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2cos^2(x)(1+2cos^2(x))=2

Solución

2 cos^{2} (x) (1 + 2 cos^{2} (x)) = 2

Solución

x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

Grados

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = - 13 5^{\circ} + 36 0^{\circ} n

Pasos de solución

2 cos^{2} (x) (1 + 2 cos^{2} (x)) = 2

Usando el método de sustitución

2 cos^{2} (x) (1 + 2 cos^{2} (x)) = 2

Sea:

cos (x) = u

2 u^{2} (1 + 2 u^{2}) = 2

2 u^{2} (1 + 2 u^{2}) = 2 : u = \frac{1}{2}, u = - \frac{1}{2}, u = i, u = - i

2 u^{2} (1 + 2 u^{2}) = 2

Desarrollar

2 u^{2} (1 + 2 u^{2}) : 2 u^{2} + 4 u^{4}

2 u^{2} (1 + 2 u^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 2 u^{2}, b = 1, c = 2 u^{2}

= 2 u^{2} \cdot 1 + 2 u^{2} \cdot 2 u^{2}

= 2 \cdot 1 \cdot u^{2} + 2 \cdot 2 u^{2} u^{2}

Simplificar

2 \cdot 1 \cdot u^{2} + 2 \cdot 2 u^{2} u^{2} : 2 u^{2} + 4 u^{4}

2 \cdot 1 \cdot u^{2} + 2 \cdot 2 u^{2} u^{2}

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

2 \cdot 1 \cdot u^{2}

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 u^{2}

2 \cdot 2 u^{2} u^{2} = 4 u^{4}

2 \cdot 2 u^{2} u^{2}

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 u^{2} u^{2}

Aplicar las leyes de los exponentes:

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

u^{2} u^{2} = u^{2 + 2}

= 4 u^{2 + 2}

Sumar:

2 + 2 = 4

= 4 u^{4}

= 2 u^{2} + 4 u^{4}

= 2 u^{2} + 4 u^{4}

2 u^{2} + 4 u^{4} = 2

Mover

2

al lado izquierdo

2 u^{2} + 4 u^{4} = 2

Restar

2

de ambos lados

2 u^{2} + 4 u^{4} - 2 = 2 - 2

Simplificar

2 u^{2} + 4 u^{4} - 2 = 0

2 u^{2} + 4 u^{4} - 2 = 0

Escribir en la forma binómica

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

4 u^{4} + 2 u^{2} - 2 = 0

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

4 v^{2} + 2 v - 2 = 0

Resolver

4 v^{2} + 2 v - 2 = 0 : v = \frac{1}{2}, v = - 1

4 v^{2} + 2 v - 2 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

4 v^{2} + 2 v - 2 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 4, b = 2, c = - 2

v_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 4 ( - 2 )}{2 \cdot 4}

v_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 4 ( - 2 )}{2 \cdot 4}

2^{2} - 4 \cdot 4 (- 2) = 6

2^{2} - 4 \cdot 4 (- 2)

Aplicar la regla

- (- a) = a

= 2^{2} + 4 \cdot 4 \cdot 2

Multiplicar los numeros:

4 \cdot 4 \cdot 2 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

Sumar:

4 + 32 = 36

= 36

Descomponer el número en factores primos:

36 = 6^{2}

= 6^{2}

Aplicar las leyes de los exponentes:

6^{2} = 6

= 6

v_{1, 2} = \frac{- 2 \pm 6}{2 \cdot 4}

Separar las soluciones

v_{1} = \frac{- 2 + 6}{2 \cdot 4}, v_{2} = \frac{- 2 - 6}{2 \cdot 4}

v = \frac{- 2 + 6}{2 \cdot 4} : \frac{1}{2}

\frac{- 2 + 6}{2 \cdot 4}

Sumar/restar lo siguiente:

- 2 + 6 = 4

= \frac{4}{2 \cdot 4}

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{4}{8}

Eliminar los terminos comunes:

4

= \frac{1}{2}

v = \frac{- 2 - 6}{2 \cdot 4} : - 1

\frac{- 2 - 6}{2 \cdot 4}

Restar:

- 2 - 6 = - 8

= \frac{- 8}{2 \cdot 4}

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{- 8}{8}

Aplicar las propiedades de las fracciones:

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

= - \frac{8}{8}

Aplicar la regla

\frac{a}{a} = 1

= - 1

Las soluciones a la ecuación de segundo grado son:

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

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

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

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

Resolver

u^{2} = - 1 : u = i, u = - i

u^{2} = - 1

Para

x^{2} = f (a)

las soluciones son

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

u = - 1, u = - - 1

Simplificar

- 1 : i

- 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= i

Simplificar

- - 1 : - i

- - 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= - i

u = i, u = - i

Las soluciones son

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

Sustituir en la ecuación

u = cos (x)

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

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

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

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

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

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

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

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

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

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

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

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

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

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

cos (x) = i :

Sin solución

cos (x) = i

Sin soluci \overset{o}{ˊ} n

cos (x) = - i :

Sin solución

cos (x) = - i

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

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

Mostrar soluciones en forma decimal

x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

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