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Popular Trigonometria >

(sin(45)sin(15))/(cos(135)cos(105))

Solução

\frac{sin ( 4 5 ^{\circ} ) sin ( 1 5 ^{\circ} )}{cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

Solução

1

Passos da solução

\frac{sin ( 4 5 ^{\circ} ) sin ( 1 5 ^{\circ} )}{cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

Reeecreva usando identidades trigonométricas:

sin (4 5^{\circ}) sin (1 5^{\circ}) = \frac{cos ( 3 0 ^{\circ} ) - cos ( 6 0 ^{\circ} )}{2}

sin (4 5^{\circ}) sin (1 5^{\circ})

Use a identidade da transformação de produto em soma:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= \frac{1}{2} (cos (4 5^{\circ} - 1 5^{\circ}) - cos (4 5^{\circ} + 1 5^{\circ}))

Simplificar

= \frac{cos ( 3 0 ^{\circ} ) - cos ( 6 0 ^{\circ} )}{2}

= \frac{\frac{c o s ( 3 0 ^{\circ} ) - c o s ( 6 0 ^{\circ} )}{2}}{cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

Simplificar

= \frac{cos ( 3 0 ^{\circ} ) - cos ( 6 0 ^{\circ} )}{2 cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

Utilizar a seguinte identidade trivial:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

Utilizar a seguinte identidade trivial:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Utilizar a seguinte identidade trivial:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

Reeecreva usando identidades trigonométricas:

cos (10 5^{\circ}) = \frac{2 ( 1 - 3 )}{4}

cos (10 5^{\circ})

Reeecreva usando identidades trigonométricas:

cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

cos (10 5^{\circ})

Escrever

cos (10 5^{\circ})

como

cos (6 0^{\circ} + 4 5^{\circ})

= cos (6 0^{\circ} + 4 5^{\circ})

Use a identidade de soma de ângulos:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

Utilizar a seguinte identidade trivial:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Utilizar a seguinte identidade trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Utilizar a seguinte identidade trivial:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Utilizar a seguinte identidade trivial:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

Simplificar

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2} : \frac{2 ( 1 - 3 )}{4}

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

Fatorar o termo comum

\frac{2}{2}

= \frac{2}{2} (\frac{1}{2} - \frac{3}{2})

\frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2}

\frac{1}{2} - \frac{3}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - 3}{2}

= \frac{2}{2} \cdot \frac{1 - 3}{2}

Multiplicar frações:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{( 1 - 3 ) 2}{2 \cdot 2}

Multiplicar os números:

2 \cdot 2 = 4

= \frac{2 ( 1 - 3 )}{4}

= \frac{2 ( 1 - 3 )}{4}

= \frac{\frac{3}{2} - \frac{1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}}

Simplificar

\frac{\frac{3}{2} - \frac{1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}} : 1

\frac{\frac{3}{2} - \frac{1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}}

Combinar as frações usando o mínimo múltiplo comum:

\frac{3 - 1}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 - 1}{2}

= \frac{\frac{3 - 1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}}

Remover os parênteses:

(- a) = - a

= \frac{\frac{3 - 1}{2}}{- 2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

Aplicar as propriedades das frações:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{3 - 1}{2}}{2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

Aplicar as propriedades das frações:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

\frac{\frac{3 - 1}{2}}{2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}} = \frac{3 - 1}{2 \cdot 2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

= - \frac{3 - 1}{2 \cdot 2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

Multiplicar os números:

2 \cdot 2 = 4

= - \frac{3 - 1}{4 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

Multiplicar

4 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4} : 1 - 3

4 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}

Multiplicar frações:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{2 2 ( 1 - 3 ) \cdot 4}{2 \cdot 4}

Eliminar o fator comum:

4

= \frac{2 2 ( 1 - 3 )}{2}

22 (1 - 3) = 2 (1 - 3)

22 (1 - 3)

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2 (1 - 3)

= \frac{2 ( 1 - 3 )}{2}

Dividir:

\frac{2}{2} = 1

= 1 - 3

= - \frac{3 - 1}{1 - 3}

1 - 3 = - (3 - 1)

= \frac{3 - 1}{- ( 3 - 1 )}

Simplificar

= - \frac{3 - 1}{3 - 1}

Eliminar o fator comum:

3 - 1

= - (- 1)

Aplicar a regra

- (- a) = a

= 1

= 1

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