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sinh(1.275)

Lösung

sinh (1.275)

\frac{e ^{\frac{51}{20}} - 1}{2 e ^{\frac{51}{40}}}

Dezimale

1.64963 \dots

Schritte zur Lösung

sinh (1.275)

= sinh (\frac{51}{40})

Umschreiben mit Hilfe von Trigonometrie-Identitäten:

\frac{e ^{\frac{51}{20}} - 1}{2 e ^{\frac{51}{40}}}

sinh (\frac{51}{40})

Hyperbolische Identität anwenden:

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

= \frac{e ^{\frac{51}{40}} - e ^{- \frac{51}{40}}}{2}

\frac{e ^{\frac{51}{40}} - e ^{- \frac{51}{40}}}{2} = \frac{e ^{\frac{51}{20}} - 1}{2 e ^{\frac{51}{40}}}

\frac{e ^{\frac{51}{40}} - e ^{- \frac{51}{40}}}{2}

Wende Exponentenregel an:

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

= \frac{e ^{\frac{51}{40}} - \frac{1}{e ^{\frac{51}{40}}}}{2}

Füge

e^{\frac{51}{40}} - \frac{1}{e ^{\frac{51}{40}}}

zusammen:

\frac{e ^{\frac{51}{20}} - 1}{e ^{\frac{51}{40}}}

e^{\frac{51}{40}} - \frac{1}{e ^{\frac{51}{40}}}

Wandle das Element in einen Bruch um:

e^{\frac{51}{40}} = \frac{e ^{\frac{51}{40}} e ^{\frac{51}{40}}}{e ^{\frac{51}{40}}}

= \frac{e ^{\frac{51}{40}} e ^{\frac{51}{40}}}{e ^{\frac{51}{40}}} - \frac{1}{e ^{\frac{51}{40}}}

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

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

= \frac{e ^{\frac{51}{40}} e ^{\frac{51}{40}} - 1}{e ^{\frac{51}{40}}}

e^{\frac{51}{40}} e^{\frac{51}{40}} - 1 = e^{2 \cdot \frac{51}{40}} - 1

e^{\frac{51}{40}} e^{\frac{51}{40}} - 1

e^{\frac{51}{40}} e^{\frac{51}{40}} = e^{2 \cdot \frac{51}{40}}

e^{\frac{51}{40}} e^{\frac{51}{40}}

Wende Exponentenregel an:

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

e^{\frac{51}{40}} e^{\frac{51}{40}} = e^{\frac{51}{40} + \frac{51}{40}}

= e^{\frac{51}{40} + \frac{51}{40}}

Addiere gleiche Elemente:

\frac{51}{40} + \frac{51}{40} = 2 \cdot \frac{51}{40}

= e^{2 \cdot \frac{51}{40}}

= e^{2 \cdot \frac{51}{40}} - 1

= \frac{e ^{2 \cdot \frac{51}{40}} - 1}{e ^{\frac{51}{40}}}

Multipliziere

2 \cdot \frac{51}{40} : \frac{51}{20}

2 \cdot \frac{51}{40}

Multipliziere Brüche:

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

= \frac{51 \cdot 2}{40}

Multipliziere die Zahlen:

51 \cdot 2 = 102

= \frac{102}{40}

Streiche die gemeinsamen Faktoren:

2

= \frac{51}{20}

= \frac{e ^{\frac{51}{20}} - 1}{e ^{\frac{51}{40}}}

= \frac{\frac{e ^{\frac{51}{20}} - 1}{e ^{\frac{51}{40}}}}{2}

Wende Bruchregel an:

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

= \frac{e ^{\frac{51}{20}} - 1}{e ^{\frac{51}{40}} \cdot 2}

= \frac{e ^{\frac{51}{20}} - 1}{2 e ^{\frac{51}{40}}}

= \frac{e ^{\frac{51}{20}} - 1}{2 e ^{\frac{51}{40}}}

arctan (\frac{20}{25})

sec (69 0^{\circ})

csc (arctan (\frac{3}{4}))

arccos (\frac{2}{21})

sin (\frac{8 π}{4})