Trigonometria Exemplos

$\sin (5 x)$

Etapa 1

Um bom método para expandir $\sin (5 x)$ é usando o teorema de De Moivre $(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . Quando $r = 1$ , $\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

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

Etapa 2

Expanda o lado direito de $\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ usando o teorema binomial.

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

Etapa 3

Use o teorema binomial.

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

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 $i \sin (x)$ .

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

Etapa 4.1.2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 4.1.3

Reescreva $i^{2}$ como $- 1$ .

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

Etapa 4.1.4

Multiplique $10$ por $- 1$ .

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

Etapa 4.1.5

Aplique a regra do produto a $i \sin (x)$ .

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

Etapa 4.1.6

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 4.1.7

Fatore $i^{2}$ .

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

Etapa 4.1.8

Reescreva $i^{2}$ como $- 1$ .

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

Etapa 4.1.9

Reescreva $- 1 i$ como $- i$ .

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

Etapa 4.1.10

Multiplique $- 1$ por $10$ .

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

Etapa 4.1.11

Aplique a regra do produto a $i \sin (x)$ .

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

Etapa 4.1.12

Reescreva $i^{4}$ como $1$ .

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Etapa 4.1.12.1

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

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

Etapa 4.1.12.2

Reescreva $i^{2}$ como $- 1$ .

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

Etapa 4.1.12.3

Eleve $- 1$ à potência de $2$ .

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

Etapa 4.1.13

Multiplique $\sin^{4} (x)$ por $1$ .

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

Etapa 4.1.14

Aplique a regra do produto a $i \sin (x)$ .

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

Etapa 4.1.15

Fatore $i^{4}$ .

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

Etapa 4.1.16

Reescreva $i^{4}$ como $1$ .

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Etapa 4.1.16.1

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

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

Etapa 4.1.16.2

Reescreva $i^{2}$ como $- 1$ .

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

Etapa 4.1.16.3

Eleve $- 1$ à potência de $2$ .

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

Etapa 4.1.17

Multiplique $i$ por $1$ .

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

Etapa 4.2

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

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

Etapa 5

Retire as expressões com a parte imaginária, que são iguais a $\sin (5 x)$ . Remova o número imaginário $i$ .

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