Trigonometria Exemplos

$\sin (6 x)$

Etapa 1

Fatore $2$ de $6 x$ .

$\sin (2 (3 x))$

Etapa 2

Aplique a fórmula do arco triplo do seno.

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

Etapa 3

Aplique a propriedade distributiva.

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

Etapa 4

Multiplique.

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Etapa 4.1

Multiplique $- 4$ por $2$ .

$(- 8 \sin^{3} (x) + 2 (3 \sin (x))) \cos (3 x)$

Etapa 4.2

Multiplique $3$ por $2$ .

$(- 8 \sin^{3} (x) + 6 \sin (x)) \cos (3 x)$

Etapa 5

Use a fórmula do arco triplo para transformar $\cos (3 x)$ em $4 \cos^{3} (x) - 3 \cos (x)$ .

$(- 8 \sin^{3} (x) + 6 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x))$

Etapa 6

Expanda $(- 8 \sin^{3} (x) + 6 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x))$ usando o método FOIL.

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Etapa 6.1

Aplique a propriedade distributiva.

$- 8 \sin^{3} (x) (4 \cos^{3} (x) - 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x) - 3 \cos (x))$

Etapa 6.2

Aplique a propriedade distributiva.

$- 8 \sin^{3} (x) (4 \cos^{3} (x)) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x) - 3 \cos (x))$

Etapa 6.3

Aplique a propriedade distributiva.

$- 8 \sin^{3} (x) (4 \cos^{3} (x)) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x))$

Etapa 7

Simplifique os termos.

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Etapa 7.1

Simplifique cada termo.

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Etapa 7.1.1

Reescreva usando a propriedade comutativa da multiplicação.

$- 8 \cdot 4 \sin^{3} (x) \cos^{3} (x) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x))$

Etapa 7.1.2

Multiplique $- 8$ por $4$ .

$- 32 \sin^{3} (x) \cos^{3} (x) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x))$

Etapa 7.1.3

Multiplique $- 3$ por $- 8$ .

$- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x))$

Etapa 7.1.4

Multiplique $4$ por $6$ .

$- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) + 6 \sin (x) (- 3 \cos (x))$

Etapa 7.1.5

Multiplique $- 3$ por $6$ .

$- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x)$

Etapa 7.2

Simplifique com fatoração.

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Etapa 7.2.1

Fatore $2 \sin (x) \cos (x)$ de $- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x)$ .

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Etapa 7.2.1.1

Fatore $2 \sin (x) \cos (x)$ de $- 32 \sin^{3} (x) \cos^{3} (x)$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x)$

Etapa 7.2.1.2

Fatore $2 \sin (x) \cos (x)$ de $24 \sin^{3} (x) \cos (x)$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x)) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x)$

Etapa 7.2.1.3

Fatore $2 \sin (x) \cos (x)$ de $24 \sin (x) \cos^{3} (x)$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x)) - 18 \sin (x) \cos (x)$

Etapa 7.2.1.4

Fatore $2 \sin (x) \cos (x)$ de $- 18 \sin (x) \cos (x)$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9)$

Etapa 7.2.1.5

Fatore $2 \sin (x) \cos (x)$ de $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x))$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9)$

Etapa 7.2.1.6

Fatore $2 \sin (x) \cos (x)$ de $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x))$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x) + 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9)$

Etapa 7.2.1.7

Fatore $2 \sin (x) \cos (x)$ de $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x) + 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9)$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x) + 12 \cos^{2} (x) - 9)$

Etapa 7.2.2

Fatore $12$ de $12 \sin^{2} (x)$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 (\sin^{2} (x)) + 12 \cos^{2} (x) - 9)$

Etapa 7.2.3

Fatore $12$ de $12 \cos^{2} (x)$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 (\sin^{2} (x)) + 12 (\cos^{2} (x)) - 9)$

Etapa 7.2.4

Fatore $12$ de $12 (\sin^{2} (x)) + 12 (\cos^{2} (x))$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 (\sin^{2} (x) + \cos^{2} (x)) - 9)$

Etapa 8

Aplique a identidade trigonométrica fundamental.

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \cdot 1 - 9)$

Etapa 9

Simplifique multiplicando.

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Etapa 9.1

Multiplique $12$ por $1$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 - 9)$

Etapa 9.2

Subtraia $9$ de $12$ .

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 3)$

Etapa 9.3

Aplique a propriedade distributiva.

$2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3$

Etapa 10

Multiplique $\sin (x)$ por $\sin^{2} (x)$ somando os expoentes.

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Etapa 10.1

Mova $\sin^{2} (x)$ .

$2 (\sin^{2} (x) \sin (x)) \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3$

Etapa 10.2

Multiplique $\sin^{2} (x)$ por $\sin (x)$ .

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Etapa 10.2.1

Eleve $\sin (x)$ à potência de $1$ .

$2 (\sin^{2} (x) \sin^{1} (x)) \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3$

Etapa 10.2.2

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

$2 {\sin (x)}^{2 + 1} \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3$

Etapa 10.3

Some $2$ e $1$ .

$2 \sin^{3} (x) \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3$

Etapa 11

Multiplique $3$ por $2$ .

$2 \sin^{3} (x) \cos (x) (- 16 \cos^{2} (x)) + 6 \sin (x) \cos (x)$

Etapa 12

Simplifique cada termo.

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Etapa 12.1

Multiplique $\cos (x)$ por $\cos^{2} (x)$ somando os expoentes.

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Etapa 12.1.1

Mova $\cos^{2} (x)$ .

$2 \sin^{3} (x) (\cos^{2} (x) \cos (x)) \cdot - 16 + 6 \sin (x) \cos (x)$

Etapa 12.1.2

Multiplique $\cos^{2} (x)$ por $\cos (x)$ .

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Etapa 12.1.2.1

Eleve $\cos (x)$ à potência de $1$ .

$2 \sin^{3} (x) (\cos^{2} (x) \cos^{1} (x)) \cdot - 16 + 6 \sin (x) \cos (x)$

Etapa 12.1.2.2

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

$2 \sin^{3} (x) {\cos (x)}^{2 + 1} \cdot - 16 + 6 \sin (x) \cos (x)$

Etapa 12.1.3

Some $2$ e $1$ .

$2 \sin^{3} (x) \cos^{3} (x) \cdot - 16 + 6 \sin (x) \cos (x)$

Etapa 12.2

Multiplique $- 16$ por $2$ .

$- 32 \sin^{3} (x) \cos^{3} (x) + 6 \sin (x) \cos (x)$