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Popular Trigonometria >

sin(x+pi/6)-sin(x-(7pi)/6)=(sqrt(3))/2

Solução

sin (x + \frac{π}{6}) - sin (x - \frac{7 π}{6}) = \frac{3}{2}

Solução

x = 2 πn + \frac{π}{2} + \frac{π}{3}, x = 2 πn + \frac{π}{2} + \frac{5 π}{3}

Graus

x = 15 0^{\circ} + 36 0^{\circ} n, x = 39 0^{\circ} + 36 0^{\circ} n

Passos da solução

sin (x + \frac{π}{6}) - sin (x - \frac{7 π}{6}) = \frac{3}{2}

Reeecreva usando identidades trigonométricas

sin (x + \frac{π}{6}) - sin (x - \frac{7 π}{6})

Use a identidade da transformação de soma em produto:

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

= 2 sin (\frac{x + \frac{π}{6} - ( x - \frac{7 π}{6} )}{2}) cos (\frac{x + \frac{π}{6} + x - \frac{7 π}{6}}{2})

Simplificar

2 sin (\frac{x + \frac{π}{6} - ( x - \frac{7 π}{6} )}{2}) cos (\frac{x + \frac{π}{6} + x - \frac{7 π}{6}}{2}) : 3 cos (\frac{2 x - π}{2})

2 sin (\frac{x + \frac{π}{6} - ( x - \frac{7 π}{6} )}{2}) cos (\frac{x + \frac{π}{6} + x - \frac{7 π}{6}}{2})

\frac{x + \frac{π}{6} - ( x - \frac{7 π}{6} )}{2} = \frac{2 π}{3}

\frac{x + \frac{π}{6} - ( x - \frac{7 π}{6} )}{2}

Simplificar

x + \frac{π}{6} - (x - \frac{7 π}{6})

em uma fração:

\frac{4 π}{3}

x + \frac{π}{6} - (x - \frac{7 π}{6})

Converter para fração:

x = \frac{x 6}{6}, (x - \frac{7 π}{6}) = \frac{( x - \frac{7 π}{6} ) 6}{6}

= \frac{x \cdot 6}{6} + \frac{π}{6} - \frac{( x - \frac{7 π}{6} ) \cdot 6}{6}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{x \cdot 6 + π - ( x - \frac{7 π}{6} ) \cdot 6}{6}

Expandir

x \cdot 6 + π - (x - \frac{7 π}{6}) \cdot 6 : 8 π

x \cdot 6 + π - (x - \frac{7 π}{6}) \cdot 6

= 6 x + π - 6 (x - \frac{7 π}{6})

Expandir

- 6 (x - \frac{7 π}{6}) : - 6 x + 7 π

- 6 (x - \frac{7 π}{6})

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = - 6, b = x, c = \frac{7 π}{6}

= - 6 x - (- 6) \frac{7 π}{6}

Aplicar as regras dos sinais

- (- a) = a

= - 6 x + 6 \cdot \frac{7 π}{6}

6 \cdot \frac{7 π}{6} = 7 π

6 \cdot \frac{7 π}{6}

Multiplicar frações:

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

= \frac{7 π 6}{6}

Eliminar o fator comum:

6

= 7 π

= - 6 x + 7 π

= x \cdot 6 + π - 6 x + 7 π

Simplificar

x \cdot 6 + π - 6 x + 7 π : 8 π

x \cdot 6 + π - 6 x + 7 π

Agrupar termos semelhantes

= 6 x - 6 x + π + 7 π

Somar elementos similares:

6 x - 6 x = 0

= π + 7 π

Somar elementos similares:

π + 7 π = 8 π

= 8 π

= 8 π

= \frac{8 π}{6}

Eliminar o fator comum:

2

= \frac{4 π}{3}

= \frac{\frac{4 π}{3}}{2}

Aplicar as propriedades das frações:

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

= \frac{4 π}{3 \cdot 2}

Multiplicar os números:

3 \cdot 2 = 6

= \frac{4 π}{6}

Eliminar o fator comum:

2

= \frac{2 π}{3}

= 2 sin (\frac{2 π}{3}) cos (\frac{x + x + \frac{π}{6} - \frac{7 π}{6}}{2})

\frac{x + \frac{π}{6} + x - \frac{7 π}{6}}{2} = \frac{2 x - π}{2}

\frac{x + \frac{π}{6} + x - \frac{7 π}{6}}{2}

Combinar as frações usando o mínimo múltiplo comum:

- π

Aplicar a regra

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

= \frac{π - 7 π}{6}

Somar elementos similares:

π - 7 π = - 6 π

= \frac{- 6 π}{6}

Aplicar as propriedades das frações:

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

= - \frac{6 π}{6}

Dividir:

\frac{6}{6} = 1

= - π

= \frac{x - π + x}{2}

x - π + x = 2 x - π

x - π + x

Agrupar termos semelhantes

= x + x - π

Somar elementos similares:

x + x = 2 x

= 2 x - π

= \frac{2 x - π}{2}

= 2 sin (\frac{2 π}{3}) cos (\frac{2 x - π}{2})

Simplificar

sin (\frac{2 π}{3}) : \frac{3}{2}

sin (\frac{2 π}{3})

Utilizar a seguinte identidade trivial:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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}

= 2 \cdot \frac{3}{2} cos (\frac{2 x - π}{2})

Multiplicar frações:

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

= \frac{2 3}{2} cos (\frac{2 x - π}{2})

Eliminar o fator comum:

2

= cos (\frac{2 x - π}{2}) 3

= 3 cos (\frac{2 x - π}{2})

3 cos (\frac{2 x - π}{2}) = \frac{3}{2}

Dividir ambos os lados por

3

3 cos (\frac{2 x - π}{2}) = \frac{3}{2}

Dividir ambos os lados por

3

\frac{3 cos ( \frac{2 x - π}{2} )}{3} = \frac{\frac{3}{2}}{3}

Simplificar

\frac{3 cos ( \frac{2 x - π}{2} )}{3} = \frac{\frac{3}{2}}{3}

Simplificar

\frac{3 cos ( \frac{2 x - π}{2} )}{3} : cos (\frac{2 x - π}{2})

\frac{3 cos ( \frac{2 x - π}{2} )}{3}

Eliminar o fator comum:

3

= cos (\frac{2 x - π}{2})

Simplificar

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

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

Aplicar as propriedades das frações:

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

= \frac{3}{2 3}

Eliminar o fator comum:

3

= \frac{1}{2}

cos (\frac{2 x - π}{2}) = \frac{1}{2}

cos (\frac{2 x - π}{2}) = \frac{1}{2}

cos (\frac{2 x - π}{2}) = \frac{1}{2}

Soluções gerais para

cos (\frac{2 x - π}{2}) = \frac{1}{2}

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{2 x - π}{2} = \frac{π}{3} + 2 πn, \frac{2 x - π}{2} = \frac{5 π}{3} + 2 πn

\frac{2 x - π}{2} = \frac{π}{3} + 2 πn, \frac{2 x - π}{2} = \frac{5 π}{3} + 2 πn

Resolver

\frac{2 x - π}{2} = \frac{π}{3} + 2 πn : x = 2 πn + \frac{π}{2} + \frac{π}{3}

\frac{2 x - π}{2} = \frac{π}{3} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x - π}{2} = \frac{π}{3} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Simplificar

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Simplificar

\frac{2 ( 2 x - π )}{2} : 2 x - π

\frac{2 ( 2 x - π )}{2}

Dividir:

\frac{2}{2} = 1

= 2 x - π

Simplificar

2 \cdot \frac{π}{3} + 2 \cdot 2 πn : \frac{2 π}{3} + 4 πn

2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Multiplicar

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiplicar frações:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= \frac{2 π}{3} + 4 πn

2 x - π = \frac{2 π}{3} + 4 πn

2 x - π = \frac{2 π}{3} + 4 πn

2 x - π = \frac{2 π}{3} + 4 πn

Mova

π

para o lado direito

2 x - π = \frac{2 π}{3} + 4 πn

Adicionar

π

a ambos os lados

2 x - π + π = \frac{2 π}{3} + 4 πn + π

Simplificar

2 x = \frac{2 π}{3} + 4 πn + π

2 x = \frac{2 π}{3} + 4 πn + π

Dividir ambos os lados por

2

2 x = \frac{2 π}{3} + 4 πn + π

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{2 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2} : 2 πn + \frac{π}{2} + \frac{π}{3}

\frac{\frac{2 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Agrupar termos semelhantes

= \frac{π}{2} + \frac{4 πn}{2} + \frac{\frac{2 π}{3}}{2}

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Dividir:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Aplicar as propriedades das frações:

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

= \frac{2 π}{3 \cdot 2}

Multiplicar os números:

3 \cdot 2 = 6

= \frac{2 π}{6}

Eliminar o fator comum:

2

= \frac{π}{3}

= \frac{π}{2} + 2 πn + \frac{π}{3}

Agrupar termos semelhantes

= 2 πn + \frac{π}{2} + \frac{π}{3}

x = 2 πn + \frac{π}{2} + \frac{π}{3}

x = 2 πn + \frac{π}{2} + \frac{π}{3}

x = 2 πn + \frac{π}{2} + \frac{π}{3}

Resolver

\frac{2 x - π}{2} = \frac{5 π}{3} + 2 πn : x = 2 πn + \frac{π}{2} + \frac{5 π}{3}

\frac{2 x - π}{2} = \frac{5 π}{3} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x - π}{2} = \frac{5 π}{3} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

Simplificar

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

Simplificar

\frac{2 ( 2 x - π )}{2} : 2 x - π

\frac{2 ( 2 x - π )}{2}

Dividir:

\frac{2}{2} = 1

= 2 x - π

Simplificar

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn : \frac{10 π}{3} + 4 πn

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{3} = \frac{10 π}{3}

2 \cdot \frac{5 π}{3}

Multiplicar frações:

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

= \frac{5 π 2}{3}

Multiplicar os números:

5 \cdot 2 = 10

= \frac{10 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= \frac{10 π}{3} + 4 πn

2 x - π = \frac{10 π}{3} + 4 πn

2 x - π = \frac{10 π}{3} + 4 πn

2 x - π = \frac{10 π}{3} + 4 πn

Mova

π

para o lado direito

2 x - π = \frac{10 π}{3} + 4 πn

Adicionar

π

a ambos os lados

2 x - π + π = \frac{10 π}{3} + 4 πn + π

Simplificar

2 x = \frac{10 π}{3} + 4 πn + π

2 x = \frac{10 π}{3} + 4 πn + π

Dividir ambos os lados por

2

2 x = \frac{10 π}{3} + 4 πn + π

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2} : 2 πn + \frac{π}{2} + \frac{5 π}{3}

\frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Agrupar termos semelhantes

= \frac{π}{2} + \frac{4 πn}{2} + \frac{\frac{10 π}{3}}{2}

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Dividir:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{10 π}{3}}{2} = \frac{5 π}{3}

\frac{\frac{10 π}{3}}{2}

Aplicar as propriedades das frações:

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

= \frac{10 π}{3 \cdot 2}

Multiplicar os números:

3 \cdot 2 = 6

= \frac{10 π}{6}

Eliminar o fator comum:

2

= \frac{5 π}{3}

= \frac{π}{2} + 2 πn + \frac{5 π}{3}

Agrupar termos semelhantes

= 2 πn + \frac{π}{2} + \frac{5 π}{3}

x = 2 πn + \frac{π}{2} + \frac{5 π}{3}

x = 2 πn + \frac{π}{2} + \frac{5 π}{3}

x = 2 πn + \frac{π}{2} + \frac{5 π}{3}

x = 2 πn + \frac{π}{2} + \frac{π}{3}, x = 2 πn + \frac{π}{2} + \frac{5 π}{3}

Gráfico

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