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2cos^2(4x)-1=0

Solución

2 cos^{2} (4 x) - 1 = 0

Solución

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}, x = \frac{3 π}{16} + \frac{πn}{2}, x = - \frac{3 π}{16} + \frac{πn}{2}

Grados

x = 11.2 5^{\circ} + 9 0^{\circ} n, x = 78.7 5^{\circ} + 9 0^{\circ} n, x = 33.7 5^{\circ} + 9 0^{\circ} n, x = - 33.7 5^{\circ} + 9 0^{\circ} n

Pasos de solución

2 cos^{2} (4 x) - 1 = 0

Usando el método de sustitución

2 cos^{2} (4 x) - 1 = 0

Sea:

cos (4 x) = u

2 u^{2} - 1 = 0

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

2 u^{2} - 1 = 0

Mover

1

al lado derecho

2 u^{2} - 1 = 0

Sumar

1

a ambos lados

2 u^{2} - 1 + 1 = 0 + 1

Simplificar

2 u^{2} = 1

2 u^{2} = 1

Dividir ambos lados entre

2

2 u^{2} = 1

Dividir ambos lados entre

2

\frac{2 u ^{2}}{2} = \frac{1}{2}

Simplificar

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = \frac{1}{2}, u = - \frac{1}{2}

Sustituir en la ecuación

u = cos (4 x)

cos (4 x) = \frac{1}{2}, cos (4 x) = - \frac{1}{2}

cos (4 x) = \frac{1}{2}, cos (4 x) = - \frac{1}{2}

cos (4 x) = \frac{1}{2} : x = \frac{π}{16} + \frac{πn}{2}, x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

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

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

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

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

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

Resolver

4 x = arccos (\frac{1}{2}) + 2 πn : x = \frac{π}{16} + \frac{πn}{2}

4 x = arccos (\frac{1}{2}) + 2 πn

Simplificar

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

arccos (\frac{1}{2}) + 2 πn

Utilizar la siguiente identidad trivial:

arccos (\frac{1}{2}) = \frac{π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{4} + 2 πn

4 x = \frac{π}{4} + 2 πn

Dividir ambos lados entre

4

4 x = \frac{π}{4} + 2 πn

Dividir ambos lados entre

4

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

Simplificar

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

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

\frac{\frac{π}{4}}{4} + \frac{2 πn}{4} : \frac{π}{16} + \frac{πn}{2}

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

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{4 \cdot 4}

Multiplicar los numeros:

4 \cdot 4 = 16

= \frac{π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

= \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

Resolver

4 x = 2 π - arccos (\frac{1}{2}) + 2 πn : x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

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

Simplificar

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

2 π - arccos (\frac{1}{2}) + 2 πn

Utilizar la siguiente identidad trivial:

arccos (\frac{1}{2}) = \frac{π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{4} + 2 πn

4 x = 2 π - \frac{π}{4} + 2 πn

Dividir ambos lados entre

4

4 x = 2 π - \frac{π}{4} + 2 πn

Dividir ambos lados entre

4

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

Simplificar

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

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

\frac{2 π}{4} - \frac{\frac{π}{4}}{4} + \frac{2 πn}{4} : \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

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

\frac{2 π}{4} = \frac{π}{2}

\frac{2 π}{4}

Eliminar los terminos comunes:

2

= \frac{π}{2}

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{4 \cdot 4}

Multiplicar los numeros:

4 \cdot 4 = 16

= \frac{π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

= \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}

cos (4 x) = - \frac{1}{2} : x = \frac{3 π}{16} + \frac{πn}{2}, x = - \frac{3 π}{16} + \frac{πn}{2}

cos (4 x) = - \frac{1}{2}

Aplicar propiedades trigonométricas inversas

cos (4 x) = - \frac{1}{2}

Soluciones generales para

cos (4 x) = - \frac{1}{2}

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

4 x = arccos (- \frac{1}{2}) + 2 πn, 4 x = - arccos (- \frac{1}{2}) + 2 πn

4 x = arccos (- \frac{1}{2}) + 2 πn, 4 x = - arccos (- \frac{1}{2}) + 2 πn

Resolver

4 x = arccos (- \frac{1}{2}) + 2 πn : x = \frac{3 π}{16} + \frac{πn}{2}

4 x = arccos (- \frac{1}{2}) + 2 πn

Simplificar

arccos (- \frac{1}{2}) + 2 πn : \frac{3 π}{4} + 2 πn

arccos (- \frac{1}{2}) + 2 πn

Utilizar la siguiente identidad trivial:

arccos (- \frac{1}{2}) = \frac{3 π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{3 π}{4} + 2 πn

4 x = \frac{3 π}{4} + 2 πn

Dividir ambos lados entre

4

4 x = \frac{3 π}{4} + 2 πn

Dividir ambos lados entre

4

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

Simplificar

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

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

\frac{\frac{3 π}{4}}{4} + \frac{2 πn}{4} : \frac{3 π}{16} + \frac{πn}{2}

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

\frac{\frac{3 π}{4}}{4} = \frac{3 π}{16}

\frac{\frac{3 π}{4}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{3 π}{4 \cdot 4}

Multiplicar los numeros:

4 \cdot 4 = 16

= \frac{3 π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

= \frac{3 π}{16} + \frac{πn}{2}

x = \frac{3 π}{16} + \frac{πn}{2}

x = \frac{3 π}{16} + \frac{πn}{2}

x = \frac{3 π}{16} + \frac{πn}{2}

Resolver

4 x = - arccos (- \frac{1}{2}) + 2 πn : x = - \frac{3 π}{16} + \frac{πn}{2}

4 x = - arccos (- \frac{1}{2}) + 2 πn

Simplificar

- arccos (- \frac{1}{2}) + 2 πn : - \frac{3 π}{4} + 2 πn

- arccos (- \frac{1}{2}) + 2 πn

Utilizar la siguiente identidad trivial:

arccos (- \frac{1}{2}) = \frac{3 π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= - \frac{3 π}{4} + 2 πn

4 x = - \frac{3 π}{4} + 2 πn

Dividir ambos lados entre

4

4 x = - \frac{3 π}{4} + 2 πn

Dividir ambos lados entre

4

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

Simplificar

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

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

- \frac{\frac{3 π}{4}}{4} + \frac{2 πn}{4} : - \frac{3 π}{16} + \frac{πn}{2}

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

\frac{\frac{3 π}{4}}{4} = \frac{3 π}{16}

\frac{\frac{3 π}{4}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{3 π}{4 \cdot 4}

Multiplicar los numeros:

4 \cdot 4 = 16

= \frac{3 π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

= - \frac{3 π}{16} + \frac{πn}{2}

x = - \frac{3 π}{16} + \frac{πn}{2}

x = - \frac{3 π}{16} + \frac{πn}{2}

x = - \frac{3 π}{16} + \frac{πn}{2}

x = \frac{3 π}{16} + \frac{πn}{2}, x = - \frac{3 π}{16} + \frac{πn}{2}

Combinar toda las soluciones

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{π}{2} - \frac{π}{16} + \frac{πn}{2}, x = \frac{3 π}{16} + \frac{πn}{2}, x = - \frac{3 π}{16} + \frac{πn}{2}

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