What is the value of 64sinh(2) ?

The value of 64sinh(2) is (32(e^4-1))/(e^2)

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64sinh(2)

64 sinh (2)

\frac{32 ( e ^{4} - 1 )}{e ^{2}}

Decimal

232.11906 \dots

Solution steps

64 sinh (2)

Rewrite using trig identities:

sinh (2) = \frac{e ^{4} - 1}{2 e ^{2}}

sinh (2)

Use the Hyperbolic identity:

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

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

\frac{e ^{2} - e ^{- 2}}{2} = \frac{e ^{4} - 1}{2 e ^{2}}

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

Apply exponent rule:

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

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

Join

e^{2} - \frac{1}{e ^{2}} : \frac{e ^{4} - 1}{e ^{2}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

e^{2} e^{2} - 1 = e^{4} - 1

e^{2} e^{2} - 1

e^{2} e^{2} = e^{4}

e^{2} e^{2}

Apply exponent rule:

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

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} - 1

= \frac{e ^{4} - 1}{e ^{2}}

= \frac{\frac{e ^{4} - 1}{e ^{2}}}{2}

Apply the fraction rule:

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

= \frac{e ^{4} - 1}{e ^{2} \cdot 2}

= \frac{e ^{4} - 1}{2 e ^{2}}

= 64 \cdot \frac{e ^{4} - 1}{2 e ^{2}}

Simplify

64 \cdot \frac{e ^{4} - 1}{2 e ^{2}} : \frac{32 ( e ^{4} - 1 )}{e ^{2}}

64 \cdot \frac{e ^{4} - 1}{2 e ^{2}}

Multiply fractions:

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

= \frac{( e ^{4} - 1 ) \cdot 64}{2 e ^{2}}

Divide the numbers:

\frac{64}{2} = 32

= \frac{32 ( e ^{4} - 1 )}{e ^{2}}

= \frac{32 ( e ^{4} - 1 )}{e ^{2}}