Pré-cálculo Exemplos

$\sin (a + b) \sin (a - b)$

Etapa 1

Aplique a fórmula da soma dos ângulos.

$(\sin (a) \cos (b) + \cos (a) \sin (b)) \sin (a - b)$

Etapa 2

Aplique a fórmula da diferença dos ângulos.

$(\sin (a) \cos (b) + \cos (a) \sin (b)) (\sin (a) \cos (b) - \cos (a) \sin (b))$

Etapa 3

Expanda $(\sin (a) \cos (b) + \cos (a) \sin (b)) (\sin (a) \cos (b) - \cos (a) \sin (b))$ usando o método FOIL.

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Etapa 3.1

Aplique a propriedade distributiva.

$\sin (a) \cos (b) (\sin (a) \cos (b) - \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b) - \cos (a) \sin (b))$

Etapa 3.2

Aplique a propriedade distributiva.

$\sin (a) \cos (b) (\sin (a) \cos (b)) + \sin (a) \cos (b) (- \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b) - \cos (a) \sin (b))$

Etapa 3.3

Aplique a propriedade distributiva.

$\sin (a) \cos (b) (\sin (a) \cos (b)) + \sin (a) \cos (b) (- \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4

Simplifique os termos.

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Etapa 4.1

Combine os termos opostos em $\sin (a) \cos (b) (\sin (a) \cos (b)) + \sin (a) \cos (b) (- \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$ .

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Etapa 4.1.1

Reorganize os fatores nos termos $\sin (a) \cos (b) (- \cos (a) \sin (b))$ e $\cos (a) \sin (b) (\sin (a) \cos (b))$ .

$\sin (a) \cos (b) (\sin (a) \cos (b)) - \cos (a) \cos (b) \sin (a) \sin (b) + \cos (a) \cos (b) \sin (a) \sin (b) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.1.2

Some $- \cos (a) \cos (b) \sin (a) \sin (b)$ e $\cos (a) \cos (b) \sin (a) \sin (b)$ .

$\sin (a) \cos (b) (\sin (a) \cos (b)) + 0 + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.1.3

Some $\sin (a) \cos (b) (\sin (a) \cos (b))$ e $0$ .

$\sin (a) \cos (b) (\sin (a) \cos (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2

Simplifique cada termo.

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Etapa 4.2.1

Multiplique $\sin (a) \cos (b) (\sin (a) \cos (b))$ .

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Etapa 4.2.1.1

Eleve $\sin (a)$ à potência de $1$ .

$\sin^{1} (a) \sin (a) \cos (b) \cos (b) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.1.2

Eleve $\sin (a)$ à potência de $1$ .

$\sin^{1} (a) \sin^{1} (a) \cos (b) \cos (b) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.1.3

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

${\sin (a)}^{1 + 1} \cos (b) \cos (b) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.1.4

Some $1$ e $1$ .

$\sin^{2} (a) \cos (b) \cos (b) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.1.5

Eleve $\cos (b)$ à potência de $1$ .

$\sin^{2} (a) (\cos^{1} (b) \cos (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.1.6

Eleve $\cos (b)$ à potência de $1$ .

$\sin^{2} (a) (\cos^{1} (b) \cos^{1} (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.1.7

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

$\sin^{2} (a) {\cos (b)}^{1 + 1} + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.1.8

Some $1$ e $1$ .

$\sin^{2} (a) \cos^{2} (b) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Etapa 4.2.2

Reescreva usando a propriedade comutativa da multiplicação.

$\sin^{2} (a) \cos^{2} (b) - \cos (a) \sin (b) \cos (a) \sin (b)$

Etapa 4.2.3

Multiplique $- \cos (a) \sin (b) \cos (a)$ .

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Etapa 4.2.3.1

Eleve $\cos (a)$ à potência de $1$ .

$\sin^{2} (a) \cos^{2} (b) - (\cos^{1} (a) \cos (a)) \sin (b) \sin (b)$

Etapa 4.2.3.2

Eleve $\cos (a)$ à potência de $1$ .

$\sin^{2} (a) \cos^{2} (b) - (\cos^{1} (a) \cos^{1} (a)) \sin (b) \sin (b)$

Etapa 4.2.3.3

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

$\sin^{2} (a) \cos^{2} (b) - {\cos (a)}^{1 + 1} \sin (b) \sin (b)$

Etapa 4.2.3.4

Some $1$ e $1$ .

$\sin^{2} (a) \cos^{2} (b) - \cos^{2} (a) \sin (b) \sin (b)$

Etapa 4.2.4

Multiplique $- \cos^{2} (a) \sin (b) \sin (b)$ .

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Etapa 4.2.4.1

Eleve $\sin (b)$ à potência de $1$ .

$\sin^{2} (a) \cos^{2} (b) - \cos^{2} (a) (\sin^{1} (b) \sin (b))$

Etapa 4.2.4.2

Eleve $\sin (b)$ à potência de $1$ .

$\sin^{2} (a) \cos^{2} (b) - \cos^{2} (a) (\sin^{1} (b) \sin^{1} (b))$

Etapa 4.2.4.3

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

$\sin^{2} (a) \cos^{2} (b) - \cos^{2} (a) {\sin (b)}^{1 + 1}$

Etapa 4.2.4.4

Some $1$ e $1$ .

$\sin^{2} (a) \cos^{2} (b) - \cos^{2} (a) \sin^{2} (b)$