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0=cos(2x)+cos(x)

Solução

0 = cos (2 x) + cos (x)

Solução

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

Graus

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n

Passos da solução

0 = cos (2 x) + cos (x)

Trocar lados

cos (2 x) + cos (x) = 0

Reeecreva usando identidades trigonométricas

cos (2 x) + cos (x)

Use a identidade da transformação de soma em produto:

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

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

Simplificar

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

2 cos (\frac{2 x + x}{2}) cos (\frac{2 x - x}{2})

Somar elementos similares:

2 x + x = 3 x

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

Somar elementos similares:

2 x - x = x

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

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

2 cos (\frac{3 x}{2}) cos (\frac{x}{2}) = 0

Resolver cada parte separadamente

cos (\frac{3 x}{2}) = 0 or cos (\frac{x}{2}) = 0

cos (\frac{3 x}{2}) = 0 : x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

cos (\frac{3 x}{2}) = 0

Soluções gerais para

cos (\frac{3 x}{2}) = 0

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

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

Resolver

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

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiplicar os números:

2 \cdot 3 = 6

= \frac{6 x}{2}

Dividir:

\frac{6}{2} = 3

= 3 x

Simplificar

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiplicar frações:

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

= \frac{π 2}{2}

Eliminar o fator comum:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

Dividir ambos os lados por

3

3 x = π + 4 πn

Dividir ambos os lados por

3

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

Simplificar

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

Resolver

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

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiplicar os números:

2 \cdot 3 = 6

= \frac{6 x}{2}

Dividir:

\frac{6}{2} = 3

= 3 x

Simplificar

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiplicar frações:

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

= \frac{3 π 2}{2}

Eliminar o fator comum:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

Dividir ambos os lados por

3

3 x = 3 π + 4 πn

Dividir ambos os lados por

3

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

Simplificar

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

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

cos (\frac{x}{2}) = 0 : x = π + 4 πn, x = 3 π + 4 πn

cos (\frac{x}{2}) = 0

Soluções gerais para

cos (\frac{x}{2}) = 0

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

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

Resolver

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

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiplicar frações:

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

= \frac{π 2}{2}

Eliminar o fator comum:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

Resolver

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

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiplicar frações:

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

= \frac{3 π 2}{2}

Eliminar o fator comum:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar os números:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = π + 4 πn, x = 3 π + 4 πn

Combinar toda as soluções

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

Junte intervalos que se sobrepoem

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

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