Trigonometria Exemplos

${(\frac{1}{4} \cdot (\cos (\frac{π}{10}) + i \sin (\frac{π}{10})))}^{5}$

Etapa 1

Aplique a propriedade distributiva.

${(\frac{1}{4} \cos (\frac{π}{10}) + \frac{1}{4} (i \sin (\frac{π}{10})))}^{5}$

Etapa 2

Combine $\frac{1}{4}$ e $\cos (\frac{π}{10})$ .

${(\frac{\cos (\frac{π}{10})}{4} + \frac{1}{4} (i \sin (\frac{π}{10})))}^{5}$

Etapa 3

Multiplique $\frac{1}{4} (i \sin (\frac{π}{10}))$ .

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Etapa 3.1

Combine $i$ e $\frac{1}{4}$ .

${(\frac{\cos (\frac{π}{10})}{4} + \frac{i}{4} \sin (\frac{π}{10}))}^{5}$

Etapa 3.2

Combine $\frac{i}{4}$ e $\sin (\frac{π}{10})$ .

${(\frac{\cos (\frac{π}{10})}{4} + \frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 4

Use o teorema binomial.

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

Etapa 5

Simplifique os termos.

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Etapa 5.1

Simplifique cada termo.

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Etapa 5.1.1

Aplique a regra do produto a $\frac{\cos (\frac{π}{10})}{4}$ .

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

Etapa 5.1.2

Eleve $4$ à potência de $5$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + 5 {(\frac{\cos (\frac{π}{10})}{4})}^{4} \frac{i \sin (\frac{π}{10})}{4} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{3} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.3

Aplique a regra do produto a $\frac{\cos (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + 5 \frac{\cos^{4} (\frac{π}{10})}{4^{4}} \frac{i \sin (\frac{π}{10})}{4} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{3} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.4

Eleve $4$ à potência de $4$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + 5 \frac{\cos^{4} (\frac{π}{10})}{256} \frac{i \sin (\frac{π}{10})}{4} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{3} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.5

Combine $5$ e $\frac{\cos^{4} (\frac{π}{10})}{256}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10})}{256} \cdot \frac{i \sin (\frac{π}{10})}{4} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{3} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.6

Combine.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) (i \sin (\frac{π}{10}))}{256 \cdot 4} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{3} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.7

Multiplique $256$ por $4$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{3} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.8

Aplique a regra do produto a $\frac{\cos (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + 10 \frac{\cos^{3} (\frac{π}{10})}{4^{3}} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.9

Eleve $4$ à potência de $3$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + 10 \frac{\cos^{3} (\frac{π}{10})}{64} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.10

Cancele o fator comum de $2$ .

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Etapa 5.1.10.1

Fatore $2$ de $10$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + 2 (5) \frac{\cos^{3} (\frac{π}{10})}{64} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.10.2

Fatore $2$ de $64$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + 2 \cdot 5 \frac{\cos^{3} (\frac{π}{10})}{2 \cdot 32} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.10.3

Cancele o fator comum.

Etapa 5.1.10.4

Reescreva a expressão.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + 5 \frac{\cos^{3} (\frac{π}{10})}{32} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.11

Combine $5$ e $\frac{\cos^{3} (\frac{π}{10})}{32}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{5 \cos^{3} (\frac{π}{10})}{32} {(\frac{i \sin (\frac{π}{10})}{4})}^{2} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.12

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 5.1.12.1

Aplique a regra do produto a $\frac{i \sin (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{5 \cos^{3} (\frac{π}{10})}{32} \cdot \frac{{(i \sin (\frac{π}{10}))}^{2}}{4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.12.2

Aplique a regra do produto a $i \sin (\frac{π}{10})$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{5 \cos^{3} (\frac{π}{10})}{32} \cdot \frac{i^{2} \sin^{2} (\frac{π}{10})}{4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.13

Combine.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{5 \cos^{3} (\frac{π}{10}) (i^{2} \sin^{2} (\frac{π}{10}))}{32 \cdot 4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.14

Simplifique o numerador.

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Etapa 5.1.14.1

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

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{5 \cos^{3} (\frac{π}{10}) \cdot - 1 \sin^{2} (\frac{π}{10})}{32 \cdot 4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.14.2

Combine expoentes.

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Etapa 5.1.14.2.1

Fatore o negativo.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- (5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10}))}{32 \cdot 4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.14.2.2

Multiplique $5$ por $- 1$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 (\cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10}))}{32 \cdot 4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{32 \cdot 4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.15

Simplifique o denominador.

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Etapa 5.1.15.1

Reescreva $32$ como $2^{5}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{2^{5} \cdot 4^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.15.2

Reescreva $4$ como $2^{2}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{2^{5} \cdot {(2^{2})}^{2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.15.3

Multiplique os expoentes em ${(2^{2})}^{2}$ .

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Etapa 5.1.15.3.1

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{2^{5} \cdot 2^{2 \cdot 2}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.15.3.2

Multiplique $2$ por $2$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{2^{5} \cdot 2^{4}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.15.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{2^{5 + 4}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.15.5

Some $5$ e $4$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{2^{9}} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.16

Eleve $2$ à potência de $9$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} + \frac{- 5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.17

Mova o número negativo para a frente da fração.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 10 {(\frac{\cos (\frac{π}{10})}{4})}^{2} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.18

Aplique a regra do produto a $\frac{\cos (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 10 \frac{\cos^{2} (\frac{π}{10})}{4^{2}} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.19

Eleve $4$ à potência de $2$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 10 \frac{\cos^{2} (\frac{π}{10})}{16} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.20

Cancele o fator comum de $2$ .

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Etapa 5.1.20.1

Fatore $2$ de $10$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 2 (5) \frac{\cos^{2} (\frac{π}{10})}{16} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.20.2

Fatore $2$ de $16$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 2 \cdot 5 \frac{\cos^{2} (\frac{π}{10})}{2 \cdot 8} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.20.3

Cancele o fator comum.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 2 \cdot 5 \frac{\cos^{2} (\frac{π}{10})}{2 \cdot 8} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.20.4

Reescreva a expressão.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + 5 \frac{\cos^{2} (\frac{π}{10})}{8} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.21

Combine $5$ e $\frac{\cos^{2} (\frac{π}{10})}{8}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{5 \cos^{2} (\frac{π}{10})}{8} {(\frac{i \sin (\frac{π}{10})}{4})}^{3} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.22

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 5.1.22.1

Aplique a regra do produto a $\frac{i \sin (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{5 \cos^{2} (\frac{π}{10})}{8} \cdot \frac{{(i \sin (\frac{π}{10}))}^{3}}{4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.22.2

Aplique a regra do produto a $i \sin (\frac{π}{10})$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{5 \cos^{2} (\frac{π}{10})}{8} \cdot \frac{i^{3} \sin^{3} (\frac{π}{10})}{4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.23

Combine.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{5 \cos^{2} (\frac{π}{10}) (i^{3} \sin^{3} (\frac{π}{10}))}{8 \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.24

Simplifique o numerador.

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Etapa 5.1.24.1

Fatore $i^{2}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{5 \cos^{2} (\frac{π}{10}) (i^{2} \cdot i) \sin^{3} (\frac{π}{10})}{8 \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.24.2

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

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{5 \cos^{2} (\frac{π}{10}) (- 1 \cdot i) \sin^{3} (\frac{π}{10})}{8 \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.24.3

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

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{5 \cos^{2} (\frac{π}{10}) \cdot - 1 i \sin^{3} (\frac{π}{10})}{8 \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.24.4

Combine expoentes.

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Etapa 5.1.24.4.1

Fatore o negativo.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- (5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10}))}{8 \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.24.4.2

Multiplique $5$ por $- 1$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 (\cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10}))}{8 \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{8 \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.25

Simplifique o denominador.

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Etapa 5.1.25.1

Reescreva $8$ como $2^{3}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{2^{3} \cdot 4^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.25.2

Reescreva $4$ como $2^{2}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{2^{3} \cdot {(2^{2})}^{3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.25.3

Multiplique os expoentes em ${(2^{2})}^{3}$ .

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Etapa 5.1.25.3.1

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{2^{3} \cdot 2^{2 \cdot 3}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.25.3.2

Multiplique $2$ por $3$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{2^{3} \cdot 2^{6}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.25.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{2^{3 + 6}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.25.5

Some $3$ e $6$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{2^{9}} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.26

Eleve $2$ à potência de $9$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{- 5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.27

Mova o número negativo para a frente da fração.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + 5 \frac{\cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.28

Combine $5$ e $\frac{\cos (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10})}{4} {(\frac{i \sin (\frac{π}{10})}{4})}^{4} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.29

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 5.1.29.1

Aplique a regra do produto a $\frac{i \sin (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10})}{4} \cdot \frac{{(i \sin (\frac{π}{10}))}^{4}}{4^{4}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.29.2

Aplique a regra do produto a $i \sin (\frac{π}{10})$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10})}{4} \cdot \frac{i^{4} \sin^{4} (\frac{π}{10})}{4^{4}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.30

Combine.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) (i^{4} \sin^{4} (\frac{π}{10}))}{4 \cdot 4^{4}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.31

Multiplique $4$ por $4^{4}$ somando os expoentes.

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Etapa 5.1.31.1

Multiplique $4$ por $4^{4}$ .

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Etapa 5.1.31.1.1

Eleve $4$ à potência de $1$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) (i^{4} \sin^{4} (\frac{π}{10}))}{4^{1} \cdot 4^{4}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.31.1.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) (i^{4} \sin^{4} (\frac{π}{10}))}{4^{1 + 4}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.31.2

Some $1$ e $4$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) (i^{4} \sin^{4} (\frac{π}{10}))}{4^{5}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.32

Simplifique o numerador.

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Etapa 5.1.32.1

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

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Etapa 5.1.32.1.1

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

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) {(i^{2})}^{2} \sin^{4} (\frac{π}{10})}{4^{5}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.32.1.2

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

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) {(- 1)}^{2} \sin^{4} (\frac{π}{10})}{4^{5}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.32.1.3

Eleve $- 1$ à potência de $2$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \cdot 1 \sin^{4} (\frac{π}{10})}{4^{5}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.32.2

Multiplique $5$ por $1$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{4^{5}} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.33

Eleve $4$ à potência de $5$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + {(\frac{i \sin (\frac{π}{10})}{4})}^{5}$

Etapa 5.1.34

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 5.1.34.1

Aplique a regra do produto a $\frac{i \sin (\frac{π}{10})}{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{{(i \sin (\frac{π}{10}))}^{5}}{4^{5}}$

Etapa 5.1.34.2

Aplique a regra do produto a $i \sin (\frac{π}{10})$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{i^{5} \sin^{5} (\frac{π}{10})}{4^{5}}$

Etapa 5.1.35

Simplifique o numerador.

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Etapa 5.1.35.1

Fatore $i^{4}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{i^{4} i \sin^{5} (\frac{π}{10})}{4^{5}}$

Etapa 5.1.35.2

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

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Etapa 5.1.35.2.1

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

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{{(i^{2})}^{2} i \sin^{5} (\frac{π}{10})}{4^{5}}$

Etapa 5.1.35.2.2

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

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{{(- 1)}^{2} i \sin^{5} (\frac{π}{10})}{4^{5}}$

Etapa 5.1.35.2.3

Eleve $- 1$ à potência de $2$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{1 i \sin^{5} (\frac{π}{10})}{4^{5}}$

Etapa 5.1.35.3

Multiplique $i$ por $1$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{i \sin^{5} (\frac{π}{10})}{4^{5}}$

Etapa 5.1.36

Eleve $4$ à potência de $5$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{i \sin^{5} (\frac{π}{10})}{1024}$

Etapa 5.2

Simplifique a expressão.

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Etapa 5.2.1

Reordene os fatores em $\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 \cos^{4} (\frac{π}{10}) i \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 \cos^{2} (\frac{π}{10}) i \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{i \sin^{5} (\frac{π}{10})}{1024}$ .

$\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 i \cos^{4} (\frac{π}{10}) \sin (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} - \frac{5 i \cos^{2} (\frac{π}{10}) \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{i \sin^{5} (\frac{π}{10})}{1024}$

Etapa 5.2.2

Reordene $\frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 i \cos^{4} (\frac{π}{10}) \sin (\frac{π}{10})}{1024}$ e $- \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512}$ .

$- \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 i \cos^{4} (\frac{π}{10}) \sin (\frac{π}{10})}{1024} - \frac{5 i \cos^{2} (\frac{π}{10}) \sin^{3} (\frac{π}{10})}{512} + \frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} + \frac{i \sin^{5} (\frac{π}{10})}{1024}$

Etapa 5.2.3

Reordene $- \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 i \cos^{4} (\frac{π}{10}) \sin (\frac{π}{10})}{1024} - \frac{5 i \cos^{2} (\frac{π}{10}) \sin^{3} (\frac{π}{10})}{512}$ e $\frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024}$ .

$\frac{5 \cos (\frac{π}{10}) \sin^{4} (\frac{π}{10})}{1024} - \frac{5 \cos^{3} (\frac{π}{10}) \sin^{2} (\frac{π}{10})}{512} + \frac{\cos^{5} (\frac{π}{10})}{1024} + \frac{5 i \cos^{4} (\frac{π}{10}) \sin (\frac{π}{10})}{1024} - \frac{5 i \cos^{2} (\frac{π}{10}) \sin^{3} (\frac{π}{10})}{512} + \frac{i \sin^{5} (\frac{π}{10})}{1024}$