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6tan^2(x)-2cos^2(x)=cos(2x)

Solución

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

Solución

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Grados

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Pasos de solución

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

Restar

cos (2 x)

de ambos lados

6 tan^{2} (x) - 2 cos^{2} (x) - cos (2 x) = 0

Expresar con seno, coseno

- cos (2 x) - 2 cos^{2} (x) + 6 tan^{2} (x)

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2}

Simplificar

- cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2} : \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

- cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2}

6 (\frac{sin ( x )}{cos ( x )})^{2} = \frac{6 sin ^{2} ( x )}{cos ^{2} ( x )}

6 (\frac{sin ( x )}{cos ( x )})^{2}

(\frac{sin ( x )}{cos ( x )})^{2} = \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

(\frac{sin ( x )}{cos ( x )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

= 6 \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Multiplicar fracciones:

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

= \frac{sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

= - cos (2 x) - 2 cos^{2} (x) + \frac{6 sin ^{2} ( x )}{cos ^{2} ( x )}

Convertir a fracción:

cos (2 x) = \frac{cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )}, 2 cos^{2} (x) = \frac{2 cos ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

= - \frac{cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} - \frac{2 cos ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- cos ( 2 x ) cos ^{2} ( x ) - 2 cos ^{2} ( x ) cos ^{2} ( x ) + sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6 = - cos^{2} (x) cos (2 x) - 2 cos^{4} (x) + 6 sin^{2} (x)

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6

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

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

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

2 + 2 = 4

= 2 cos^{4} (x)

= - cos^{2} (x) cos (2 x) - 2 cos^{4} (x) + 6 sin^{2} (x)

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{- 2 cos ^{4} ( x ) + 6 sin ^{2} ( x ) - cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x) = 0

Re-escribir usando identidades trigonométricas

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x)

Utilizar la identidad trigonométrica del ángulo doble:

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

= - 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

Simplificar

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x)) : - 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - cos^{2} (x), b = cos^{2} (x), c = sin^{2} (x)

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

Aplicar las reglas de los signos

- (- a) = a

= - cos^{2} (x) cos^{2} (x) + cos^{2} (x) sin^{2} (x)

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

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

Aplicar las leyes de los exponentes:

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

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

= cos^{2 + 2} (x)

Sumar:

2 + 2 = 4

= cos^{4} (x)

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

= - 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{4} (x) + cos^{2} (x) sin^{2} (x)

Sumar elementos similares:

- 2 cos^{4} (x) - cos^{4} (x) = - 3 cos^{4} (x)

= - 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

= - 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

Utilizar la identidad pitagórica:

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

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

= - 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x))

Simplificar

- 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x)) : - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

- 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x))

Expandir

6 (1 - cos^{2} (x)) : 6 - 6 cos^{2} (x)

6 (1 - cos^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 6, b = 1, c = cos^{2} (x)

= 6 \cdot 1 - 6 cos^{2} (x)

Multiplicar los numeros:

6 \cdot 1 = 6

= 6 - 6 cos^{2} (x)

= - 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) (1 - cos^{2} (x))

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

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

Simplificar

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

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

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

1 \cdot cos^{2} (x)

Multiplicar:

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

= cos^{2} (x)

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

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

Aplicar las leyes de los exponentes:

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

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

= cos^{2 + 2} (x)

Sumar:

2 + 2 = 4

= cos^{4} (x)

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

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

= - 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x)

Simplificar

- 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x) : - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

- 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x)

Agrupar términos semejantes

= - 3 cos^{4} (x) - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x) + 6

Sumar elementos similares:

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

= - 3 cos^{4} (x) - 5 cos^{2} (x) - cos^{4} (x) + 6

Sumar elementos similares:

- 3 cos^{4} (x) - cos^{4} (x) = - 4 cos^{4} (x)

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

Usando el método de sustitución

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

Sea:

cos (x) = u

6 - 4 u^{4} - 5 u^{2} = 0

6 - 4 u^{4} - 5 u^{2} = 0 : u = 2 i, u = - 2 i, u = \frac{3}{2}, u = - \frac{3}{2}

6 - 4 u^{4} - 5 u^{2} = 0

Escribir en la forma binómica

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

- 4 u^{4} - 5 u^{2} + 6 = 0

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

- 4 v^{2} - 5 v + 6 = 0

Resolver

- 4 v^{2} - 5 v + 6 = 0 : v = - 2, v = \frac{3}{4}

- 4 v^{2} - 5 v + 6 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

- 4 v^{2} - 5 v + 6 = 0

Formula general para ecuaciones de segundo grado:

Para

a = - 4, b = - 5, c = 6

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

(- 5)^{2} - 4 (- 4) \cdot 6 = 11

(- 5)^{2} - 4 (- 4) \cdot 6

Aplicar la regla

- (- a) = a

= (- 5)^{2} + 4 \cdot 4 \cdot 6

Aplicar las leyes de los exponentes:

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

n

es par

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 4 \cdot 6

Multiplicar los numeros:

4 \cdot 4 \cdot 6 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

Sumar:

25 + 96 = 121

= 121

Descomponer el número en factores primos:

121 = 1 1^{2}

= 1 1^{2}

Aplicar las leyes de los exponentes:

1 1^{2} = 11

= 11

v_{1, 2} = \frac{- ( - 5 ) \pm 11}{2 ( - 4 )}

Separar las soluciones

v_{1} = \frac{- ( - 5 ) + 11}{2 ( - 4 )}, v_{2} = \frac{- ( - 5 ) - 11}{2 ( - 4 )}

v = \frac{- ( - 5 ) + 11}{2 ( - 4 )} : - 2

\frac{- ( - 5 ) + 11}{2 ( - 4 )}

Quitar los parentesis:

(- a) = - a, - (- a) = a

= \frac{5 + 11}{- 2 \cdot 4}

Sumar:

5 + 11 = 16

= \frac{16}{- 2 \cdot 4}

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{16}{- 8}

Aplicar las propiedades de las fracciones:

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

= - \frac{16}{8}

Dividir:

\frac{16}{8} = 2

= - 2

v = \frac{- ( - 5 ) - 11}{2 ( - 4 )} : \frac{3}{4}

\frac{- ( - 5 ) - 11}{2 ( - 4 )}

Quitar los parentesis:

(- a) = - a, - (- a) = a

= \frac{5 - 11}{- 2 \cdot 4}

Restar:

5 - 11 = - 6

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

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{- 6}{- 8}

Aplicar las propiedades de las fracciones:

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

= \frac{6}{8}

Eliminar los terminos comunes:

2

= \frac{3}{4}

Las soluciones a la ecuación de segundo grado son:

v = - 2, v = \frac{3}{4}

v = - 2, v = \frac{3}{4}

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

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

u^{2} = - 2

Para

x^{2} = f (a)

las soluciones son

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

u = - 2, u = - - 2

Simplificar

- 2 : 2 i

- 2

Aplicar las leyes de los exponentes:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= 2 i

Simplificar

- - 2 : - 2 i

- - 2

Simplificar

- 2 : 2 i

- 2

Aplicar las leyes de los exponentes:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

Resolver

u^{2} = \frac{3}{4} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{4}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{3}{4}

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{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Simplificar

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{3}{4}

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{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

Las soluciones son

u = 2 i, u = - 2 i, u = \frac{3}{2}, u = - \frac{3}{2}

Sustituir en la ecuación

u = cos (x)

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

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

cos (x) = 2 i :

Sin solución

cos (x) = 2 i

Sin soluci \overset{o}{ˊ} n

cos (x) = - 2 i :

Sin solución

cos (x) = - 2 i

Sin soluci \overset{o}{ˊ} n

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

Soluciones generales para

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

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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

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

Soluciones generales para

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

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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combinar toda las soluciones

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Ejemplos populares

2cos^2(x)-3sin(x)-3=0

2 cos^{2} (x) - 3 sin (x) - 3 = 0

6tan(θ)+5=0

6 tan (θ) + 5 = 0

solvefor x,y=arcsin(x)resolver para

x, y = arcsin (x)

tan(a)= 5/12

tan (a) = \frac{5}{12}

2sin(x)-3=-csc(x)

2 sin (x) - 3 = - csc (x)