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Popular Trigonometria >

cos(2x)=cos(1/2)

Solução

cos (2 x) = cos (\frac{1}{2})

Solução

x = \frac{1}{4} + πn, x = π - \frac{1}{4} + πn

Graus

x = 14.32394 \dots^{\circ} + 18 0^{\circ} n, x = 165.67605 \dots^{\circ} + 18 0^{\circ} n

Passos da solução

cos (2 x) = cos (\frac{1}{2})

Aplique as propriedades trigonométricas inversas

cos (2 x) = cos (\frac{1}{2})

Soluções gerais para

cos (2 x) = cos (\frac{1}{2})

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 x = arccos (cos (\frac{1}{2})) + 2 πn, 2 x = 2 π - arccos (cos (\frac{1}{2})) + 2 πn

2 x = arccos (cos (\frac{1}{2})) + 2 πn, 2 x = 2 π - arccos (cos (\frac{1}{2})) + 2 πn

Resolver

2 x = arccos (cos (\frac{1}{2})) + 2 πn : x = \frac{1}{4} + πn

2 x = arccos (cos (\frac{1}{2})) + 2 πn

Dividir ambos os lados por

2

2 x = arccos (cos (\frac{1}{2})) + 2 πn

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2} : \frac{1}{4} + πn

\frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2}

\frac{arccos ( cos ( \frac{1}{2} ) )}{2} = \frac{1}{4}

\frac{arccos ( cos ( \frac{1}{2} ) )}{2}

arccos (cos (\frac{1}{2})) = \frac{1}{2}

arccos (cos (\frac{1}{2}))

Para

0 \leq x \leq π

a rccos (cos (x)) = x

0 \leq \frac{1}{2} \leq π

= \frac{1}{2}

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

Aplicar as propriedades das frações:

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

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

Multiplicar os números:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{1}{4} + \frac{2 πn}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{1}{4} + πn

x = \frac{1}{4} + πn

x = \frac{1}{4} + πn

x = \frac{1}{4} + πn

Resolver

2 x = 2 π - arccos (cos (\frac{1}{2})) + 2 πn : x = π - \frac{1}{4} + πn

2 x = 2 π - arccos (cos (\frac{1}{2})) + 2 πn

Dividir ambos os lados por

2

2 x = 2 π - arccos (cos (\frac{1}{2})) + 2 πn

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{2 π}{2} - \frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2} : π - \frac{1}{4} + πn

\frac{2 π}{2} - \frac{arccos ( cos ( \frac{1}{2} ) )}{2} + \frac{2 πn}{2}

\frac{2 π}{2} = π

\frac{2 π}{2}

Dividir:

\frac{2}{2} = 1

= π

\frac{arccos ( cos ( \frac{1}{2} ) )}{2} = \frac{1}{4}

\frac{arccos ( cos ( \frac{1}{2} ) )}{2}

arccos (cos (\frac{1}{2})) = \frac{1}{2}

arccos (cos (\frac{1}{2}))

Para

0 \leq x \leq π

a rccos (cos (x)) = x

0 \leq \frac{1}{2} \leq π

= \frac{1}{2}

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

Aplicar as propriedades das frações:

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

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

Multiplicar os números:

2 \cdot 2 = 4

= \frac{1}{4}

= π - \frac{1}{4} + \frac{2 πn}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= π - \frac{1}{4} + πn

x = π - \frac{1}{4} + πn

x = π - \frac{1}{4} + πn

x = π - \frac{1}{4} + πn

x = \frac{1}{4} + πn, x = π - \frac{1}{4} + πn

Gráfico

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