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Popular Trigonometria >

sin^3(x)=-2/3

Solução

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

Solução

x = - 1.06251 \dots + 2 πn, x = π + 1.06251 \dots + 2 πn

Graus

x = - 60.87741 \dots^{\circ} + 36 0^{\circ} n, x = 240.87741 \dots^{\circ} + 36 0^{\circ} n

Passos da solução

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

Usando o método de substituição

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

Sea:

sin (x) = u

u^{3} = - \frac{2}{3}

u^{3} = - \frac{2}{3} : u = - 3 \frac{2}{3}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

u^{3} = - \frac{2}{3}

Para

x^{3} = f (a)

as soluções são

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 - \frac{2}{3}, u = 3 - \frac{2}{3} \frac{- 1 + 3 i}{2}, u = 3 - \frac{2}{3} \frac{- 1 - 3 i}{2}

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

3 - \frac{2}{3}

Aplicar as propriedades dos radicais:

n - a = - n a,

n

é ímpar

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

= - 3 \frac{2}{3}

Simplificar

3 - \frac{2}{3} \frac{- 1 + 3 i}{2} : \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}

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

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

3 - \frac{2}{3}

Aplicar as propriedades dos radicais:

n - a = - n a,

n

é ímpar

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

= - 3 \frac{2}{3}

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

Multiplicar frações:

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

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

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

3 \frac{2}{3}

Aplicar a seguinte propriedade dos radicais:

n \frac{a}{b} = \frac{n a}{n b},

assumindo que

a \geq 0, b \geq 0

= \frac{3 2}{3 3}

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

Multiplicar

(- 1 + 3 i) \frac{3 2}{3 3} : \frac{3 2 ( - 1 + 3 i )}{3 3}

(- 1 + 3 i) \frac{3 2}{3 3}

Multiplicar frações:

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

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

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

Aplicar as propriedades das frações:

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

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

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

32 = 2^{\frac{1}{3}}

= \frac{2 ^{\frac{1}{3}} ( - 1 + 3 i )}{2 3 3}

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3 i}{3 3 \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrair:

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

= - \frac{- 1 + 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}}

Racionalizar

- \frac{- 1 + 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}} : - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

- \frac{- 1 + 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}}

Multiplicar pelo conjugado

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

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

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}} = 3 \cdot 2^{\frac{2}{3}}

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 3^{1}

Aplicar a regra

a^{1} = a

= 3

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

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

Multiplicar pelo conjugado

\frac{3 2}{3 2}

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

3 \cdot 2^{\frac{2}{3}} 32 = 6

3 \cdot 2^{\frac{2}{3}} 32

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

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

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 2^{1}

Aplicar a regra

a^{1} = a

= 2

= 3 \cdot 2

Multiplicar os números:

3 \cdot 2 = 6

= 6

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

Reescrever

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

na forma complexa padrão:

\frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} - \frac{6 3 3 2}{2} i

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

Cancelar

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6} : \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

Fatorar

6 : 2 \cdot 3

Fatorar

6 = 2 \cdot 3

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

Cancelar

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{2 \cdot 3} : \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{2 \cdot 3}

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtrair:

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

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

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

32 = 2^{\frac{1}{3}}

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

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3 i}{3 ^{\frac{1}{3}} \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrair:

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

= \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

= \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

3^{\frac{1}{3}} = 33

Aplicar as propriedades dos radicais:

a^{\frac{1}{n}} = n a

3^{\frac{1}{3}} = 33

= \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

= - \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

= - \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

Aplicar as propriedades das frações:

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

\frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3} = - (- \frac{1}{2 ^{\frac{2}{3}} 3 3}) - (\frac{3 i}{2 ^{\frac{2}{3}} 3 3})

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

Remover os parênteses:

(a) = a, - (- a) = a

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

Cancelar

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Cancelar

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

33 = 3^{\frac{1}{3}}, 3 = 3^{\frac{1}{2}}

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

Aplicar as propriedades dos expoentes:

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

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

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

Subtrair:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2 ^{\frac{2}{3}}}

Aplicar as propriedades dos radicais:

a^{\frac{1}{n}} = n a

3^{\frac{1}{6}} = 63

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} - \frac{6 3 i}{2 ^{\frac{2}{3}}}

- \frac{6 3}{2 ^{\frac{2}{3}}} = - \frac{6 3 3 2}{2}

- \frac{6 3}{2 ^{\frac{2}{3}}}

Multiplicar pelo conjugado

\frac{3 2}{3 2}

= - \frac{6 3 3 2}{2 ^{\frac{2}{3}} 3 2}

2^{\frac{2}{3}} 32 = 2

2^{\frac{2}{3}} 32

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 2^{\frac{2}{3} + \frac{1}{3}}

Simplificar

\frac{2}{3} + \frac{1}{3}

em uma fração:

1

\frac{2}{3} + \frac{1}{3}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 2^{1}

Aplicar a regra

a^{1} = a

= 2

= - \frac{6 3 3 2}{2}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} - \frac{6 3 3 2}{2} i

\frac{1}{2 ^{\frac{2}{3}} 3 3} = \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

\frac{1}{2 ^{\frac{2}{3}} 3 3}

Multiplicar pelo conjugado

\frac{3 2}{3 2}

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

1 \cdot 32 = 32

2^{\frac{2}{3}} 3332 = 233

2^{\frac{2}{3}} 3332

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 33 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 2^{1}

Aplicar a regra

a^{1} = a

= 2

= 233

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

Multiplicar pelo conjugado

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

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

233 \cdot 3^{\frac{2}{3}} = 6

233 \cdot 3^{\frac{2}{3}}

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 3^{1}

Aplicar a regra

a^{1} = a

= 3

= 2 \cdot 3

Multiplicar os números:

2 \cdot 3 = 6

= 6

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} - \frac{6 3 3 2}{2} i

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} - \frac{6 3 3 2}{2} i

Simplificar

3 - \frac{2}{3} \frac{- 1 - 3 i}{2} : \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

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

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

3 - \frac{2}{3}

Aplicar as propriedades dos radicais:

n - a = - n a,

n

é ímpar

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

= - 3 \frac{2}{3}

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

Multiplicar frações:

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

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

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

3 \frac{2}{3}

Aplicar a seguinte propriedade dos radicais:

n \frac{a}{b} = \frac{n a}{n b},

assumindo que

a \geq 0, b \geq 0

= \frac{3 2}{3 3}

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

Multiplicar

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

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

Multiplicar frações:

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

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

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

Aplicar as propriedades das frações:

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

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

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

32 = 2^{\frac{1}{3}}

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

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 - 3 i}{3 3 \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrair:

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

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

Racionalizar

- \frac{- 1 - 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}} : - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

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

Multiplicar pelo conjugado

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

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

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}} = 3 \cdot 2^{\frac{2}{3}}

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 3^{1}

Aplicar a regra

a^{1} = a

= 3

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

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

Multiplicar pelo conjugado

\frac{3 2}{3 2}

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

3 \cdot 2^{\frac{2}{3}} 32 = 6

3 \cdot 2^{\frac{2}{3}} 32

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

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

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 2^{1}

Aplicar a regra

a^{1} = a

= 2

= 3 \cdot 2

Multiplicar os números:

3 \cdot 2 = 6

= 6

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

Reescrever

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

na forma complexa padrão:

\frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} + \frac{6 3 3 2}{2} i

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

Cancelar

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6} : \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} 3 3}

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

Fatorar

6 : 2 \cdot 3

Fatorar

6 = 2 \cdot 3

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

Cancelar

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

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{2 \cdot 3}

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtrair:

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

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

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

32 = 2^{\frac{1}{3}}

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

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 - 3 i}{3 ^{\frac{1}{3}} \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrair:

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

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

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

3^{\frac{1}{3}} = 33

Aplicar as propriedades dos radicais:

a^{\frac{1}{n}} = n a

3^{\frac{1}{3}} = 33

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

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

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

Aplicar as propriedades das frações:

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

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

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

Aplicar a regra

- (- a) = a

= \frac{1}{2 ^{\frac{2}{3}} 3 3} + \frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Cancelar

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Cancelar

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

33 = 3^{\frac{1}{3}}, 3 = 3^{\frac{1}{2}}

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

Aplicar as propriedades dos expoentes:

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

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

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

Subtrair:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2 ^{\frac{2}{3}}}

Aplicar as propriedades dos radicais:

a^{\frac{1}{n}} = n a

3^{\frac{1}{6}} = 63

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} + \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{6 3}{2 ^{\frac{2}{3}}} = \frac{6 3 3 2}{2}

\frac{6 3}{2 ^{\frac{2}{3}}}

Multiplicar pelo conjugado

\frac{3 2}{3 2}

= \frac{6 3 3 2}{2 ^{\frac{2}{3}} 3 2}

2^{\frac{2}{3}} 32 = 2

2^{\frac{2}{3}} 32

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 2^{\frac{2}{3} + \frac{1}{3}}

Simplificar

\frac{2}{3} + \frac{1}{3}

em uma fração:

1

\frac{2}{3} + \frac{1}{3}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 2^{1}

Aplicar a regra

a^{1} = a

= 2

= \frac{6 3 3 2}{2}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} + \frac{6 3 3 2}{2} i

\frac{1}{2 ^{\frac{2}{3}} 3 3} = \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

\frac{1}{2 ^{\frac{2}{3}} 3 3}

Multiplicar pelo conjugado

\frac{3 2}{3 2}

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

1 \cdot 32 = 32

2^{\frac{2}{3}} 3332 = 233

2^{\frac{2}{3}} 3332

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 33 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 2^{1}

Aplicar a regra

a^{1} = a

= 2

= 233

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

Multiplicar pelo conjugado

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

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

233 \cdot 3^{\frac{2}{3}} = 6

233 \cdot 3^{\frac{2}{3}}

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Combinar as frações usando o mínimo múltiplo comum:

1

Aplicar a regra

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

= \frac{2 + 1}{3}

Somar:

2 + 1 = 3

= \frac{3}{3}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 3^{1}

Aplicar a regra

a^{1} = a

= 3

= 2 \cdot 3

Multiplicar os números:

2 \cdot 3 = 6

= 6

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} + \frac{6 3 3 2}{2} i

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} + \frac{6 3 3 2}{2} i

u = - 3 \frac{2}{3}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

Substituir na equação

u = sin (x)

sin (x) = - 3 \frac{2}{3}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

sin (x) = - 3 \frac{2}{3}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

sin (x) = - 3 \frac{2}{3} : x = arcsin (- 3 \frac{2}{3}) + 2 πn, x = π + arcsin (3 \frac{2}{3}) + 2 πn

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

Aplique as propriedades trigonométricas inversas

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

Soluções gerais para

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

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

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

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

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2} :

Sem solução

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}

Sem solu \overset{c}{\c} \tilde{a} o

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2} :

Sem solução

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

Sem solu \overset{c}{\c} \tilde{a} o

Combinar toda as soluções

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

Mostrar soluções na forma decimal

x = - 1.06251 \dots + 2 πn, x = π + 1.06251 \dots + 2 πn

Gráfico

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