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Popular Trigonometria >

provar tan(120)=tan(180-60)

Solução

provar

tan (12 0^{\circ}) = tan (18 0^{\circ} - 6 0^{\circ})

Solução

Verdadeiro

Passos da solução

tan (12 0^{\circ}) = tan (18 0^{\circ} - 6 0^{\circ})

Manipular o lado direito

tan (12 0^{\circ})

Simplificar

tan (12 0^{\circ}) : - 3

tan (12 0^{\circ})

Reeecreva usando identidades trigonométricas:

\frac{sin ( 12 0 ^{\circ} )}{cos ( 12 0 ^{\circ} )}

tan (12 0^{\circ})

Utilizar a Identidade Básica da Trigonometria:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( 12 0 ^{\circ} )}{cos ( 12 0 ^{\circ} )}

= \frac{sin ( 12 0 ^{\circ} )}{cos ( 12 0 ^{\circ} )}

Utilizar a seguinte identidade trivial:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 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:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 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}

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

Simplificar

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

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

Aplicar as propriedades das frações:

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

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

Dividir frações:

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

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

Simplificar

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

Eliminar o fator comum:

2

= - 3

= - 3

= - 3

Manipular o lado esquerdo

tan (18 0^{\circ} - 6 0^{\circ})

Reeecreva usando identidades trigonométricas

tan (18 0^{\circ} - 6 0^{\circ})

Utilizar a Identidade Básica da Trigonometria:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( 18 0 ^{\circ} - 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} - 6 0 ^{\circ} )}

Use a identidade de diferença de ângulos:

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

= \frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} - 6 0 ^{\circ} )}

Use a identidade de diferença de ângulos:

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

= \frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}

\frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )} = - 3

\frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}

sin (18 0^{\circ}) cos (6 0^{\circ}) - cos (18 0^{\circ}) sin (6 0^{\circ}) = \frac{3}{2}

sin (18 0^{\circ}) cos (6 0^{\circ}) - cos (18 0^{\circ}) sin (6 0^{\circ})

sin (18 0^{\circ}) cos (6 0^{\circ}) = 0

sin (18 0^{\circ}) cos (6 0^{\circ})

Simplificar

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Utilizar a seguinte identidade trivial:

sin (18 0^{\circ}) = 0

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}

= 0

= 0 \cdot cos (6 0^{\circ})

Simplificar

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

cos (6 0^{\circ})

Utilizar a seguinte identidade trivial:

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

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}

= 0 \cdot \frac{1}{2}

Aplicar a regra

0 \cdot a = 0

= 0

cos (18 0^{\circ}) sin (6 0^{\circ}) = - \frac{3}{2}

cos (18 0^{\circ}) sin (6 0^{\circ})

Simplificar

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Utilizar a seguinte identidade trivial:

cos (18 0^{\circ}) = (- 1)

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}

= - 1

= - 1 \cdot sin (6 0^{\circ})

Simplificar

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

sin (6 0^{\circ})

Utilizar a seguinte identidade trivial:

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

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}

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

Multiplicar:

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

= - \frac{3}{2}

= 0 - (- \frac{3}{2})

Aplicar a regra

- (- a) = a

= 0 + \frac{3}{2}

0 + \frac{3}{2} = \frac{3}{2}

= \frac{3}{2}

= \frac{\frac{3}{2}}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}

cos (18 0^{\circ}) cos (6 0^{\circ}) + sin (18 0^{\circ}) sin (6 0^{\circ}) = - \frac{1}{2}

cos (18 0^{\circ}) cos (6 0^{\circ}) + sin (18 0^{\circ}) sin (6 0^{\circ})

cos (18 0^{\circ}) cos (6 0^{\circ}) = - \frac{1}{2}

cos (18 0^{\circ}) cos (6 0^{\circ})

Simplificar

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Utilizar a seguinte identidade trivial:

cos (18 0^{\circ}) = (- 1)

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}

= - 1

= - 1 \cdot cos (6 0^{\circ})

Simplificar

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

cos (6 0^{\circ})

Utilizar a seguinte identidade trivial:

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

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}

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

Multiplicar:

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

= - \frac{1}{2}

= - \frac{1}{2} + sin (18 0^{\circ}) sin (6 0^{\circ})

sin (18 0^{\circ}) sin (6 0^{\circ}) = 0

sin (18 0^{\circ}) sin (6 0^{\circ})

Simplificar

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Utilizar a seguinte identidade trivial:

sin (18 0^{\circ}) = 0

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}

= 0

= 0 \cdot sin (6 0^{\circ})

Simplificar

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

sin (6 0^{\circ})

Utilizar a seguinte identidade trivial:

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

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}

= 0 \cdot \frac{3}{2}

Aplicar a regra

0 \cdot a = 0

= 0

= - \frac{1}{2} + 0

- \frac{1}{2} + 0 = - \frac{1}{2}

= - \frac{1}{2}

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

Simplificar

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

Aplicar as propriedades das frações:

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

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

Dividir frações:

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

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

Simplificar

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

Eliminar o fator comum:

2

= - 3

= - 3

= - 3

= - 3

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

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