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sin^2(x)+cos^4(x)=n

Solución

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

Solución

x = arccos \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 + 4 n - 3}{2} + 2 πk, x = arccos - \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 + 4 n - 3}{2} + 2 πk, x = arccos \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 - 4 n - 3}{2} + 2 πk, x = arccos - \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 - 4 n - 3}{2} + 2 πk

Pasos de solución

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

Restar

n

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Usando el método de sustitución

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

Sea:

cos (x) = u

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

1 - u^{2} + u^{4} - n = 0 : u = \frac{1 + 4 n - 3}{2}, u = - \frac{1 + 4 n - 3}{2}, u = \frac{1 - 4 n - 3}{2}, u = - \frac{1 - 4 n - 3}{2}

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

Escribir en la forma binómica

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

u^{4} - u^{2} + 1 - n = 0

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

v^{2} - v + 1 - n = 0

Resolver

v^{2} - v + 1 - n = 0 : v = \frac{1 + 4 n - 3}{2}, v = \frac{1 - 4 n - 3}{2}

v^{2} - v + 1 - n = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

v^{2} - v + 1 - n = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = - 1, c = 1 - n

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

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

Simplificar

(- 1)^{2} - 4 \cdot 1 \cdot (1 - n) : 4 n - 3

(- 1)^{2} - 4 \cdot 1 \cdot (1 - n)

(- 1)^{2} = 1

(- 1)^{2}

Aplicar las leyes de los exponentes:

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

n

es par

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

= 1^{2}

Aplicar la regla

1^{a} = 1

= 1

4 \cdot 1 \cdot (1 - n) = 4 (1 - n)

4 \cdot 1 \cdot (1 - n)

Multiplicar los numeros:

4 \cdot 1 = 4

= 4 (- n + 1)

= 1 - 4 (- n + 1)

Expandir

1 - 4 (1 - n) : 4 n - 3

1 - 4 (1 - n)

Expandir

- 4 (1 - n) : - 4 + 4 n

- 4 (1 - n)

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 4, b = 1, c = n

= - 4 \cdot 1 - (- 4) n

Aplicar las reglas de los signos

- (- a) = a

= - 4 \cdot 1 + 4 n

Multiplicar los numeros:

4 \cdot 1 = 4

= - 4 + 4 n

= 1 - 4 + 4 n

Restar:

1 - 4 = - 3

= 4 n - 3

= 4 n - 3

v_{1, 2} = \frac{- ( - 1 ) \pm 4 n - 3}{2 \cdot 1}

Separar las soluciones

v_{1} = \frac{- ( - 1 ) + 4 n - 3}{2 \cdot 1}, v_{2} = \frac{- ( - 1 ) - 4 n - 3}{2 \cdot 1}

v = \frac{- ( - 1 ) + 4 n - 3}{2 \cdot 1} : \frac{1 + 4 n - 3}{2}

\frac{- ( - 1 ) + 4 n - 3}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{1 + 4 n - 3}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{1 + 4 n - 3}{2}

v = \frac{- ( - 1 ) - 4 n - 3}{2 \cdot 1} : \frac{1 - 4 n - 3}{2}

\frac{- ( - 1 ) - 4 n - 3}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{1 - 4 n - 3}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{1 - 4 n - 3}{2}

Las soluciones a la ecuación de segundo grado son:

v = \frac{1 + 4 n - 3}{2}, v = \frac{1 - 4 n - 3}{2}

v = \frac{1 + 4 n - 3}{2}, v = \frac{1 - 4 n - 3}{2}

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

u^{2} = \frac{1 + 4 n - 3}{2} : u = \frac{1 + 4 n - 3}{2}, u = - \frac{1 + 4 n - 3}{2}

u^{2} = \frac{1 + 4 n - 3}{2}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{1 + 4 n - 3}{2}, u = - \frac{1 + 4 n - 3}{2}

Resolver

u^{2} = \frac{1 - 4 n - 3}{2} : u = \frac{1 - 4 n - 3}{2}, u = - \frac{1 - 4 n - 3}{2}

u^{2} = \frac{1 - 4 n - 3}{2}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{1 - 4 n - 3}{2}, u = - \frac{1 - 4 n - 3}{2}

Las soluciones son

u = \frac{1 + 4 n - 3}{2}, u = - \frac{1 + 4 n - 3}{2}, u = \frac{1 - 4 n - 3}{2}, u = - \frac{1 - 4 n - 3}{2}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{1 + 4 n - 3}{2}, cos (x) = - \frac{1 + 4 n - 3}{2}, cos (x) = \frac{1 - 4 n - 3}{2}, cos (x) = - \frac{1 - 4 n - 3}{2}

cos (x) = \frac{1 + 4 n - 3}{2}, cos (x) = - \frac{1 + 4 n - 3}{2}, cos (x) = \frac{1 - 4 n - 3}{2}, cos (x) = - \frac{1 - 4 n - 3}{2}

cos (x) = \frac{1 + 4 n - 3}{2} : x = arccos \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 + 4 n - 3}{2} + 2 πk

cos (x) = \frac{1 + 4 n - 3}{2}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{1 + 4 n - 3}{2}

Soluciones generales para

cos (x) = \frac{1 + 4 n - 3}{2}

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

x = arccos \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 + 4 n - 3}{2} + 2 πk

x = arccos \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 + 4 n - 3}{2} + 2 πk

cos (x) = - \frac{1 + 4 n - 3}{2} : x = arccos - \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 + 4 n - 3}{2} + 2 πk

cos (x) = - \frac{1 + 4 n - 3}{2}

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{1 + 4 n - 3}{2}

Soluciones generales para

cos (x) = - \frac{1 + 4 n - 3}{2}

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

x = arccos - \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 + 4 n - 3}{2} + 2 πk

x = arccos - \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 + 4 n - 3}{2} + 2 πk

cos (x) = \frac{1 - 4 n - 3}{2} : x = arccos \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 - 4 n - 3}{2} + 2 πk

cos (x) = \frac{1 - 4 n - 3}{2}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{1 - 4 n - 3}{2}

Soluciones generales para

cos (x) = \frac{1 - 4 n - 3}{2}

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

x = arccos \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 - 4 n - 3}{2} + 2 πk

x = arccos \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 - 4 n - 3}{2} + 2 πk

cos (x) = - \frac{1 - 4 n - 3}{2} : x = arccos - \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 - 4 n - 3}{2} + 2 πk

cos (x) = - \frac{1 - 4 n - 3}{2}

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{1 - 4 n - 3}{2}

Soluciones generales para

cos (x) = - \frac{1 - 4 n - 3}{2}

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

x = arccos - \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 - 4 n - 3}{2} + 2 πk

x = arccos - \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 - 4 n - 3}{2} + 2 πk

Combinar toda las soluciones

x = arccos \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 + 4 n - 3}{2} + 2 πk, x = arccos - \frac{1 + 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 + 4 n - 3}{2} + 2 πk, x = arccos \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos \frac{1 - 4 n - 3}{2} + 2 πk, x = arccos - \frac{1 - 4 n - 3}{2} + 2 πk, x = - arccos - \frac{1 - 4 n - 3}{2} + 2 πk

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