Trigonometria Exemplos

$\cos (80 + x) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1

Simplifique cada termo.

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Etapa 1.1

Reorganize os termos.

$\cos (x + 80) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.2

Double negation

$\cos (x - (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.3

Aplique a fórmula da diferença dos ângulos $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$(\cos (x) \cos (- 80) + \sin (x) \sin (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.4

Simplifique cada termo.

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Etapa 1.4.1

Avalie $\cos (- 80)$ .

$(\cos (x) \cdot - 0.11038724 + \sin (x) \sin (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.4.2

Mova $- 0.11038724$ para a esquerda de $\cos (x)$ .

$(- 0.11038724 \cdot \cos (x) + \sin (x) \sin (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.4.3

Avalie $\sin (- 80)$ .

$(- 0.11038724 \cos (x) + \sin (x) \cdot 0.99388865) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.4.4

Mova $0.99388865$ para a esquerda de $\sin (x)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.5

Reorganize os termos.

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot \cos (x + 2) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.6

Double negation

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot \cos (x - (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.7

Aplique a fórmula da diferença dos ângulos $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (\cos (x) \cos (- 2) + \sin (x) \sin (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.8

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Etapa 1.8.1

Avalie $\cos (- 2)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (\cos (x) \cdot - 0.41614683 + \sin (x) \sin (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.8.2

Mova $- 0.41614683$ para a esquerda de $\cos (x)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (- 0.41614683 \cdot \cos (x) + \sin (x) \sin (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.8.3

Avalie $\sin (- 2)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (- 0.41614683 \cos (x) + \sin (x) \cdot - 0.90929742) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.8.4

Mova $- 0.90929742$ para a esquerda de $\sin (x)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.9

Expanda $(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) (- 0.41614683 \cos (x) - 0.90929742 \sin (x))$ usando o método FOIL.

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Etapa 1.9.1

Aplique a propriedade distributiva.

$- 0.11038724 \cos (x) (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.9.2

Aplique a propriedade distributiva.

$- 0.11038724 \cos (x) (- 0.41614683 \cos (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.9.3

Aplique a propriedade distributiva.

$- 0.11038724 \cos (x) (- 0.41614683 \cos (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10

Simplifique e combine termos semelhantes.

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Etapa 1.10.1

Simplifique cada termo.

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Etapa 1.10.1.1

Multiplique $- 0.11038724 \cos (x) (- 0.41614683 \cos (x))$ .

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Etapa 1.10.1.1.1

Multiplique $- 0.41614683$ por $- 0.11038724$ .

$0.0459373 \cos (x) \cos (x) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.1.2

Eleve $\cos (x)$ à potência de $1$ .

$0.0459373 (\cos^{1} (x) \cos (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.1.3

Eleve $\cos (x)$ à potência de $1$ .

$0.0459373 (\cos^{1} (x) \cos^{1} (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.1.4

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

$0.0459373 {\cos (x)}^{1 + 1} - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.1.5

Some $1$ e $1$ .

$0.0459373 \cos^{2} (x) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.2

Multiplique $- 0.90929742$ por $- 0.11038724$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.3

Multiplique $- 0.41614683$ por $0.99388865$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.4

Multiplique $0.99388865 \sin (x) (- 0.90929742 \sin (x))$ .

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Etapa 1.10.1.4.1

Multiplique $- 0.90929742$ por $0.99388865$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 \sin (x) \sin (x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.4.2

Eleve $\sin (x)$ à potência de $1$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 (\sin^{1} (x) \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.4.3

Eleve $\sin (x)$ à potência de $1$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 (\sin^{1} (x) \sin^{1} (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.4.4

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

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 {\sin (x)}^{1 + 1} + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.1.4.5

Some $1$ e $1$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 \sin^{2} (x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.2

Reordene os fatores de $- 0.41360361 \sin (x) \cos (x)$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.10.3

Subtraia $0.41360361 \cos (x) \sin (x)$ de $0.10037483 \cos (x) \sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (80 + x) \cdot \sin (20 + x)$

Etapa 1.11

Reorganize os termos.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (x + 80) \cdot \sin (20 + x)$

Etapa 1.12

Double negation

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (x - (- 80)) \cdot \sin (20 + x)$

Etapa 1.13

Aplique a fórmula da diferença dos ângulos.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (\sin (x) \cos (- 80) - \cos (x) \sin (- 80)) \cdot \sin (20 + x)$

Etapa 1.14

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Etapa 1.14.1

Avalie $\cos (- 80)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (\sin (x) \cdot - 0.11038724 - \cos (x) \sin (- 80)) \cdot \sin (20 + x)$

Etapa 1.14.2

Mova $- 0.11038724$ para a esquerda de $\sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \cdot \sin (x) - \cos (x) \sin (- 80)) \cdot \sin (20 + x)$

Etapa 1.14.3

Avalie $\sin (- 80)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - \cos (x) \cdot 0.99388865) \cdot \sin (20 + x)$

Etapa 1.14.4

Multiplique $0.99388865$ por $- 1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot \sin (20 + x)$

Etapa 1.15

Reorganize os termos.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot \sin (x + 20)$

Etapa 1.16

Double negation

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot \sin (x - (- 20))$

Etapa 1.17

Aplique a fórmula da diferença dos ângulos.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (\sin (x) \cos (- 20) - \cos (x) \sin (- 20))$

Etapa 1.18

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Etapa 1.18.1

Avalie $\cos (- 20)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (\sin (x) \cdot 0.40808206 - \cos (x) \sin (- 20))$

Etapa 1.18.2

Mova $0.40808206$ para a esquerda de $\sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (0.40808206 \cdot \sin (x) - \cos (x) \sin (- 20))$

Etapa 1.18.3

Avalie $\sin (- 20)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (0.40808206 \sin (x) - \cos (x) \cdot - 0.91294525)$

Etapa 1.18.4

Multiplique $- 0.91294525$ por $- 1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (0.40808206 \sin (x) + 0.91294525 \cos (x))$

Etapa 1.19

Expanda $(- 0.11038724 \sin (x) - 0.99388865 \cos (x)) (0.40808206 \sin (x) + 0.91294525 \cos (x))$ usando o método FOIL.

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Etapa 1.19.1

Aplique a propriedade distributiva.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.11038724 \sin (x) (0.40808206 \sin (x) + 0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x) + 0.91294525 \cos (x))$

Etapa 1.19.2

Aplique a propriedade distributiva.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.11038724 \sin (x) (0.40808206 \sin (x)) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x) + 0.91294525 \cos (x))$

Etapa 1.19.3

Aplique a propriedade distributiva.

Etapa 1.20

Simplifique e combine termos semelhantes.

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Etapa 1.20.1

Simplifique cada termo.

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Etapa 1.20.1.1

Multiplique $- 0.11038724 \sin (x) (0.40808206 \sin (x))$ .

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Etapa 1.20.1.1.1

Multiplique $0.40808206$ por $- 0.11038724$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin (x) \sin (x) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Etapa 1.20.1.1.2

Eleve $\sin (x)$ à potência de $1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 (\sin^{1} (x) \sin (x)) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Etapa 1.20.1.1.3

Eleve $\sin (x)$ à potência de $1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 (\sin^{1} (x) \sin^{1} (x)) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Etapa 1.20.1.1.4

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

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 {\sin (x)}^{1 + 1} - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Etapa 1.20.1.1.5

Some $1$ e $1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Etapa 1.20.1.2

Multiplique $0.91294525$ por $- 0.11038724$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Etapa 1.20.1.3

Multiplique $0.40808206$ por $- 0.99388865$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.40558813 \cos (x) \sin (x) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Etapa 1.20.1.4

Multiplique $- 0.99388865 \cos (x) (0.91294525 \cos (x))$ .

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Etapa 1.20.1.4.1

Multiplique $0.91294525$ por $- 0.99388865$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.40558813 \cos (x) \sin (x) - 0.90736592 \cos (x) \cos (x)$

Etapa 1.20.1.4.2

Eleve $\cos (x)$ à potência de $1$ .

Etapa 1.20.1.4.3

Eleve $\cos (x)$ à potência de $1$ .

Etapa 1.20.1.4.4

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

Etapa 1.20.1.4.5

Some $1$ e $1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.40558813 \cos (x) \sin (x) - 0.90736592 \cos^{2} (x)$

Etapa 1.20.2

Reordene os fatores de $- 0.10077751 \sin (x) \cos (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \cos (x) \sin (x) - 0.40558813 \cos (x) \sin (x) - 0.90736592 \cos^{2} (x)$

Etapa 1.20.3

Subtraia $0.40558813 \cos (x) \sin (x)$ de $- 0.10077751 \cos (x) \sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.50636564 \cos (x) \sin (x) - 0.90736592 \cos^{2} (x)$

Etapa 2

Simplifique somando os termos.

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Etapa 2.1

Subtraia $0.90736592 \cos^{2} (x)$ de $0.0459373 \cos^{2} (x)$ .

$- 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.50636564 \cos (x) \sin (x) - 0.86142862 \cos^{2} (x)$

Etapa 2.2

Subtraia $0.50636564 \cos (x) \sin (x)$ de $- 0.31322878 \cos (x) \sin (x)$ .

$- 0.81959442 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.86142862 \cos^{2} (x)$

Etapa 2.3

Subtraia $0.04504705 \sin^{2} (x)$ de $- 0.90374039 \sin^{2} (x)$ .

$- 0.81959442 \cos (x) \sin (x) - 0.94878744 \sin^{2} (x) - 0.86142862 \cos^{2} (x)$