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3sin^4(x)+cos^4(x)=1

Solución

3 sin^{4} (x) + cos^{4} (x) = 1

Solución

x = 2 πn, x = π + 2 πn, 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 = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, 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

3 sin^{4} (x) + cos^{4} (x) = 1

Restar

1

de ambos lados

3 sin^{4} (x) + cos^{4} (x) - 1 = 0

Aplicar regla de los exponentes:

a^{b} = a^{2} a^{b - 2}

- 1 + cos^{4} (x) + 3 sin^{2} (x) sin^{2} (x) = 0

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

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

3 (1 - cos^{2} (x)) (1 - cos^{2} (x))

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

1 + 1 = 2

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

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

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

Aplicar la formula del binomio al cuadrado:

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

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

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

Simplificar

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

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

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 cos^{2} (x)

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

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

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)

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

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

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

Expandir

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

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

Aplicar la siguiente regla de productos notables

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

Aplicar las reglas de los signos

+ (- a) = - a

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

Simplificar

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

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

Multiplicar los numeros:

3 \cdot 1 = 3

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

Multiplicar los numeros:

3 \cdot 2 = 6

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

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

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

Simplificar

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

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

Agrupar términos semejantes

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

Sumar elementos similares:

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

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

Sumar/restar lo siguiente:

- 1 + 3 = 2

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

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

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

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

Usando el método de sustitución

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

Sea:

cos (x) = u

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

2 + 4 u^{4} - 6 u^{2} = 0 : u = 1, u = - 1, u = \frac{1}{2}, u = - \frac{1}{2}

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

Escribir en la forma binómica

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

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

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

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

Resolver

4 v^{2} - 6 v + 2 = 0 : v = 1, v = \frac{1}{2}

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

Resolver con la fórmula general para ecuaciones de segundo grado:

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

Formula general para ecuaciones de segundo grado:

Para

a = 4, b = - 6, c = 2

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

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

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

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

Aplicar las leyes de los exponentes:

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

n

es par

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

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

Multiplicar los numeros:

4 \cdot 4 \cdot 2 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

Restar:

36 - 32 = 4

= 4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

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

Separar las soluciones

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

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

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

Aplicar la regla

- (- a) = a

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

Sumar:

6 + 2 = 8

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

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{8}{8}

Aplicar la regla

\frac{a}{a} = 1

= 1

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

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

Aplicar la regla

- (- a) = a

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

Restar:

6 - 2 = 4

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

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{4}{8}

Eliminar los terminos comunes:

4

= \frac{1}{2}

Las soluciones a la ecuación de segundo grado son:

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

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

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

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

u^{2} = 1

Para

x^{2} = f (a)

las soluciones son

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

u = 1, u = - 1

1 = 1

1

Aplicar la regla

1 = 1

= 1

- 1 = - 1

- 1

Aplicar la regla

1 = 1

= - 1

u = 1, u = - 1

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}

Las soluciones son

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

Sustituir en la ecuación

u = cos (x)

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

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

cos (x) = 1 : x = 2 πn

cos (x) = 1

Soluciones generales para

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

Soluciones generales para

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

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

Combinar toda las soluciones

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

Mostrar soluciones en forma decimal

x = 2 πn, x = π + 2 πn, 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

Ejemplos populares

sec(3x)=5

sec (3 x) = 5

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

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

3tan^2(y)=5sec(y)-1

3 tan^{2} (y) = 5 sec (y) - 1

((1-tan^2(a)))/((tan(a)))=2cot^2(a)

\frac{( 1 - tan ^{2} ( a ) )}{( tan ( a ) )} = 2 cot^{2} (a)

cos(6x)=cos(2x)

cos (6 x) = cos (2 x)