What is the value of sinh(-5) ?

The value of sinh(-5) is -(e^{10}-1)/(2e^5)

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sinh(-5)

sinh (- 5)

- \frac{e ^{10} - 1}{2 e ^{5}}

Decimal Notation

- 74.20321 \dots

Solution steps

sinh (- 5)

Use the following property:

sinh (- x) = - sinh (x)

sinh (- 5) = - sinh (5)

= - sinh (5)

Rewrite using trig identities:

sinh (5) = \frac{e ^{10} - 1}{2 e ^{5}}

sinh (5)

Use the Hyperbolic identity:

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

= \frac{e ^{5} - e ^{- 5}}{2}

\frac{e ^{5} - e ^{- 5}}{2} = \frac{e ^{10} - 1}{2 e ^{5}}

\frac{e ^{5} - e ^{- 5}}{2}

Apply exponent rule:

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

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

Join

e^{5} - \frac{1}{e ^{5}} : \frac{e ^{10} - 1}{e ^{5}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

e^{5} e^{5} - 1 = e^{10} - 1

e^{5} e^{5} - 1

e^{5} e^{5} = e^{10}

e^{5} e^{5}

Apply exponent rule:

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

e^{5} e^{5} = e^{5 + 5}

= e^{5 + 5}

Add the numbers:

5 + 5 = 10

= e^{10}

= e^{10} - 1

= \frac{e ^{10} - 1}{e ^{5}}

= \frac{\frac{e ^{10} - 1}{e ^{5}}}{2}

Apply the fraction rule:

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

= \frac{e ^{10} - 1}{e ^{5} \cdot 2}

= \frac{e ^{10} - 1}{2 e ^{5}}

= - \frac{e ^{10} - 1}{2 e ^{5}}

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