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tanh(1.4)

Lösung

tanh (1.4)

\frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

Dezimale

0.88535 \dots

Schritte zur Lösung

tanh (1.4)

= tanh (\frac{7}{5})

Umschreiben mit Hilfe von Trigonometrie-Identitäten:

\frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

tanh (\frac{7}{5})

Hyperbolische Identität anwenden:

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

= \frac{e ^{\frac{7}{5}} - e ^{- \frac{7}{5}}}{e ^{\frac{7}{5}} + e ^{- \frac{7}{5}}}

\frac{e ^{\frac{7}{5}} - e ^{- \frac{7}{5}}}{e ^{\frac{7}{5}} + e ^{- \frac{7}{5}}} = \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

\frac{e ^{\frac{7}{5}} - e ^{- \frac{7}{5}}}{e ^{\frac{7}{5}} + e ^{- \frac{7}{5}}}

Wende Exponentenregel an:

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

e^{- \frac{7}{5}} = \frac{1}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}}}{e ^{\frac{7}{5}} + \frac{1}{e ^{\frac{7}{5}}}}

Füge

e^{\frac{7}{5}} + \frac{1}{e ^{\frac{7}{5}}}

zusammen:

\frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} + \frac{1}{e ^{\frac{7}{5}}}

Wandle das Element in einen Bruch um:

e^{\frac{7}{5}} = \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}} + \frac{1}{e ^{\frac{7}{5}}}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}} + 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} e^{\frac{7}{5}} + 1 = e^{2 \cdot \frac{7}{5}} + 1

e^{\frac{7}{5}} e^{\frac{7}{5}} + 1

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{2 \cdot \frac{7}{5}}

e^{\frac{7}{5}} e^{\frac{7}{5}}

Wende Exponentenregel an:

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

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{\frac{7}{5} + \frac{7}{5}}

= e^{\frac{7}{5} + \frac{7}{5}}

Addiere gleiche Elemente:

\frac{7}{5} + \frac{7}{5} = 2 \cdot \frac{7}{5}

= e^{2 \cdot \frac{7}{5}}

= e^{2 \cdot \frac{7}{5}} + 1

= \frac{e ^{2 \cdot \frac{7}{5}} + 1}{e ^{\frac{7}{5}}}

Multipliziere

2 \cdot \frac{7}{5} : \frac{14}{5}

2 \cdot \frac{7}{5}

Multipliziere Brüche:

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

= \frac{7 \cdot 2}{5}

Multipliziere die Zahlen:

7 \cdot 2 = 14

= \frac{14}{5}

= \frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}}}{\frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}}

Füge

e^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}}

zusammen:

\frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}}

Wandle das Element in einen Bruch um:

e^{\frac{7}{5}} = \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}} - \frac{1}{e ^{\frac{7}{5}}}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}} - 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} e^{\frac{7}{5}} - 1 = e^{2 \cdot \frac{7}{5}} - 1

e^{\frac{7}{5}} e^{\frac{7}{5}} - 1

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{2 \cdot \frac{7}{5}}

e^{\frac{7}{5}} e^{\frac{7}{5}}

Wende Exponentenregel an:

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

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{\frac{7}{5} + \frac{7}{5}}

= e^{\frac{7}{5} + \frac{7}{5}}

Addiere gleiche Elemente:

\frac{7}{5} + \frac{7}{5} = 2 \cdot \frac{7}{5}

= e^{2 \cdot \frac{7}{5}}

= e^{2 \cdot \frac{7}{5}} - 1

= \frac{e ^{2 \cdot \frac{7}{5}} - 1}{e ^{\frac{7}{5}}}

Multipliziere

2 \cdot \frac{7}{5} : \frac{14}{5}

2 \cdot \frac{7}{5}

Multipliziere Brüche:

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

= \frac{7 \cdot 2}{5}

Multipliziere die Zahlen:

7 \cdot 2 = 14

= \frac{14}{5}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{7}{5}}}

= \frac{\frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{7}{5}}}}{\frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}}

Teile Brüche:

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

= \frac{( e ^{\frac{14}{5}} - 1 ) e ^{\frac{7}{5}}}{e ^{\frac{7}{5}} ( e ^{\frac{14}{5}} + 1 )}

Streiche die gemeinsamen Faktoren:

e^{\frac{7}{5}}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

csc^{2} (\frac{3 π}{2})

\frac{cos ( 1 ) - 1}{1}

sin^{2} (2 3^{\circ})

sec (60 0^{\circ}) + cot (31 5^{\circ})

2 sin (\frac{3 π}{7}) cos (\frac{3 π}{7})