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cos^2(x)+sec^2(x)=11

Solución

cos^{2} (x) + sec^{2} (x) = 11

Solución

x = 1.26319 \dots + 2 πn, x = 2 π - 1.26319 \dots + 2 πn, x = 1.87839 \dots + 2 πn, x = - 1.87839 \dots + 2 πn

Grados

x = 72.37560 \dots^{\circ} + 36 0^{\circ} n, x = 287.62439 \dots^{\circ} + 36 0^{\circ} n, x = 107.62439 \dots^{\circ} + 36 0^{\circ} n, x = - 107.62439 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

cos^{2} (x) + sec^{2} (x) = 11

Restar

11

de ambos lados

cos^{2} (x) + sec^{2} (x) - 11 = 0

Re-escribir usando identidades trigonométricas

- 11 + cos^{2} (x) + sec^{2} (x)

Utilizar la identidad trigonométrica básica:

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

= - 11 + (\frac{1}{sec ( x )})^{2} + sec^{2} (x)

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{sec ^{2} ( x )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

= - 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x)

- 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x) = 0

Usando el método de sustitución

- 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x) = 0

Sea:

sec (x) = u

- 11 + \frac{1}{u ^{2}} + u^{2} = 0

- 11 + \frac{1}{u ^{2}} + u^{2} = 0 : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

- 11 + \frac{1}{u ^{2}} + u^{2} = 0

Multiplicar ambos lados por

u^{2}

- 11 + \frac{1}{u ^{2}} + u^{2} = 0

Multiplicar ambos lados por

u^{2}

- 11 u^{2} + \frac{1}{u ^{2}} u^{2} + u^{2} u^{2} = 0 \cdot u^{2}

Simplificar

- 11 u^{2} + \frac{1}{u ^{2}} u^{2} + u^{2} u^{2} = 0 \cdot u^{2}

Simplificar

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

u^{2}

= 1

Simplificar

u^{2} u^{2} : u^{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}

= u^{2 + 2}

Sumar:

2 + 2 = 4

= u^{4}

Simplificar

0 \cdot u^{2} : 0

0 \cdot u^{2}

Aplicar la regla

0 \cdot a = 0

= 0

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

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

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

Resolver

- 11 u^{2} + 1 + u^{4} = 0 : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

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

Escribir en la forma binómica

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

u^{4} - 11 u^{2} + 1 = 0

Re-escribir la ecuación con

v = u^{2}

v^{2} = u^{4}

v^{2} - 11 v + 1 = 0

Resolver

v^{2} - 11 v + 1 = 0 : v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

v^{2} - 11 v + 1 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

v^{2} - 11 v + 1 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = - 11, c = 1

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

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

(- 11)^{2} - 4 \cdot 1 \cdot 1 = 313

(- 11)^{2} - 4 \cdot 1 \cdot 1

Aplicar las leyes de los exponentes:

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

n

es par

(- 11)^{2} = 1 1^{2}

= 1 1^{2} - 4 \cdot 1 \cdot 1

Multiplicar los numeros:

4 \cdot 1 \cdot 1 = 4

= 1 1^{2} - 4

1 1^{2} = 121

= 121 - 4

Restar:

121 - 4 = 117

= 117

Descomposición en factores primos de

117 : 3^{2} \cdot 13

117

117

divida por

3117 = 39 \cdot 3

= 3 \cdot 39

39

divida por

339 = 13 \cdot 3

= 3 \cdot 3 \cdot 13

3, 13

son números primos, por lo tanto, no se pueden factorizar mas

= 3 \cdot 3 \cdot 13

= 3^{2} \cdot 13

= 3^{2} \cdot 13

Aplicar las leyes de los exponentes:

= 13 3^{2}

Aplicar las leyes de los exponentes:

3^{2} = 3

= 313

v_{1, 2} = \frac{- ( - 11 ) \pm 3 13}{2 \cdot 1}

Separar las soluciones

v_{1} = \frac{- ( - 11 ) + 3 13}{2 \cdot 1}, v_{2} = \frac{- ( - 11 ) - 3 13}{2 \cdot 1}

v = \frac{- ( - 11 ) + 3 13}{2 \cdot 1} : \frac{11 + 3 13}{2}

\frac{- ( - 11 ) + 3 13}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{11 + 3 13}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{11 + 3 13}{2}

v = \frac{- ( - 11 ) - 3 13}{2 \cdot 1} : \frac{11 - 3 13}{2}

\frac{- ( - 11 ) - 3 13}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{11 - 3 13}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{11 - 3 13}{2}

Las soluciones a la ecuación de segundo grado son:

v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

Sustituir hacia atrás la

v = u^{2},

resolver para

u

Resolver

u^{2} = \frac{11 + 3 13}{2} : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}

u^{2} = \frac{11 + 3 13}{2}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}

Resolver

u^{2} = \frac{11 - 3 13}{2} : u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

u^{2} = \frac{11 - 3 13}{2}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

Las soluciones son

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

Verificar las soluciones

Encontrar los puntos no definidos (singularidades):

u = 0

Tomar el(los) denominador(es) de

- 11 + \frac{1}{u ^{2}} + u^{2}

y comparar con cero

Resolver

u^{2} = 0 : u = 0

u^{2} = 0

Aplicar la regla

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

u = 0

Los siguientes puntos no están definidos

u = 0

Combinar los puntos no definidos con las soluciones:

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

Sustituir en la ecuación

u = sec (x)

sec (x) = \frac{11 + 3 13}{2}, sec (x) = - \frac{11 + 3 13}{2}, sec (x) = \frac{11 - 3 13}{2}, sec (x) = - \frac{11 - 3 13}{2}

sec (x) = \frac{11 + 3 13}{2}, sec (x) = - \frac{11 + 3 13}{2}, sec (x) = \frac{11 - 3 13}{2}, sec (x) = - \frac{11 - 3 13}{2}

sec (x) = \frac{11 + 3 13}{2} : x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

sec (x) = \frac{11 + 3 13}{2}

Aplicar propiedades trigonométricas inversas

sec (x) = \frac{11 + 3 13}{2}

Soluciones generales para

sec (x) = \frac{11 + 3 13}{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

sec (x) = - \frac{11 + 3 13}{2} : x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

sec (x) = - \frac{11 + 3 13}{2}

Aplicar propiedades trigonométricas inversas

sec (x) = - \frac{11 + 3 13}{2}

Soluciones generales para

sec (x) = - \frac{11 + 3 13}{2}

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

sec (x) = \frac{11 - 3 13}{2} :

Sin solución

sec (x) = \frac{11 - 3 13}{2}

sec (x) \leq - 1 or sec (x) \geq 1

Sin soluci \overset{o}{ˊ} n

sec (x) = - \frac{11 - 3 13}{2} :

Sin solución

sec (x) = - \frac{11 - 3 13}{2}

sec (x) \leq - 1 or sec (x) \geq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn, x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

Mostrar soluciones en forma decimal

x = 1.26319 \dots + 2 πn, x = 2 π - 1.26319 \dots + 2 πn, x = 1.87839 \dots + 2 πn, x = - 1.87839 \dots + 2 πn

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