What is the value of sinh(9) ?

The value of sinh(9) is (e^{18}-1)/(2e^9)

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sinh(9)

sinh (9)

\frac{e ^{18} - 1}{2 e ^{9}}

Decimal Notation

4051.54190 \dots

Solution steps

sinh (9)

Use the Hyperbolic identity:

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

= \frac{e ^{9} - e ^{- 9}}{2}

\frac{e ^{9} - e ^{- 9}}{2} = \frac{e ^{18} - 1}{2 e ^{9}}

\frac{e ^{9} - e ^{- 9}}{2}

Apply exponent rule:

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

= \frac{e ^{9} - \frac{1}{e ^{9}}}{2}

Join

e^{9} - \frac{1}{e ^{9}} : \frac{e ^{18} - 1}{e ^{9}}

e^{9} - \frac{1}{e ^{9}}

Convert element to fraction:

e^{9} = \frac{e ^{9} e ^{9}}{e ^{9}}

= \frac{e ^{9} e ^{9}}{e ^{9}} - \frac{1}{e ^{9}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{9} e ^{9} - 1}{e ^{9}}

e^{9} e^{9} - 1 = e^{18} - 1

e^{9} e^{9} - 1

e^{9} e^{9} = e^{18}

e^{9} e^{9}

Apply exponent rule:

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

e^{9} e^{9} = e^{9 + 9}

= e^{9 + 9}

Add the numbers:

9 + 9 = 18

= e^{18}

= e^{18} - 1

= \frac{e ^{18} - 1}{e ^{9}}

= \frac{\frac{e ^{18} - 1}{e ^{9}}}{2}

Apply the fraction rule:

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

= \frac{e ^{18} - 1}{e ^{9} \cdot 2}

= \frac{e ^{18} - 1}{2 e ^{9}}

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