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solvefor b,a=cos(-1(a/(sqrt(a^2+b^2))))

Solución

resolver para

b, a = cos (- 1 (\frac{a}{a ^{2} + b ^{2}}))

Solución

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Pasos de solución

a = cos (- 1 \cdot (\frac{a}{a ^{2} + b ^{2}}))

Intercambiar lados

cos (- 1 \cdot \frac{a}{a ^{2} + b ^{2}}) = a

Aplicar propiedades trigonométricas inversas

cos (- 1 \cdot \frac{a}{a ^{2} + b ^{2}}) = a

Soluciones generales para

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

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

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn, - 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn, - 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

Resolver

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn

Desarrollar

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} : - \frac{a}{a ^{2} + b ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}}

Multiplicar:

1 \cdot \frac{a}{a ^{2} + b ^{2}} = \frac{a}{a ^{2} + b ^{2}}

= - \frac{a}{a ^{2} + b ^{2}}

- \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn

Elevar al cuadrado ambos lados:

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

- \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (arccos (a) + 2 πn)^{2}

Desarrollar

(- \frac{a}{a ^{2} + b ^{2}})^{2} : \frac{a ^{2}}{a ^{2} + b ^{2}}

(- \frac{a}{a ^{2} + b ^{2}})^{2}

Aplicar las leyes de los exponentes:

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

n

es par

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (\frac{a}{a ^{2} + b ^{2}})^{2}

= (\frac{a}{a ^{2} + b ^{2}})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{a ^{2}}{( a ^{2} + b ^{2} ) ^{2}}

(a^{2} + b^{2})^{2} : a^{2} + b^{2}

Aplicar las leyes de los exponentes:

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

= ((a^{2} + b^{2})^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= (a^{2} + b^{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

= a^{2} + b^{2}

= \frac{a ^{2}}{a ^{2} + b ^{2}}

Desarrollar

(arccos (a) + 2 πn)^{2} : arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

(arccos (a) + 2 πn)^{2}

Aplicar la formula del binomio al cuadrado:

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

a = arccos (a), b = 2 πn

= arccos^{2} (a) + 2 arccos (a) \cdot 2 πn + (2 πn)^{2}

Simplificar

arccos^{2} (a) + 2 arccos (a) \cdot 2 πn + (2 πn)^{2} : arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

arccos^{2} (a) + 2 arccos (a) \cdot 2 πn + (2 πn)^{2}

2 arccos (a) \cdot 2 πn = 4 πn arccos (a)

2 arccos (a) \cdot 2 πn

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 πn arccos (a)

(2 πn)^{2} = 4 π^{2} n^{2}

(2 πn)^{2}

Aplicar las leyes de los exponentes:

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

= 2^{2} π^{2} n^{2}

2^{2} = 4

= 4 π^{2} n^{2}

= arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

= arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

Resolver

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2} : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

Multiplicar ambos lados por

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

Multiplicar ambos lados por

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} (a^{2} + b^{2}) = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Simplificar

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Resolver

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Intercambiar lados

arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expandir

arccos^{2} (a) (a^{2} + b^{2}) : a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

arccos^{2} (a) (a^{2} + b^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = arccos^{2} (a), b = a^{2}, c = b^{2}

= arccos^{2} (a) a^{2} + arccos^{2} (a) b^{2}

= a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expandir

4 πn arccos (a) (a^{2} + b^{2}) : 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2}

4 πn arccos (a) (a^{2} + b^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 4 πn arccos (a), b = a^{2}, c = b^{2}

= 4 πn arccos (a) a^{2} + 4 πn arccos (a) b^{2}

= 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expandir

4 π^{2} n^{2} (a^{2} + b^{2}) : 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

4 π^{2} n^{2} (a^{2} + b^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 4 π^{2} n^{2}, b = a^{2}, c = b^{2}

= 4 π^{2} n^{2} a^{2} + 4 π^{2} n^{2} b^{2}

= 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Mover

a^{2} arccos^{2} (a)

al lado derecho

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Restar

a^{2} arccos^{2} (a)

de ambos lados

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - a^{2} arccos^{2} (a) = a^{2} - a^{2} arccos^{2} (a)

Simplificar

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Mover

4 π a^{2} n arccos (a)

al lado derecho

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Restar

4 π a^{2} n arccos (a)

de ambos lados

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - 4 π a^{2} n arccos (a) = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

Simplificar

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

Mover

4 π^{2} a^{2} n^{2}

al lado derecho

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

Restar

4 π^{2} a^{2} n^{2}

de ambos lados

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - 4 π^{2} a^{2} n^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Simplificar

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Factorizar

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} : b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2})

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2}

Factorizar el termino común

b^{2}

= b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2})

b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Dividir ambos lados entre

arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Dividir ambos lados entre

arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{b ^{2} ( arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2} )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplificar

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Para

x^{2} = f (a)

las soluciones son

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

b = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplificar

\frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Aplicar la regla

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

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 π a ^{2} n arccos ( a ) - 4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplificar

- \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Aplicar la regla

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

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Resolver

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

Desarrollar

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} : - \frac{a}{a ^{2} + b ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}}

Multiplicar:

1 \cdot \frac{a}{a ^{2} + b ^{2}} = \frac{a}{a ^{2} + b ^{2}}

= - \frac{a}{a ^{2} + b ^{2}}

- \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

Elevar al cuadrado ambos lados:

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

- \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (- arccos (a) + 2 πn)^{2}

Desarrollar

(- \frac{a}{a ^{2} + b ^{2}})^{2} : \frac{a ^{2}}{a ^{2} + b ^{2}}

(- \frac{a}{a ^{2} + b ^{2}})^{2}

Aplicar las leyes de los exponentes:

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

n

es par

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (\frac{a}{a ^{2} + b ^{2}})^{2}

= (\frac{a}{a ^{2} + b ^{2}})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{a ^{2}}{( a ^{2} + b ^{2} ) ^{2}}

(a^{2} + b^{2})^{2} : a^{2} + b^{2}

Aplicar las leyes de los exponentes:

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

= ((a^{2} + b^{2})^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= (a^{2} + b^{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

= a^{2} + b^{2}

= \frac{a ^{2}}{a ^{2} + b ^{2}}

Desarrollar

(- arccos (a) + 2 πn)^{2} : arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

(- arccos (a) + 2 πn)^{2}

Aplicar la formula del binomio al cuadrado:

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

a = - arccos (a), b = 2 πn

= (- arccos (a))^{2} + 2 (- arccos (a)) \cdot 2 πn + (2 πn)^{2}

Simplificar

(- arccos (a))^{2} + 2 (- arccos (a)) \cdot 2 πn + (2 πn)^{2} : arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

(- arccos (a))^{2} + 2 (- arccos (a)) \cdot 2 πn + (2 πn)^{2}

Quitar los parentesis:

(- a) = - a

= (- arccos (a))^{2} - 2 arccos (a) \cdot 2 πn + (2 πn)^{2}

(- arccos (a))^{2} = arccos^{2} (a)

(- arccos (a))^{2}

Aplicar las leyes de los exponentes:

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

n

es par

(- arccos (a))^{2} = arccos^{2} (a)

= arccos^{2} (a)

2 arccos (a) \cdot 2 πn = 4 πn arccos (a)

2 arccos (a) \cdot 2 πn

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 πn arccos (a)

(2 πn)^{2} = 4 π^{2} n^{2}

(2 πn)^{2}

Aplicar las leyes de los exponentes:

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

= 2^{2} π^{2} n^{2}

2^{2} = 4

= 4 π^{2} n^{2}

= arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

= arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

Resolver

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2} : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

Multiplicar ambos lados por

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

Multiplicar ambos lados por

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} (a^{2} + b^{2}) = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Simplificar

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Resolver

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Intercambiar lados

arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expandir

arccos^{2} (a) (a^{2} + b^{2}) : a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

arccos^{2} (a) (a^{2} + b^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = arccos^{2} (a), b = a^{2}, c = b^{2}

= arccos^{2} (a) a^{2} + arccos^{2} (a) b^{2}

= a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expandir

- 4 πn arccos (a) (a^{2} + b^{2}) : - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2}

- 4 πn arccos (a) (a^{2} + b^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = - 4 πn arccos (a), b = a^{2}, c = b^{2}

= - 4 πn arccos (a) a^{2} + (- 4 πn arccos (a)) b^{2}

Aplicar las reglas de los signos

+ (- a) = - a

= - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expandir

4 π^{2} n^{2} (a^{2} + b^{2}) : 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

4 π^{2} n^{2} (a^{2} + b^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 4 π^{2} n^{2}, b = a^{2}, c = b^{2}

= 4 π^{2} n^{2} a^{2} + 4 π^{2} n^{2} b^{2}

= 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Mover

a^{2} arccos^{2} (a)

al lado derecho

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Restar

a^{2} arccos^{2} (a)

de ambos lados

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - a^{2} arccos^{2} (a) = a^{2} - a^{2} arccos^{2} (a)

Simplificar

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Mover

4 π a^{2} n arccos (a)

al lado derecho

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Sumar

4 π a^{2} n arccos (a)

a ambos lados

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} + 4 π a^{2} n arccos (a) = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

Simplificar

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

Mover

4 π^{2} a^{2} n^{2}

al lado derecho

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

Restar

4 π^{2} a^{2} n^{2}

de ambos lados

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - 4 π^{2} a^{2} n^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Simplificar

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Factorizar

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} : b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2})

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2}

Factorizar el termino común

b^{2}

= b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2})

b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Dividir ambos lados entre

arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Dividir ambos lados entre

arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{b ^{2} ( arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2} )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplificar

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Para

x^{2} = f (a)

las soluciones son

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

b = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplificar

\frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Aplicar la regla

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

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 π a ^{2} n arccos ( a ) - 4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplificar

- \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Aplicar la regla

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

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= - \frac{a ^{2} + 4 πn a ^{2} arccos ( a ) - a ^{2} arccos ^{2} ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Ejemplos populares

(1/2 sin(2x))^2= 1/4 sin^2(4x)

(\frac{1}{2} sin (2 x))^{2} = \frac{1}{4} sin^{2} (4 x)

8sin^2(x)-3sin(x)-2=0

8 sin^{2} (x) - 3 sin (x) - 2 = 0

2sin^2(x)3=7sin(x),0<= x<= 2pi

2 sin^{2} (x) 3 = 7 sin (x), 0 \leq x \leq 2 π

6sin(x)+12=15

6 sin (x) + 12 = 15

cot^2(x)cos(x)=cos(x)

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