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solvefor x,y=(sec(x))/(cos^1(x))+8y^2

Solución

resolver para

x, y = \frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2}

Solución

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn, x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

Pasos de solución

y = \frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2}

Intercambiar lados

\frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2} = y

Restar

y

de ambos lados

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y = 0

Simplificar

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y : \frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )}

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y

Convertir a fracción:

8 y^{2} = \frac{8 y ^{2} cos ( x )}{cos ( x )}, y = \frac{y cos ( x )}{cos ( x )}

= \frac{sec ( x )}{cos ( x )} + \frac{8 y ^{2} cos ( x )}{cos ( x )} - \frac{y cos ( x )}{cos ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )}

\frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )} = 0

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

sec (x) + 8 y^{2} cos (x) - y cos (x) = 0

Re-escribir usando identidades trigonométricas

sec (x) - cos (x) y + 8 cos (x) y^{2}

Utilizar la identidad trigonométrica básica:

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

= sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2}

Simplificar

sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2} : sec (x) + \frac{- y + 8 y ^{2}}{sec ( x )}

sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2}

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

\frac{1}{sec ( x )} y

Multiplicar fracciones:

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

= \frac{1 \cdot y}{sec ( x )}

Multiplicar:

1 \cdot y = y

= \frac{y}{sec ( x )}

8 \cdot \frac{1}{sec ( x )} y^{2} = \frac{8 y ^{2}}{sec ( x )}

8 \cdot \frac{1}{sec ( x )} y^{2}

Multiplicar fracciones:

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

= \frac{1 \cdot 8 y ^{2}}{sec ( x )}

Multiplicar los numeros:

1 \cdot 8 = 8

= \frac{8 y ^{2}}{sec ( x )}

= sec (x) - \frac{y}{sec ( x )} + \frac{8 y ^{2}}{sec ( x )}

Combinar las fracciones usando el mínimo común denominador:

\frac{- y + 8 y ^{2}}{sec ( x )}

Aplicar la regla

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

= \frac{- y + 8 y ^{2}}{sec ( x )}

= sec (x) + \frac{8 y ^{2} - y}{sec ( x )}

= sec (x) + \frac{- y + 8 y ^{2}}{sec ( x )}

\frac{- y + 8 y ^{2}}{sec ( x )} + sec (x) = 0

Usando el método de sustitución

\frac{- y + 8 y ^{2}}{sec ( x )} + sec (x) = 0

Sea:

sec (x) = u

\frac{- y + 8 y ^{2}}{u} + u = 0

\frac{- y + 8 y ^{2}}{u} + u = 0 : u = y - 8 y^{2}, u = - y - 8 y^{2}

\frac{- y + 8 y ^{2}}{u} + u = 0

Multiplicar ambos lados por

u

\frac{- y + 8 y ^{2}}{u} + u = 0

Multiplicar ambos lados por

u

\frac{- y + 8 y ^{2}}{u} u + uu = 0 \cdot u

Simplificar

\frac{- y + 8 y ^{2}}{u} u + uu = 0 \cdot u

Simplificar

\frac{- y + 8 y ^{2}}{u} u : - y + 8 y^{2}

\frac{- y + 8 y ^{2}}{u} u

Multiplicar fracciones:

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

= \frac{( - y + 8 y ^{2} ) u}{u}

Eliminar los terminos comunes:

u

= - - y + 8 y^{2}

Simplificar

uu : u^{2}

uu

Aplicar las leyes de los exponentes:

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

uu = u^{1 + 1}

= u^{1 + 1}

Sumar:

1 + 1 = 2

= u^{2}

Simplificar

0 \cdot u : 0

0 \cdot u

Aplicar la regla

0 \cdot a = 0

= 0

- y + 8 y^{2} + u^{2} = 0

- y + 8 y^{2} + u^{2} = 0

- y + 8 y^{2} + u^{2} = 0

Resolver

- y + 8 y^{2} + u^{2} = 0 : u = y - 8 y^{2}, u = - y - 8 y^{2}

- y + 8 y^{2} + u^{2} = 0

Mover

y

al lado derecho

- y + 8 y^{2} + u^{2} = 0

Sumar

y

a ambos lados

- y + 8 y^{2} + u^{2} + y = 0 + y

Simplificar

8 y^{2} + u^{2} = y

8 y^{2} + u^{2} = y

Mover

8 y^{2}

al lado derecho

8 y^{2} + u^{2} = y

Restar

8 y^{2}

de ambos lados

8 y^{2} + u^{2} - 8 y^{2} = y - 8 y^{2}

Simplificar

u^{2} = y - 8 y^{2}

u^{2} = y - 8 y^{2}

Para

x^{2} = f (a)

las soluciones son

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

u = y - 8 y^{2}, u = - y - 8 y^{2}

u = y - 8 y^{2}, u = - y - 8 y^{2}

Sustituir en la ecuación

u = sec (x)

sec (x) = y - 8 y^{2}, sec (x) = - y - 8 y^{2}

sec (x) = y - 8 y^{2}, sec (x) = - y - 8 y^{2}

sec (x) = y - 8 y^{2} : x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

sec (x) = y - 8 y^{2}

Aplicar propiedades trigonométricas inversas

sec (x) = y - 8 y^{2}

Soluciones generales para

sec (x) = y - 8 y^{2}

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

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

sec (x) = - y - 8 y^{2} : x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

sec (x) = - y - 8 y^{2}

Aplicar propiedades trigonométricas inversas

sec (x) = - y - 8 y^{2}

Soluciones generales para

sec (x) = - y - 8 y^{2}

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

x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

Combinar toda las soluciones

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn, x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

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