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Popular Trigonometría >

solvefor t,x=arccos(1/(sqrt(1+t^2)))

Solución

resolver para

t, x = arccos (\frac{1}{1 + t ^{2}})

Solución

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

Pasos de solución

x = arccos (\frac{1}{1 + t ^{2}})

Intercambiar lados

arccos (\frac{1}{1 + t ^{2}}) = x

Aplicar propiedades trigonométricas inversas

arccos (\frac{1}{1 + t ^{2}}) = x

arccos (x) = a \Rightarrow x = cos (a)

\frac{1}{1 + t ^{2}} = cos (x)

\frac{1}{1 + t ^{2}} = cos (x)

Resolver

\frac{1}{1 + t ^{2}} = cos (x) : t = \frac{1 - cos ^{2} ( x )}{cos ( x )} {x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn}, t = - \frac{1 - cos ^{2} ( x )}{cos ( x )} {x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn}

\frac{1}{1 + t ^{2}} = cos (x)

Multiplicar ambos lados por

1 + t^{2}

\frac{1}{1 + t ^{2}} 1 + t^{2} = cos (x) 1 + t^{2}

Simplificar

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

Intercambiar lados

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

Dividir ambos lados entre

cos (x)

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

Dividir ambos lados entre

cos (x)

\frac{cos ( x ) 1 + t ^{2}}{cos ( x )} = \frac{1}{cos ( x )}

Simplificar

1 + t^{2} = \frac{1}{cos ( x )}

1 + t^{2} = \frac{1}{cos ( x )}

Elevar al cuadrado ambos lados:

1 + t^{2} = \frac{1}{cos ^{2} ( x )}

1 + t^{2} = \frac{1}{cos ( x )}

(1 + t^{2})^{2} = (\frac{1}{cos ( x )})^{2}

Desarrollar

(1 + t^{2})^{2} : 1 + t^{2}

(1 + t^{2})^{2}

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= ((1 + t^{2})^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= (1 + t^{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 1 + t^{2}

Desarrollar

(\frac{1}{cos ( x )})^{2} : \frac{1}{cos ^{2} ( x )}

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

1 + t^{2} = \frac{1}{cos ^{2} ( x )}

1 + t^{2} = \frac{1}{cos ^{2} ( x )}

Resolver

1 + t^{2} = \frac{1}{cos ^{2} ( x )} : t = \frac{1 - cos ^{2} ( x )}{cos ( x )}, t = - \frac{1 - cos ^{2} ( x )}{cos ( x )}

1 + t^{2} = \frac{1}{cos ^{2} ( x )}

Desplace

1

a la derecha

1 + t^{2} = \frac{1}{cos ^{2} ( x )}

Restar

1

de ambos lados

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

Simplificar

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

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

Para

x^{2} = f (a)

las soluciones son

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

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

Simplificar

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

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

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

Multiplicar:

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

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

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

Aplicar la siguiente propiedad de los radicales:

n \frac{a}{b} = \frac{n a}{n b},

asumiendo que

a \geq 0, b \geq 0

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

Aplicar la siguiente propiedad de los radicales:

n a^{n} = a,

asumiendo que

a \geq 0

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

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

Simplificar

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

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

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

Multiplicar:

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

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

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

Simplificar

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

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

Aplicar la siguiente propiedad de los radicales:

n \frac{a}{b} = \frac{n a}{n b},

asumiendo que

a \geq 0, b \geq 0

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

Aplicar la siguiente propiedad de los radicales:

n a^{n} = a,

asumiendo que

a \geq 0

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

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

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

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

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

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

Verificar las soluciones:

t = \frac{1 - cos ^{2} ( x )}{cos ( x )} {x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn}, t = - \frac{1 - cos ^{2} ( x )}{cos ( x )} {x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn}

Verificar las soluciones sustituyéndolas en

\frac{1}{1 + t ^{2}} = cos (x)

Quitar las que no concuerden con la ecuación.

Inserir

t = \frac{1 - cos ^{2} ( x )}{cos ( x )} : \frac{1}{1 + ( \frac{1 - c o s ^{2} ( x )}{c o s ( x )} ) ^{2}} = cos (x) \Rightarrow x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

\frac{1}{1 + ( \frac{1 - c o s ^{2} ( x )}{c o s ( x )} ) ^{2}} = cos (x)

Usando el método de sustitución

\frac{1}{1 + ( \frac{1 - c o s ^{2} ( x )}{c o s ( x )} ) ^{2}} = cos (x)

Sea:

cos (x) = u

\frac{1}{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}} = u

\frac{1}{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}} = u : u = 1

\frac{1}{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}} = u

Multiplicar ambos lados por

1 + (\frac{1 - u ^{2}}{u})^{2}

\frac{1}{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}} 1 + (\frac{1 - u ^{2}}{u})^{2} = u 1 + (\frac{1 - u ^{2}}{u})^{2}

Simplificar

1 = u 1 + (\frac{1 - u ^{2}}{u})^{2}

Elevar al cuadrado ambos lados:

1 = 1

1 = u 1 + (\frac{1 - u ^{2}}{u})^{2}

1^{2} = u 1 + (\frac{1 - u ^{2}}{u})^{2}^{2}

Desarrollar

1^{2} : 1

1^{2}

Aplicar la regla

1^{a} = 1

= 1

Desarrollar

u 1 + (\frac{1 - u ^{2}}{u})^{2}^{2} : 1

u 1 + (\frac{1 - u ^{2}}{u})^{2}^{2}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

= u^{2} 1 + (\frac{1 - u ^{2}}{u})^{2}^{2}

1 + (\frac{1 - u ^{2}}{u})^{2}^{2} : 1 + (\frac{1 - u ^{2}}{u})^{2}

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= 1 + (\frac{1 - u ^{2}}{u})^{2}^{\frac{1}{2}}^{2}

Aplicar las leyes de los exponentes:

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

= 1 + (\frac{1 - u ^{2}}{u})^{2}^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 1 + (\frac{1 - u ^{2}}{u})^{2}

= u^{2} 1 + (\frac{1 - u ^{2}}{u})^{2}

Desarrollar

1 + (\frac{1 - u ^{2}}{u})^{2} u^{2} : 1

1 + (\frac{1 - u ^{2}}{u})^{2} u^{2}

(\frac{1 - u ^{2}}{u})^{2} = \frac{1 - u ^{2}}{u ^{2}}

(\frac{1 - u ^{2}}{u})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{( 1 - u ^{2} ) ^{2}}{u ^{2}}

(1 - u^{2})^{2} : 1 - u^{2}

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= ((1 - u^{2})^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= (1 - u^{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 1 - u^{2}

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

= u^{2} (\frac{- u ^{2} + 1}{u ^{2}} + 1)

= u^{2} (1 + \frac{1 - u ^{2}}{u ^{2}})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = u^{2}, b = 1, c = \frac{1 - u ^{2}}{u ^{2}}

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

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

Simplificar

1 \cdot u^{2} + \frac{1 - u ^{2}}{u ^{2}} u^{2} : 1

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

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

1 \cdot u^{2}

Multiplicar:

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

= u^{2}

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

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

Multiplicar fracciones:

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

= \frac{( 1 - u ^{2} ) u ^{2}}{u ^{2}}

Eliminar los terminos comunes:

u^{2}

= 1 - u^{2}

= u^{2} + 1 - u^{2}

Agrupar términos semejantes

= u^{2} - u^{2} + 1

Sumar elementos similares:

u^{2} - u^{2} = 0

= 1

= 1

= 1

1 = 1

1 = 1

Los lados son iguales

Verdadero para todo u

Verificar las soluciones:

u < - 1

Falso

, u = - 1

Falso

, - 1 < u < 1

Falso

, u = 1

Verdadero

, u > 1

Falso

Verificar las soluciones sustituyéndolas en

\frac{1}{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}} = u

Quitar las que no concuerden con la ecuación.

Sustituir

u < - 1 : \frac{1}{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}} \neq = u \Rightarrow

Falso

La solución es

u = 1

Sustituir en la ecuación

u = cos (x)

cos (x) = 1

cos (x) = 1

cos (x) = u = 1 : x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

cos (x) = u = 1

Aplicar propiedades trigonométricas inversas

cos (x) = u = 1

Soluciones generales para

cos (x) = u = 1

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

x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

Combinar toda las soluciones

x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

Inserir

t = - \frac{1 - cos ^{2} ( x )}{cos ( x )} : \frac{1}{1 + ( - \frac{1 - c o s ^{2} ( x )}{c o s ( x )} ) ^{2}} = cos (x) \Rightarrow x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

\frac{1}{1 + ( - \frac{1 - c o s ^{2} ( x )}{c o s ( x )} ) ^{2}} = cos (x)

Usando el método de sustitución

\frac{1}{1 + ( - \frac{1 - c o s ^{2} ( x )}{c o s ( x )} ) ^{2}} = cos (x)

Sea:

cos (x) = u

\frac{1}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} = u

\frac{1}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} = u : u = 1

\frac{1}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} = u

Multiplicar ambos lados por

1 + (\frac{1 - u ^{2}}{u})^{2}

\frac{1}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} 1 + (\frac{1 - u ^{2}}{u})^{2} = u 1 + (\frac{1 - u ^{2}}{u})^{2}

Simplificar

\frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} = u 1 + (\frac{1 - u ^{2}}{u})^{2}

Desarrollar

\frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} : 1

\frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}}

Combinar los exponentes similares:

\frac{x}{y} = \frac{x}{y}

= \frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}}

Desarrollar

\frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} : 1

\frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}}

\frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} = 1

\frac{1 + ( \frac{1 - u ^{2}}{u} ) ^{2}}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}}

1 + (- \frac{1 - u ^{2}}{u})^{2} = 1 + (\frac{1 - u ^{2}}{u})^{2}

1 + (- \frac{1 - u ^{2}}{u})^{2}

Aplicar las leyes de los exponentes:

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

n

es par

(- \frac{- u ^{2} + 1}{u})^{2} = (\frac{1 - u ^{2}}{u})^{2}

= 1 + (\frac{- u ^{2} + 1}{u})^{2}

= \frac{1 + ( \frac{- u ^{2} + 1}{u} ) ^{2}}{1 + ( \frac{- u ^{2} + 1}{u} ) ^{2}}

Aplicar la regla

\frac{a}{a} = 1

= 1

= 1

Aplicar la regla

1 = 1

= 1

= 1

1 = u 1 + (\frac{1 - u ^{2}}{u})^{2}

Elevar al cuadrado ambos lados:

1 = 1

1 = u 1 + (\frac{1 - u ^{2}}{u})^{2}

1^{2} = u 1 + (\frac{1 - u ^{2}}{u})^{2}^{2}

Desarrollar

1^{2} : 1

1^{2}

Aplicar la regla

1^{a} = 1

= 1

Desarrollar

u 1 + (\frac{1 - u ^{2}}{u})^{2}^{2} : 1

u 1 + (\frac{1 - u ^{2}}{u})^{2}^{2}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

= u^{2} 1 + (\frac{1 - u ^{2}}{u})^{2}^{2}

1 + (\frac{1 - u ^{2}}{u})^{2}^{2} : 1 + (\frac{1 - u ^{2}}{u})^{2}

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= 1 + (\frac{1 - u ^{2}}{u})^{2}^{\frac{1}{2}}^{2}

Aplicar las leyes de los exponentes:

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

= 1 + (\frac{1 - u ^{2}}{u})^{2}^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 1 + (\frac{1 - u ^{2}}{u})^{2}

= u^{2} 1 + (\frac{1 - u ^{2}}{u})^{2}

Desarrollar

1 + (\frac{1 - u ^{2}}{u})^{2} u^{2} : 1

1 + (\frac{1 - u ^{2}}{u})^{2} u^{2}

(\frac{1 - u ^{2}}{u})^{2} = \frac{1 - u ^{2}}{u ^{2}}

(\frac{1 - u ^{2}}{u})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{( 1 - u ^{2} ) ^{2}}{u ^{2}}

(1 - u^{2})^{2} : 1 - u^{2}

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= ((1 - u^{2})^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= (1 - u^{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 1 - u^{2}

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

= u^{2} (\frac{- u ^{2} + 1}{u ^{2}} + 1)

= u^{2} (1 + \frac{1 - u ^{2}}{u ^{2}})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = u^{2}, b = 1, c = \frac{1 - u ^{2}}{u ^{2}}

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

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

Simplificar

1 \cdot u^{2} + \frac{1 - u ^{2}}{u ^{2}} u^{2} : 1

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

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

1 \cdot u^{2}

Multiplicar:

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

= u^{2}

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

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

Multiplicar fracciones:

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

= \frac{( 1 - u ^{2} ) u ^{2}}{u ^{2}}

Eliminar los terminos comunes:

u^{2}

= 1 - u^{2}

= u^{2} + 1 - u^{2}

Agrupar términos semejantes

= u^{2} - u^{2} + 1

Sumar elementos similares:

u^{2} - u^{2} = 0

= 1

= 1

= 1

1 = 1

1 = 1

Los lados son iguales

Verdadero para todo u

Verificar las soluciones:

u < - 1

Falso

, u = - 1

Falso

, - 1 < u < 1

Falso

, u = 1

Verdadero

, u > 1

Falso

Verificar las soluciones sustituyéndolas en

\frac{1}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} = u

Quitar las que no concuerden con la ecuación.

Sustituir

u < - 1 : \frac{1}{1 + ( - \frac{1 - u ^{2}}{u} ) ^{2}} \neq = u \Rightarrow

Falso

La solución es

u = 1

Sustituir en la ecuación

u = cos (x)

cos (x) = 1

cos (x) = 1

cos (x) = u = 1 : x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

cos (x) = u = 1

Aplicar propiedades trigonométricas inversas

cos (x) = u = 1

Soluciones generales para

cos (x) = u = 1

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

x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

Combinar toda las soluciones

x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn

Las soluciones son

t = \frac{1 - cos ^{2} ( x )}{cos ( x )} {x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn}, t = - \frac{1 - cos ^{2} ( x )}{cos ( x )} {x = arccos (u = 1) + 2 πn, x = - arccos (u = 1) + 2 πn}

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

Gráfica

Ejemplos populares

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2 cos^{2} (x) + 15 sin (x) - 15 = 0

cos(2x)=sin(x),-2pi<= x<= 2pi

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5sec^2(θ)sin(θ)-cos(θ)=0

5 sec^{2} (θ) sin (θ) - cos (θ) = 0

cos(x)= 8/11

cos (x) = \frac{8}{11}