Trigonometria Exemplos

sin (5 x)

$\sin (5 x)$

Etapa 1

Um bom método para expandir

sin (5 x)

$\sin (5 x)$ 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))}^{5}

${(\cos (x) + i \cdot \sin (x))}^{5}$

Etapa 3

Use o teorema binomial.

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

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

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

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

Etapa 4.1.2

Reescreva usando a propriedade comutativa da multiplicação.

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

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot i^{2} \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Etapa 4.1.3

Reescreva

i^{2}

$i^{2}$ como

- 1

$- 1$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) + 10 \cdot - 1 {cos}^{3} (x) {sin}^{2} (x) + 10 {cos}^{2} (x) {(i sin (x))}^{3} + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot - 1 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Etapa 4.1.4

Multiplique

10

$10$ por

- 1

$- 1$ .

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

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

Etapa 4.1.5

Aplique a regra do produto a

i sin (x)

$i \sin (x)$ .

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

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

Etapa 4.1.6

Reescreva usando a propriedade comutativa da multiplicação.

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

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

Etapa 4.1.7

Fatore

i^{2}

$i^{2}$ .

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

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

Etapa 4.1.8

Reescreva

i^{2}

$i^{2}$ como

- 1

$- 1$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) + 10 \cdot (- 1 \cdot i) {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (- 1 \cdot i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Etapa 4.1.9

Reescreva

- 1 i

$- 1 i$ como

- i

$- i$ .

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

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

Etapa 4.1.10

Multiplique

- 1

$- 1$ por

10

$10$ .

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

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

Etapa 4.1.11

Aplique a regra do produto a

i sin (x)

$i \sin (x)$ .

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

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

Etapa 4.1.12

Reescreva

i^{4}

$i^{4}$ como

1

$1$ .

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Etapa 4.1.12.1

Reescreva

i^{4}

$i^{4}$ como

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

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

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

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

Etapa 4.1.12.2

Reescreva

i^{2}

$i^{2}$ como

- 1

$- 1$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) ({(- 1)}^{2} {sin}^{4} (x)) + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) ({(- 1)}^{2} \sin^{4} (x)) + {(i \sin (x))}^{5}$

Etapa 4.1.12.3

Eleve

- 1

$- 1$ à potência de

2

$2$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) (1 {sin}^{4} (x)) + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (1 \sin^{4} (x)) + {(i \sin (x))}^{5}$

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) (1 {sin}^{4} (x)) + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (1 \sin^{4} (x)) + {(i \sin (x))}^{5}$

Etapa 4.1.13

Multiplique

{sin}^{4} (x)

$\sin^{4} (x)$ por

1

$1$ .

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

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

Etapa 4.1.14

Aplique a regra do produto a

i sin (x)

$i \sin (x)$ .

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

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

Etapa 4.1.15

Fatore

i^{4}

$i^{4}$ .

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

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

Etapa 4.1.16

Reescreva

i^{4}

$i^{4}$ como

1

$1$ .

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Etapa 4.1.16.1

Reescreva

i^{4}

$i^{4}$ como

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

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

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

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

Etapa 4.1.16.2

Reescreva

i^{2}

$i^{2}$ como

- 1

$- 1$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {sin}^{4} (x) + {(- 1)}^{2} i {sin}^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(- 1)}^{2} i \sin^{5} (x)$

Etapa 4.1.16.3

Eleve

- 1

$- 1$ à potência de

2

$2$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {sin}^{4} (x) + 1 i {sin}^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + 1 i \sin^{5} (x)$

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {sin}^{4} (x) + 1 i {sin}^{5} (x)

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + 1 i \sin^{5} (x)$

Etapa 4.1.17

Multiplique

i

$i$ por

1

$1$ .

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

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

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

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

Etapa 4.2

Reordene os fatores em

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

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

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

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

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

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

Etapa 5

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

sin (5 x)

$\sin (5 x)$ . Remova o número imaginário

i

$i$ .

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

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

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