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Popular Trigonometria >

tanh(pi/4)

Solução

tanh (\frac{π}{4})

Solução

\frac{( e ^{\frac{π}{2}} - 1 ) ^{2}}{e ^{π} - 1}

Decimal

0.65579 \dots

Passos da solução

tanh (\frac{π}{4})

Use a identidade hiperbólica:

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

= \frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{e ^{\frac{π}{4}} + e ^{- \frac{π}{4}}}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{e ^{\frac{π}{4}} + e ^{- \frac{π}{4}}} = \frac{( e ^{\frac{π}{2}} - 1 ) ^{2}}{e ^{π} - 1}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{e ^{\frac{π}{4}} + e ^{- \frac{π}{4}}}

Aplicar as propriedades dos expoentes:

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

e^{- \frac{π}{4}} = \frac{1}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}}{e ^{\frac{π}{4}} + \frac{1}{e ^{\frac{π}{4}}}}

Simplificar

e^{\frac{π}{4}} + \frac{1}{e ^{\frac{π}{4}}}

em uma fração:

\frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} + \frac{1}{e ^{\frac{π}{4}}}

Converter para fração:

e^{\frac{π}{4}} = \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}} + \frac{1}{e ^{\frac{π}{4}}}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}} + 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} e^{\frac{π}{4}} + 1 = e^{2 \cdot \frac{π}{4}} + 1

e^{\frac{π}{4}} e^{\frac{π}{4}} + 1

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{2 \cdot \frac{π}{4}}

e^{\frac{π}{4}} e^{\frac{π}{4}}

Aplicar as propriedades dos expoentes:

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

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{\frac{π}{4} + \frac{π}{4}}

= e^{\frac{π}{4} + \frac{π}{4}}

Somar elementos similares:

\frac{π}{4} + \frac{π}{4} = 2 \cdot \frac{π}{4}

= e^{2 \cdot \frac{π}{4}}

= e^{2 \cdot \frac{π}{4}} + 1

= \frac{e ^{2 \cdot \frac{π}{4}} + 1}{e ^{\frac{π}{4}}}

Multiplicar

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

Multiplicar frações:

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

= \frac{π 2}{4}

Eliminar o fator comum:

2

= \frac{π}{2}

= \frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}}{\frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}}

Simplificar

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}

em uma fração:

\frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}

Converter para fração:

e^{\frac{π}{4}} = \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}} - \frac{1}{e ^{\frac{π}{4}}}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1 = e^{2 \cdot \frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{2 \cdot \frac{π}{4}}

e^{\frac{π}{4}} e^{\frac{π}{4}}

Aplicar as propriedades dos expoentes:

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

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{\frac{π}{4} + \frac{π}{4}}

= e^{\frac{π}{4} + \frac{π}{4}}

Somar elementos similares:

\frac{π}{4} + \frac{π}{4} = 2 \cdot \frac{π}{4}

= e^{2 \cdot \frac{π}{4}}

= e^{2 \cdot \frac{π}{4}} - 1

= \frac{e ^{2 \cdot \frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

Multiplicar

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

Multiplicar frações:

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

= \frac{π 2}{4}

Eliminar o fator comum:

2

= \frac{π}{2}

= \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

= \frac{\frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}}{\frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}}

Dividir frações:

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

= \frac{( e ^{\frac{π}{2}} - 1 ) e ^{\frac{π}{4}}}{e ^{\frac{π}{4}} ( e ^{\frac{π}{2}} + 1 )}

Eliminar o fator comum:

e^{\frac{π}{4}}

= \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{2}} + 1}

Racionalizar

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

\frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{2}} + 1}

Multiplicar pelo conjugado

\frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{2}} - 1}

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

(e^{\frac{π}{2}} - 1) (e^{\frac{π}{2}} - 1) = (e^{\frac{π}{2}} - 1)^{2}

(e^{\frac{π}{2}} - 1) (e^{\frac{π}{2}} - 1)

Aplicar as propriedades dos expoentes:

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

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

= (e^{\frac{π}{2}} - 1)^{1 + 1}

Somar:

1 + 1 = 2

= (e^{\frac{π}{2}} - 1)^{2}

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

(e^{\frac{π}{2}} + 1) (e^{\frac{π}{2}} - 1)

Aplicar a regra da diferença de quadrados:

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

a = e^{\frac{π}{2}}, b = 1

= (e^{\frac{π}{2}})^{2} - 1^{2}

Simplificar

(e^{\frac{π}{2}})^{2} - 1^{2} : e^{π} - 1

(e^{\frac{π}{2}})^{2} - 1^{2}

Aplicar a regra

1^{a} = 1

1^{2} = 1

= (e^{\frac{π}{2}})^{2} - 1

(e^{\frac{π}{2}})^{2} = e^{π}

(e^{\frac{π}{2}})^{2}

Aplicar as propriedades dos expoentes:

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

= e^{\frac{π}{2} \cdot 2}

\frac{π}{2} \cdot 2 = π

\frac{π}{2} \cdot 2

Multiplicar frações:

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

= \frac{π 2}{2}

Eliminar o fator comum:

2

= π

= e^{π}

= e^{π} - 1

= e^{π} - 1

= \frac{( e ^{\frac{π}{2}} - 1 ) ^{2}}{e ^{π} - 1}

= \frac{( e ^{\frac{π}{2}} - 1 ) ^{2}}{e ^{π} - 1}

= \frac{( e ^{\frac{π}{2}} - 1 ) ^{2}}{e ^{π} - 1}

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