What is the value of sinh(7/17) ?

The value of sinh(7/17) is (e^{14/17}-1)/(2e^{7/17)}

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sinh(7/17)

sinh (\frac{7}{17})

\frac{e ^{\frac{14}{17}} - 1}{2 e ^{\frac{7}{17}}}

Decimal

0.42349 \dots

Solution steps

sinh (\frac{7}{17})

Use the Hyperbolic identity:

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

= \frac{e ^{\frac{7}{17}} - e ^{- \frac{7}{17}}}{2}

\frac{e ^{\frac{7}{17}} - e ^{- \frac{7}{17}}}{2} = \frac{e ^{\frac{14}{17}} - 1}{2 e ^{\frac{7}{17}}}

\frac{e ^{\frac{7}{17}} - e ^{- \frac{7}{17}}}{2}

Apply exponent rule:

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

= \frac{e ^{\frac{7}{17}} - \frac{1}{e ^{\frac{7}{17}}}}{2}

Join

e^{\frac{7}{17}} - \frac{1}{e ^{\frac{7}{17}}} : \frac{e ^{\frac{14}{17}} - 1}{e ^{\frac{7}{17}}}

e^{\frac{7}{17}} - \frac{1}{e ^{\frac{7}{17}}}

Convert element to fraction:

e^{\frac{7}{17}} = \frac{e ^{\frac{7}{17}} e ^{\frac{7}{17}}}{e ^{\frac{7}{17}}}

= \frac{e ^{\frac{7}{17}} e ^{\frac{7}{17}}}{e ^{\frac{7}{17}}} - \frac{1}{e ^{\frac{7}{17}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{7}{17}} e ^{\frac{7}{17}} - 1}{e ^{\frac{7}{17}}}

e^{\frac{7}{17}} e^{\frac{7}{17}} - 1 = e^{2 \cdot \frac{7}{17}} - 1

e^{\frac{7}{17}} e^{\frac{7}{17}} - 1

e^{\frac{7}{17}} e^{\frac{7}{17}} = e^{2 \cdot \frac{7}{17}}

e^{\frac{7}{17}} e^{\frac{7}{17}}

Apply exponent rule:

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

e^{\frac{7}{17}} e^{\frac{7}{17}} = e^{\frac{7}{17} + \frac{7}{17}}

= e^{\frac{7}{17} + \frac{7}{17}}

Add similar elements:

\frac{7}{17} + \frac{7}{17} = 2 \cdot \frac{7}{17}

= e^{2 \cdot \frac{7}{17}}

= e^{2 \cdot \frac{7}{17}} - 1

= \frac{e ^{2 \cdot \frac{7}{17}} - 1}{e ^{\frac{7}{17}}}

Multiply

2 \cdot \frac{7}{17} : \frac{14}{17}

2 \cdot \frac{7}{17}

Multiply fractions:

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

= \frac{7 \cdot 2}{17}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14}{17}

= \frac{e ^{\frac{14}{17}} - 1}{e ^{\frac{7}{17}}}

= \frac{\frac{e ^{\frac{14}{17}} - 1}{e ^{\frac{7}{17}}}}{2}

Apply the fraction rule:

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

= \frac{e ^{\frac{14}{17}} - 1}{e ^{\frac{7}{17}} \cdot 2}

= \frac{e ^{\frac{14}{17}} - 1}{2 e ^{\frac{7}{17}}}

What is the value of sinh(7/17) ?
The value of sinh(7/17) is (e^{14/17}-1)/(2e^{7/17)}