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arctan(x/(12))-arctan(x)=0.001

Solución

arctan (\frac{x}{12}) - arctan (x) = 0.001

Solución

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Pasos de solución

arctan (\frac{x}{12}) - arctan (x) = 0.001

Re-escribir usando identidades trigonométricas

arctan (\frac{x}{12}) - arctan (x)

Utilizar la identidad suma-producto:

arctan (s) - arctan (t) = arctan (\frac{s - t}{1 + s t})

= arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x})

arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}) = 0.001

Aplicar propiedades trigonométricas inversas

arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}) = 0.001

arctan (x) = a \Rightarrow x = tan (a)

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (0.001)

tan (0.001) = tan (\frac{1}{1000})

tan (0.001)

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

Resolver

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000}) : x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

Simplificar

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} : - \frac{11 x}{12 + x ^{2}}

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}

\frac{x}{12} x = \frac{x ^{2}}{12}

\frac{x}{12} x

Multiplicar fracciones:

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

= \frac{xx}{12}

xx = x^{2}

xx

Aplicar las leyes de los exponentes:

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

xx = x^{1 + 1}

= x^{1 + 1}

Sumar:

1 + 1 = 2

= x^{2}

= \frac{x ^{2}}{12}

= \frac{\frac{x}{12} - x}{1 + \frac{x ^{2}}{12}}

Simplificar

\frac{x}{12} - x

en una fracción:

- \frac{11 x}{12}

\frac{x}{12} - x

Convertir a fracción:

x = \frac{x 12}{12}

= \frac{x}{12} - \frac{x \cdot 12}{12}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{x - x \cdot 12}{12}

Sumar elementos similares:

x - 12 x = - 11 x

= \frac{- 11 x}{12}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{11 x}{12}

= \frac{- \frac{11 x}{12}}{1 + \frac{x ^{2}}{12}}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{11 x}{12}}{1 + \frac{x ^{2}}{12}}

Aplicar las propiedades de las fracciones:

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

\frac{\frac{11 x}{12}}{1 + \frac{x ^{2}}{12}} = \frac{11 x}{12 ( 1 + \frac{x ^{2}}{12} )}

= - \frac{11 x}{12 ( 1 + \frac{x ^{2}}{12} )}

Simplificar

1 + \frac{x ^{2}}{12}

en una fracción:

\frac{12 + x ^{2}}{12}

1 + \frac{x ^{2}}{12}

Convertir a fracción:

1 = \frac{1 \cdot 12}{12}

= \frac{1 \cdot 12}{12} + \frac{x ^{2}}{12}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 \cdot 12 + x ^{2}}{12}

Multiplicar los numeros:

1 \cdot 12 = 12

= \frac{12 + x ^{2}}{12}

= - \frac{11 x}{12 \cdot \frac{x ^{2} + 12}{12}}

Multiplicar

12 \cdot \frac{12 + x ^{2}}{12} : 12 + x^{2}

12 \cdot \frac{12 + x ^{2}}{12}

Multiplicar fracciones:

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

= \frac{( 12 + x ^{2} ) \cdot 12}{12}

Eliminar los terminos comunes:

12

= 12 + x^{2}

= - \frac{11 x}{x ^{2} + 12}

- \frac{11 x}{12 + x ^{2}} = tan (\frac{1}{1000})

Multiplicar ambos lados por

12 + x^{2}

- \frac{11 x}{12 + x ^{2}} = tan (\frac{1}{1000})

Multiplicar ambos lados por

12 + x^{2}

- \frac{11 x}{12 + x ^{2}} (12 + x^{2}) = tan (\frac{1}{1000}) (12 + x^{2})

Simplificar

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

Resolver

- 11 x = tan (\frac{1}{1000}) (12 + x^{2}) : x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

Desarrollar

tan (\frac{1}{1000}) (12 + x^{2}) : 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

tan (\frac{1}{1000}) (12 + x^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = tan (\frac{1}{1000}), b = 12, c = x^{2}

= tan (\frac{1}{1000}) \cdot 12 + tan (\frac{1}{1000}) x^{2}

= 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

- 11 x = 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

Intercambiar lados

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} = - 11 x

Mover

11 x

al lado izquierdo

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} = - 11 x

Sumar

11 x

a ambos lados

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = - 11 x + 11 x

Simplificar

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = 0

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

tan (\frac{1}{1000}) x^{2} + 11 x + 12 tan (\frac{1}{1000}) = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

tan (\frac{1}{1000}) x^{2} + 11 x + 12 tan (\frac{1}{1000}) = 0

Formula general para ecuaciones de segundo grado:

Para

a = tan (\frac{1}{1000}), b = 11, c = 12 tan (\frac{1}{1000})

x_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 tan ( \frac{1}{1000} ) \cdot 12 tan ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 tan ( \frac{1}{1000} ) \cdot 12 tan ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

1 1^{2} - 4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000}) = 121 - 48 tan^{2} (\frac{1}{1000})

1 1^{2} - 4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000})

4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000}) = 48 tan^{2} (\frac{1}{1000})

4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000})

Multiplicar los numeros:

4 \cdot 12 = 48

= 48 tan (\frac{1}{1000}) tan (\frac{1}{1000})

Aplicar las leyes de los exponentes:

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

tan (\frac{1}{1000}) tan (\frac{1}{1000}) = tan^{1 + 1} (\frac{1}{1000})

= 48 tan^{1 + 1} (\frac{1}{1000})

Sumar:

1 + 1 = 2

= 48 tan^{2} (\frac{1}{1000})

= 1 1^{2} - 48 tan^{2} (\frac{1}{1000})

1 1^{2} = 121

= 121 - 48 tan^{2} (\frac{1}{1000})

x_{1, 2} = \frac{- 11 \pm 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Separar las soluciones

x_{1} = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x_{2} = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Las soluciones a la ecuación de segundo grado son:

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

arctan (\frac{x}{12}) - arctan (x) = 0.001

Quitar las que no concuerden con la ecuación.

Verificar la solución

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} :

Verdadero

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Sustituir

n = 1

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Multiplicar

arctan (\frac{x}{12}) - arctan (x) = 0.001

por

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

arctan \frac{\frac{- 11 + 121 - 48 t a n ^{2} ( \frac{1}{1000} )}{2 t a n ( \frac{1}{1000} )}}{12} - arctan \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} = 0.001

Simplificar

0.00099 \dots = 0.001

\Rightarrow Verdadero

Verificar la solución

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} :

Verdadero

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Sustituir

n = 1

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Multiplicar

arctan (\frac{x}{12}) - arctan (x) = 0.001

por

x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

arctan \frac{\frac{- 11 - 121 - 48 t a n ^{2} ( \frac{1}{1000} )}{2 t a n ( \frac{1}{1000} )}}{12} - arctan \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} = 0.001

Simplificar

0.001 = 0.001

\Rightarrow Verdadero

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

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