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Popular Trigonometria >

provar cos(θ+30)-sin(θ+60)=-sin(θ)

Solução

provar

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

Solução

Verdadeiro

Passos da solução

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

Manipular o lado direito

cos (θ + 3 0^{\circ}) - sin (θ + 6 0^{\circ})

Reeecreva usando identidades trigonométricas

sin (θ + 6 0^{\circ})

Use a identidade de soma de ângulos:

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

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

Simplificar

sin (θ) cos (6 0^{\circ}) + cos (θ) sin (6 0^{\circ}) : \frac{1}{2} sin (θ) + \frac{3}{2} cos (θ)

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

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

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}

= \frac{1}{2} sin (θ) + \frac{3}{2} cos (θ)

= \frac{1}{2} sin (θ) + \frac{3}{2} cos (θ)

= cos (θ + 3 0^{\circ}) - (\frac{1}{2} sin (θ) + \frac{3}{2} cos (θ))

Reeecreva usando identidades trigonométricas

cos (θ + 3 0^{\circ})

Use a identidade de soma de ângulos:

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

= cos (θ) cos (3 0^{\circ}) - sin (θ) sin (3 0^{\circ})

Simplificar

cos (θ) cos (3 0^{\circ}) - sin (θ) sin (3 0^{\circ}) : \frac{3}{2} cos (θ) - \frac{1}{2} sin (θ)

cos (θ) cos (3 0^{\circ}) - sin (θ) sin (3 0^{\circ})

Simplificar

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

cos (3 0^{\circ})

Utilizar a seguinte identidade trivial:

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

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

Simplificar

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Utilizar a seguinte identidade trivial:

sin (3 0^{\circ}) = \frac{1}{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{1}{2}

= \frac{3}{2} cos (θ) - \frac{1}{2} sin (θ)

= \frac{3}{2} cos (θ) - \frac{1}{2} sin (θ)

= \frac{3}{2} cos (θ) - \frac{1}{2} sin (θ) - (\frac{1}{2} sin (θ) + \frac{3}{2} cos (θ))

Simplificar

\frac{3}{2} cos (θ) - \frac{1}{2} sin (θ) - (\frac{1}{2} sin (θ) + \frac{3}{2} cos (θ)) : - sin (θ)

\frac{3}{2} cos (θ) - \frac{1}{2} sin (θ) - (\frac{1}{2} sin (θ) + \frac{3}{2} cos (θ))

- (\frac{1}{2} sin (θ) + \frac{3}{2} cos (θ)) : - \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ)

- (\frac{1}{2} sin (θ) + \frac{3}{2} cos (θ))

Colocar os parênteses

= - (\frac{1}{2} sin (θ)) - (\frac{3}{2} cos (θ))

Aplicar as regras dos sinais

+ (- a) = - a

= - \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ)

= \frac{3}{2} cos (θ) - \frac{1}{2} sin (θ) - \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ)

Simplificar

\frac{3}{2} cos (θ) - \frac{1}{2} sin (θ) - \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ) : - sin (θ)

\frac{3}{2} cos (θ) - \frac{1}{2} sin (θ) - \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ)

Agrupar termos semelhantes

= - \frac{1}{2} sin (θ) - \frac{1}{2} sin (θ) + \frac{3}{2} cos (θ) - \frac{3}{2} cos (θ)

Somar elementos similares:

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

\frac{3}{2} cos (θ) - \frac{3}{2} cos (θ)

Fatorar o termo comum

cos (θ)

= cos (θ) (\frac{3}{2} - \frac{3}{2})

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

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

Já que os denominadores são iguais, combinar as frações:

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

= \frac{3 - 3}{2}

Fatorar

3 - 3 : 0

3 - 3

Fatorar o termo comum

3

= 3 (1 - 1)

Simplificar

= 0

= \frac{0}{2}

Aplicar a regra

\frac{0}{a} = 0, a \neq = 0

= 0

= 0

= - \frac{1}{2} sin (θ) - \frac{1}{2} sin (θ)

Somar elementos similares:

- \frac{1}{2} sin (θ) - \frac{1}{2} sin (θ) = - sin (θ)

- \frac{1}{2} sin (θ) - \frac{1}{2} sin (θ)

Fatorar o termo comum

sin (θ)

= sin (θ) (- \frac{1}{2} - \frac{1}{2})

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

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

Aplicar a regra

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

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

Subtrair:

- 1 - 1 = - 2

= \frac{- 2}{2}

Aplicar as propriedades das frações:

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

= - \frac{2}{2}

Aplicar a regra

\frac{a}{a} = 1

= - 1

= - sin (θ)

= - sin (θ)

= - sin (θ)

= - sin (θ)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

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