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2-2cosh(0.1-2)

Solución

2 - 2 cosh (0.1 - 2)

Solución

\frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

Notación decimal

- 4.83546 \dots

Pasos de solución

2 - 2 cosh (0.1 - 2)

= 2 - 2 cosh (\frac{1}{10} - 2)

Simplificar:

\frac{1}{10} - 2 = - \frac{19}{10}

\frac{1}{10} - 2

Convertir a fracción:

2 = \frac{2 \cdot 10}{10}

= \frac{1}{10} - \frac{2 \cdot 10}{10}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 - 2 \cdot 10}{10}

1 - 2 \cdot 10 = - 19

1 - 2 \cdot 10

Multiplicar los numeros:

2 \cdot 10 = 20

= 1 - 20

Restar:

1 - 20 = - 19

= - 19

= \frac{- 19}{10}

Aplicar las propiedades de las fracciones:

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

= - \frac{19}{10}

= 2 - 2 cosh (- \frac{19}{10})

Utilizar la siguiente propiedad:

cosh (- x) = cosh (x)

cosh (- \frac{19}{10}) = cosh (\frac{19}{10})

= 2 - 2 cosh (\frac{19}{10})

Re-escribir usando identidades trigonométricas:

cosh (\frac{19}{10}) = \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

cosh (\frac{19}{10})

Utilizar la identidad hiperbólica:

cosh (x) = \frac{e ^{x} + e ^{- x}}{2}

= \frac{e ^{\frac{19}{10}} + e ^{- \frac{19}{10}}}{2}

\frac{e ^{\frac{19}{10}} + e ^{- \frac{19}{10}}}{2} = \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

\frac{e ^{\frac{19}{10}} + e ^{- \frac{19}{10}}}{2}

Aplicar las leyes de los exponentes:

a^{- b} = \frac{1}{a ^{b}}

= \frac{e ^{\frac{19}{10}} + \frac{1}{e ^{\frac{19}{10}}}}{2}

Simplificar

e^{\frac{19}{10}} + \frac{1}{e ^{\frac{19}{10}}}

en una fracción:

\frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

e^{\frac{19}{10}} + \frac{1}{e ^{\frac{19}{10}}}

Convertir a fracción:

e^{\frac{19}{10}} = \frac{e ^{\frac{19}{10}} e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}}

= \frac{e ^{\frac{19}{10}} e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}} + \frac{1}{e ^{\frac{19}{10}}}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{e ^{\frac{19}{10}} e ^{\frac{19}{10}} + 1}{e ^{\frac{19}{10}}}

e^{\frac{19}{10}} e^{\frac{19}{10}} + 1 = e^{2 \cdot \frac{19}{10}} + 1

e^{\frac{19}{10}} e^{\frac{19}{10}} + 1

e^{\frac{19}{10}} e^{\frac{19}{10}} = e^{2 \cdot \frac{19}{10}}

e^{\frac{19}{10}} e^{\frac{19}{10}}

Aplicar las leyes de los exponentes:

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

e^{\frac{19}{10}} e^{\frac{19}{10}} = e^{\frac{19}{10} + \frac{19}{10}}

= e^{\frac{19}{10} + \frac{19}{10}}

Sumar elementos similares:

\frac{19}{10} + \frac{19}{10} = 2 \cdot \frac{19}{10}

= e^{2 \cdot \frac{19}{10}}

= e^{2 \cdot \frac{19}{10}} + 1

= \frac{e ^{2 \cdot \frac{19}{10}} + 1}{e ^{\frac{19}{10}}}

Multiplicar

2 \cdot \frac{19}{10} : \frac{19}{5}

2 \cdot \frac{19}{10}

Multiplicar fracciones:

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

= \frac{19 \cdot 2}{10}

Multiplicar los numeros:

19 \cdot 2 = 38

= \frac{38}{10}

Eliminar los terminos comunes:

2

= \frac{19}{5}

= \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

= \frac{\frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}} \cdot 2}

= \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

= 2 - 2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

2 - 2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}} = \frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

2 - 2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}} = \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

Multiplicar fracciones:

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

= \frac{( e ^{\frac{19}{5}} + 1 ) \cdot 2}{2 e ^{\frac{19}{10}}}

Eliminar los terminos comunes:

2

= \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

= 2 - \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

Convertir a fracción:

2 = \frac{2 e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}}

= \frac{2 e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}} - \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{2 e ^{\frac{19}{10}} - ( e ^{\frac{19}{5}} + 1 )}{e ^{\frac{19}{10}}}

- (e^{\frac{19}{5}} + 1) : - e^{\frac{19}{5}} - 1

- (e^{\frac{19}{5}} + 1)

Poner los parentesis

= - (e^{\frac{19}{5}}) - (1)

Aplicar las reglas de los signos

+ (- a) = - a

= - e^{\frac{19}{5}} - 1

= \frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

= \frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

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