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Popular Trigonometría >

5cos^2(2x)+4cos^2(x)-5=0

Solución

5 cos^{2} (2 x) + 4 cos^{2} (x) - 5 = 0

Solución

x = 0.46364 \dots + 2 πn, x = 2 π - 0.46364 \dots + 2 πn, x = 2.67794 \dots + 2 πn, x = - 2.67794 \dots + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Grados

x = 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 333.43494 \dots^{\circ} + 36 0^{\circ} n, x = 153.43494 \dots^{\circ} + 36 0^{\circ} n, x = - 153.43494 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Pasos de solución

5 cos^{2} (2 x) + 4 cos^{2} (x) - 5 = 0

Re-escribir usando identidades trigonométricas

- 5 + 4 cos^{2} (x) + 5 cos^{2} (2 x)

Utilizar la identidad trigonométrica del ángulo doble:

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

= - 5 + 4 cos^{2} (x) + 5 (2 cos^{2} (x) - 1)^{2}

Simplificar

- 5 + 4 cos^{2} (x) + 5 (2 cos^{2} (x) - 1)^{2} : 20 cos^{4} (x) - 16 cos^{2} (x)

- 5 + 4 cos^{2} (x) + 5 (2 cos^{2} (x) - 1)^{2}

(2 cos^{2} (x) - 1)^{2} : 4 cos^{4} (x) - 4 cos^{2} (x) + 1

Aplicar la formula del binomio al cuadrado:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2 cos^{2} (x), b = 1

= (2 cos^{2} (x))^{2} - 2 \cdot 2 cos^{2} (x) \cdot 1 + 1^{2}

Simplificar

(2 cos^{2} (x))^{2} - 2 \cdot 2 cos^{2} (x) \cdot 1 + 1^{2} : 4 cos^{4} (x) - 4 cos^{2} (x) + 1

(2 cos^{2} (x))^{2} - 2 \cdot 2 cos^{2} (x) \cdot 1 + 1^{2}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= (2 cos^{2} (x))^{2} - 2 \cdot 2 \cdot 1 \cdot cos^{2} (x) + 1

(2 cos^{2} (x))^{2} = 4 cos^{4} (x)

(2 cos^{2} (x))^{2}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

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

(cos^{2} (x))^{2} : cos^{4} (x)

Aplicar las leyes de los exponentes:

(a^{b})^{c} = a^{b c}

= cos^{2 \cdot 2} (x)

Multiplicar los numeros:

2 \cdot 2 = 4

= cos^{4} (x)

= 2^{2} cos^{4} (x)

2^{2} = 4

= 4 cos^{4} (x)

2 \cdot 2 cos^{2} (x) \cdot 1 = 4 cos^{2} (x)

2 \cdot 2 cos^{2} (x) \cdot 1

Multiplicar los numeros:

2 \cdot 2 \cdot 1 = 4

= 4 cos^{2} (x)

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

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

= - 5 + 4 cos^{2} (x) + 5 (4 cos^{4} (x) - 4 cos^{2} (x) + 1)

Expandir

5 (4 cos^{4} (x) - 4 cos^{2} (x) + 1) : 20 cos^{4} (x) - 20 cos^{2} (x) + 5

5 (4 cos^{4} (x) - 4 cos^{2} (x) + 1)

Aplicar la siguiente regla de productos notables

= 5 \cdot 4 cos^{4} (x) + 5 (- 4 cos^{2} (x)) + 5 \cdot 1

Aplicar las reglas de los signos

+ (- a) = - a

= 5 \cdot 4 cos^{4} (x) - 5 \cdot 4 cos^{2} (x) + 5 \cdot 1

Simplificar

5 \cdot 4 cos^{4} (x) - 5 \cdot 4 cos^{2} (x) + 5 \cdot 1 : 20 cos^{4} (x) - 20 cos^{2} (x) + 5

5 \cdot 4 cos^{4} (x) - 5 \cdot 4 cos^{2} (x) + 5 \cdot 1

Multiplicar los numeros:

5 \cdot 4 = 20

= 20 cos^{4} (x) - 20 cos^{2} (x) + 5 \cdot 1

Multiplicar los numeros:

5 \cdot 1 = 5

= 20 cos^{4} (x) - 20 cos^{2} (x) + 5

= 20 cos^{4} (x) - 20 cos^{2} (x) + 5

= - 5 + 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) + 5

Simplificar

- 5 + 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) + 5 : 20 cos^{4} (x) - 16 cos^{2} (x)

- 5 + 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) + 5

Agrupar términos semejantes

= 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) - 5 + 5

Sumar elementos similares:

4 cos^{2} (x) - 20 cos^{2} (x) = - 16 cos^{2} (x)

= - 16 cos^{2} (x) + 20 cos^{4} (x) - 5 + 5

- 5 + 5 = 0

= 20 cos^{4} (x) - 16 cos^{2} (x)

= 20 cos^{4} (x) - 16 cos^{2} (x)

= 20 cos^{4} (x) - 16 cos^{2} (x)

- 16 cos^{2} (x) + 20 cos^{4} (x) = 0

Usando el método de sustitución

- 16 cos^{2} (x) + 20 cos^{4} (x) = 0

Sea:

cos (x) = u

- 16 u^{2} + 20 u^{4} = 0

- 16 u^{2} + 20 u^{4} = 0 : u = \frac{2 5}{5}, u = - \frac{2 5}{5}, u = 0

- 16 u^{2} + 20 u^{4} = 0

Escribir en la forma binómica

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

20 u^{4} - 16 u^{2} = 0

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

20 v^{2} - 16 v = 0

Resolver

20 v^{2} - 16 v = 0 : v = \frac{4}{5}, v = 0

20 v^{2} - 16 v = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

20 v^{2} - 16 v = 0

Formula general para ecuaciones de segundo grado:

Para

a = 20, b = - 16, c = 0

v_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 20 \cdot 0}{2 \cdot 20}

v_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 20 \cdot 0}{2 \cdot 20}

(- 16)^{2} - 4 \cdot 20 \cdot 0 = 16

(- 16)^{2} - 4 \cdot 20 \cdot 0

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

(- 16)^{2} = 1 6^{2}

= 1 6^{2} - 4 \cdot 20 \cdot 0

Aplicar la regla

0 \cdot a = 0

= 1 6^{2} - 0

1 6^{2} - 0 = 1 6^{2}

= 1 6^{2}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0

= 16

v_{1, 2} = \frac{- ( - 16 ) \pm 16}{2 \cdot 20}

Separar las soluciones

v_{1} = \frac{- ( - 16 ) + 16}{2 \cdot 20}, v_{2} = \frac{- ( - 16 ) - 16}{2 \cdot 20}

v = \frac{- ( - 16 ) + 16}{2 \cdot 20} : \frac{4}{5}

\frac{- ( - 16 ) + 16}{2 \cdot 20}

Aplicar la regla

- (- a) = a

= \frac{16 + 16}{2 \cdot 20}

Sumar:

16 + 16 = 32

= \frac{32}{2 \cdot 20}

Multiplicar los numeros:

2 \cdot 20 = 40

= \frac{32}{40}

Eliminar los terminos comunes:

8

= \frac{4}{5}

v = \frac{- ( - 16 ) - 16}{2 \cdot 20} : 0

\frac{- ( - 16 ) - 16}{2 \cdot 20}

Aplicar la regla

- (- a) = a

= \frac{16 - 16}{2 \cdot 20}

Restar:

16 - 16 = 0

= \frac{0}{2 \cdot 20}

Multiplicar los numeros:

2 \cdot 20 = 40

= \frac{0}{40}

Aplicar la regla

\frac{0}{a} = 0, a \neq = 0

= 0

Las soluciones a la ecuación de segundo grado son:

v = \frac{4}{5}, v = 0

v = \frac{4}{5}, v = 0

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

u^{2} = \frac{4}{5} : u = \frac{2 5}{5}, u = - \frac{2 5}{5}

u^{2} = \frac{4}{5}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{4}{5}, u = - \frac{4}{5}

\frac{4}{5} = \frac{2 5}{5}

\frac{4}{5}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{4}{5}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{2}{5}

Racionalizar

\frac{2}{5} : \frac{2 5}{5}

\frac{2}{5}

Multiplicar por el conjugado

\frac{5}{5}

= \frac{2 5}{5 5}

55 = 5

55

Aplicar las leyes de los exponentes:

a a = a

55 = 5

= 5

= \frac{2 5}{5}

= \frac{2 5}{5}

- \frac{4}{5} = - \frac{2 5}{5}

- \frac{4}{5}

Simplificar

\frac{4}{5} : \frac{2}{5}

\frac{4}{5}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{4}{5}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{2}{5}

= - \frac{2}{5}

Racionalizar

- \frac{2}{5} : - \frac{2 5}{5}

- \frac{2}{5}

Multiplicar por el conjugado

\frac{5}{5}

= - \frac{2 5}{5 5}

55 = 5

55

Aplicar las leyes de los exponentes:

a a = a

55 = 5

= 5

= - \frac{2 5}{5}

= - \frac{2 5}{5}

u = \frac{2 5}{5}, u = - \frac{2 5}{5}

Resolver

u^{2} = 0 : u = 0

u^{2} = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

u = 0

Las soluciones son

u = \frac{2 5}{5}, u = - \frac{2 5}{5}, u = 0

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{2 5}{5}, cos (x) = - \frac{2 5}{5}, cos (x) = 0

cos (x) = \frac{2 5}{5}, cos (x) = - \frac{2 5}{5}, cos (x) = 0

cos (x) = \frac{2 5}{5} : x = arccos (\frac{2 5}{5}) + 2 πn, x = 2 π - arccos (\frac{2 5}{5}) + 2 πn

cos (x) = \frac{2 5}{5}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{2 5}{5}

Soluciones generales para

cos (x) = \frac{2 5}{5}

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

x = arccos (\frac{2 5}{5}) + 2 πn, x = 2 π - arccos (\frac{2 5}{5}) + 2 πn

x = arccos (\frac{2 5}{5}) + 2 πn, x = 2 π - arccos (\frac{2 5}{5}) + 2 πn

cos (x) = - \frac{2 5}{5} : x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

cos (x) = - \frac{2 5}{5}

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{2 5}{5}

Soluciones generales para

cos (x) = - \frac{2 5}{5}

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

x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

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

cos (x) = 0

Soluciones generales para

cos (x) = 0

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

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

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

Combinar toda las soluciones

x = arccos (\frac{2 5}{5}) + 2 πn, x = 2 π - arccos (\frac{2 5}{5}) + 2 πn, x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Mostrar soluciones en forma decimal

x = 0.46364 \dots + 2 πn, x = 2 π - 0.46364 \dots + 2 πn, x = 2.67794 \dots + 2 πn, x = - 2.67794 \dots + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Gráfica

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