Trigonometria Exemplos

$(\cos (x) + \sin (x)) (1 - \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Etapa 1

Simplifique $(\cos (x) + \sin (x)) (1 - \cos (x) \sin (x))$ .

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Etapa 1.1

Expanda $(\cos (x) + \sin (x)) (1 - \cos (x) \sin (x))$ usando o método FOIL.

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Etapa 1.1.1

Aplique a propriedade distributiva.

$\cos (x) (1 - \cos (x) \sin (x)) + \sin (x) (1 - \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Etapa 1.1.2

Aplique a propriedade distributiva.

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

Etapa 1.1.3

Aplique a propriedade distributiva.

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

Etapa 1.2

Simplifique cada termo.

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Etapa 1.2.1

Multiplique $\cos (x)$ por $1$ .

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

Etapa 1.2.2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 1.2.3

Multiplique $- \cos (x) \cos (x)$ .

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Etapa 1.2.3.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 1.2.3.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 1.2.3.3

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

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

Etapa 1.2.3.4

Some $1$ e $1$ .

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

Etapa 1.2.4

Multiplique $\sin (x)$ por $1$ .

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

Etapa 1.2.5

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 1.2.6

Multiplique $- \sin (x) \cos (x) \sin (x)$ .

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Etapa 1.2.6.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.2.6.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.2.6.3

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

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

Etapa 1.2.6.4

Some $1$ e $1$ .

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

Etapa 2

Simplifique $\sin^{3} (x) + \cos^{3} (x)$ .

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Etapa 2.1

Como os dois termos são cubos perfeitos, fatore usando a fórmula da soma de cubos, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ em que $a = \sin (x)$ e $b = \cos (x)$ .

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

Etapa 2.2

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

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

Etapa 2.3

Aplique a identidade trigonométrica fundamental.

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = (\sin (x) + \cos (x)) (1 - \sin (x) \cos (x))$

Etapa 2.4

Expanda $(\sin (x) + \cos (x)) (1 - \sin (x) \cos (x))$ usando o método FOIL.

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Etapa 2.4.1

Aplique a propriedade distributiva.

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) (1 - \sin (x) \cos (x)) + \cos (x) (1 - \sin (x) \cos (x))$

Etapa 2.4.2

Aplique a propriedade distributiva.

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

Etapa 2.4.3

Aplique a propriedade distributiva.

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

Etapa 2.5

Simplifique cada termo.

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Etapa 2.5.1

Multiplique $\sin (x)$ por $1$ .

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

Etapa 2.5.2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 2.5.3

Multiplique $- \sin (x) \sin (x)$ .

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Etapa 2.5.3.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 2.5.3.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 2.5.3.3

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

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

Etapa 2.5.3.4

Some $1$ e $1$ .

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

Etapa 2.5.4

Multiplique $\cos (x)$ por $1$ .

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

Etapa 2.5.5

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 2.5.6

Multiplique $- \cos (x) \sin (x) \cos (x)$ .

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Etapa 2.5.6.1

Eleve $\cos (x)$ à potência de $1$ .

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin^{2} (x) \cos (x) + \cos (x) - (\cos^{1} (x) \cos (x)) \sin (x)$

Etapa 2.5.6.2

Eleve $\cos (x)$ à potência de $1$ .

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin^{2} (x) \cos (x) + \cos (x) - (\cos^{1} (x) \cos^{1} (x)) \sin (x)$

Etapa 2.5.6.3

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

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin^{2} (x) \cos (x) + \cos (x) - {\cos (x)}^{1 + 1} \sin (x)$

Etapa 2.5.6.4

Some $1$ e $1$ .

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

Etapa 3

Represente cada lado da equação em um gráfico. A solução é o valor x do ponto de intersecção.

$x \approx 0, - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, - \frac{2}{5}$

Etapa 4