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Popular Trigonometria >

cos^3(x)-1=0

Solução

cos^{3} (x) - 1 = 0

Solução

x = 2 πn

+1

Graus

x = 0^{\circ} + 36 0^{\circ} n

Passos da solução

cos^{3} (x) - 1 = 0

Usando o método de substituição

cos^{3} (x) - 1 = 0

Sea:

cos (x) = u

u^{3} - 1 = 0

u^{3} - 1 = 0 : u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

u^{3} - 1 = 0

Mova

1

para o lado direito

u^{3} - 1 = 0

Adicionar

1

a ambos os lados

u^{3} - 1 + 1 = 0 + 1

Simplificar

u^{3} = 1

u^{3} = 1

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

Simplificar

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

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

Reescrever

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

na forma complexa padrão:

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

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

Aplicar as propriedades das frações:

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

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

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

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

Simplificar

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

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

Reescrever

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

na forma complexa padrão:

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

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

Aplicar as propriedades das frações:

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

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

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

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

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

Substituir na equação

u = cos (x)

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

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

cos (x) = 1 : x = 2 πn

cos (x) = 1

Soluções gerais para

cos (x) = 1

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}

x = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

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

Sem solução

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

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

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

Sem solução

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

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

Combinar toda as soluções

x = 2 πn

Gráfico

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