What is the value of sinh(6) ?

The value of sinh(6) is (e^{12}-1)/(2e^6)

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sinh(6)

sinh (6)

\frac{e ^{12} - 1}{2 e ^{6}}

Decimal Notation

201.71315 \dots

Solution steps

sinh (6)

Use the Hyperbolic identity:

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

= \frac{e ^{6} - e ^{- 6}}{2}

\frac{e ^{6} - e ^{- 6}}{2} = \frac{e ^{12} - 1}{2 e ^{6}}

\frac{e ^{6} - e ^{- 6}}{2}

Apply exponent rule:

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

= \frac{e ^{6} - \frac{1}{e ^{6}}}{2}

Join

e^{6} - \frac{1}{e ^{6}} : \frac{e ^{12} - 1}{e ^{6}}

e^{6} - \frac{1}{e ^{6}}

Convert element to fraction:

e^{6} = \frac{e ^{6} e ^{6}}{e ^{6}}

= \frac{e ^{6} e ^{6}}{e ^{6}} - \frac{1}{e ^{6}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{6} e ^{6} - 1}{e ^{6}}

e^{6} e^{6} - 1 = e^{12} - 1

e^{6} e^{6} - 1

e^{6} e^{6} = e^{12}

e^{6} e^{6}

Apply exponent rule:

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

e^{6} e^{6} = e^{6 + 6}

= e^{6 + 6}

Add the numbers:

6 + 6 = 12

= e^{12}

= e^{12} - 1

= \frac{e ^{12} - 1}{e ^{6}}

= \frac{\frac{e ^{12} - 1}{e ^{6}}}{2}

Apply the fraction rule:

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

= \frac{e ^{12} - 1}{e ^{6} \cdot 2}

= \frac{e ^{12} - 1}{2 e ^{6}}

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