}$ .

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

Etapa 4.1.18

Reescreva

$i^{4}$ como

$1$ .

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Etapa 4.1.18.1

Reescreva

$i^{4}$ como

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

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

Etapa 4.1.18.2

Reescreva

$i^{2}$ como

$- 1$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) - 35 i \cos^{4} (x) \sin^{3} (x) + 35 \cos^{3} (x) \sin^{4} (x) + 21 \cdot ({(- 1)}^{2} i) \cos^{2} (x) \sin^{5} (x) + 7 \cos (x) {(i \sin (x))}^{6} + {(i \sin (x))}^{7}$

Etapa 4.1.18.3

Eleve

$- 1$ à potência de

$2$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) - 35 i \cos^{4} (x) \sin^{3} (x) + 35 \cos^{3} (x) \sin^{4} (x) + 21 \cdot (1 i) \cos^{2} (x) \sin^{5} (x) + 7 \cos (x) {(i \sin (x))}^{6} + {(i \sin (x))}^{7}$

Etapa 4.1.19

Multiplique

$i$ por

$1$ .

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

Etapa 4.1.20

Aplique a regra do produto a

$i \sin (x)$ .

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

Etapa 4.1.21

Fatore

$i^{4}$ .

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

Etapa 4.1.22

Reescreva

$i^{4}$ como

$1$ .

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Etapa 4.1.22.1

Reescreva

$i^{4}$ como

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

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

Etapa 4.1.22.2

Reescreva

$i^{2}$ como

$- 1$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) - 35 i \cos^{4} (x) \sin^{3} (x) + 35 \cos^{3} (x) \sin^{4} (x) + 21 i \cos^{2} (x) \sin^{5} (x) + 7 \cos (x) ({(- 1)}^{2} i^{2} \sin^{6} (x)) + {(i \sin (x))}^{7}$

Etapa 4.1.22.3

Eleve

$- 1$ à potência de

$2$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) - 35 i \cos^{4} (x) \sin^{3} (x) + 35 \cos^{3} (x) \sin^{4} (x) + 21 i \cos^{2} (x) \sin^{5} (x) + 7 \cos (x) (1 i^{2} \sin^{6} (x)) + {(i \sin (x))}^{7}$

Etapa 4.1.23

Multiplique

$i^{2}$ por

$1$ .

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

Etapa 4.1.24

Reescreva

$i^{2}$ como

$- 1$ .

$\cos^{7} (x) + 7 \cos^{6} (x) i \sin (x) - 21 \cos^{5} (x) \sin^{2} (x) - 35 i \cos^{4} (x) \sin^{3} (x) + 35 \cos^{3} (x) \sin^{4} (x) + 21 i \cos^{2} (x) \sin^{5} (x) + 7 \cos (x) (- 1 \sin^{6} (x)) + {(i \sin (x))}^{7}$

Etapa 4.1.25

Reescreva

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

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

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

Etapa 4.1.26

Multiplique

$- 1$ por

$7$ .

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

Etapa 4.1.27

Aplique a regra do produto a

$i \sin (x)$ .

Etapa 4.1.28

Reescreva

$i^{7}$ como

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

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Etapa 4.1.28.1

Fatore

$i^{4}$ .

Etapa 4.1.28.2

Fatore

$i^{2}$ .

Etapa 4.1.29

Reescreva

$i^{4}$ como

$1$ .

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Etapa 4.1.29.1

Reescreva

$i^{4}$ como

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

Etapa 4.1.29.2

Reescreva

$i^{2}$ como

$- 1$ .

Etapa 4.1.29.3

Eleve

$- 1$ à potência de

$2$ .

Etapa 4.1.30

Multiplique

$i^{2} \cdot i$ por

$1$ .

Etapa 4.1.31

Reescreva

$i^{2}$ como

$- 1$ .

Etapa 4.1.32

Reescreva

$- 1 i$ como

$- i$ .

Etapa 4.2

Reordene os fatores em

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

Etapa 5

Retire as expressões com a parte imaginária, que são iguais a

$\sin (7 x)$ . Remova o número imaginário

$i$ .

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